text: 18 (a) LES (b) CWBL model FIG. 15: Comparison between the (a) LES and (b) model averaged normalized hub-height velocity in a staggered wind-farm with a streamwise spacing of s = 7.85 and a spanwise spacing s = 5.24. x y (a) LES (b) CWBL model FIG. 16: Comparison between the (a) LES and (b) model averaged normalized hub-height velocity in an aligned wind-farm with a streamwise spacing of s = 3.49 and a spanwise spacing s = 7.85. x y VI. CONCLUSIONS In this paper we have introduced the CWBL model, a framework for predicting the power output in both the entrance and fully developed regions of wind-farms. The method combines two well-known approaches, the wake model and the “top-down” boundary layer model thus resulting in the proposed coupled model. Both of the constitutive approaches have one parameter that needs to be determined. For the wake model this is the wake expansion coeﬃcient k and for the “top-down” model this is the eﬀective spanwise spacing s . In the CWBL model, w ye the eﬀective spanwise spacing is obtained from the wake model and is then used in the “top-down” model. These results are then coupled through an iterative procedure to obtain the wake expansion coeﬃcient k that ensures that w the turbine velocity is matched in both models. A detailed comparison with LES results for a variety of cases reveals that the model represents the LES data quite well for both the fully developed region and the entrance region of the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: LES

text: (b)

text: CWBL

text: model

text: FIG.

text: Comparison

label: caption

text: between

text: the

text: (a)

text: LES

text: and

text: (b)

text: model

text: averaged

text: normalized

text: hub-height

text: velocity

text: in

text: a

text: staggered

text: wind-farm

text: with

text: a

text: streamwise

text: spacing

text: of

text: s

text: and

text: a

text: spanwise

text: spacing

text: s

text: x

text: y

text: (a)

text: LES

text: (b)

text: CWBL

text: model

text: FIG.

text: Comparison

text: between

text: the

text: (a)

text: LES

text: and

text: (b)

text: model

text: averaged

text: normalized

text: hub-height

text: velocity

text: in

text: an

text: aligned

text: wind-farm

text: with

text: a

text: streamwise

text: spacing

text: of

text: s

text: and

text: a

text: spanwise

text: spacing

text: s

text: x

text: y

text: VI.

text: CONCLUSIONS

text: In

text: this

text: paper

text: we

text: have

text: introduced

text: the

text: CWBL

text: model,

text: a

text: framework

text: for

text: predicting

text: the

text: power

text: output

text: in

text: both

text: the

text: entrance

text: and

text: fully

text: developed

text: regions

text: of

text: wind-farms.

text: The

text: method

text: combines

text: two

text: well-known

text: approaches,

text: the

text: wake

text: model

text: and

text: the

text: “top-down”

text: boundary

text: layer

text: model

text: thus

text: resulting

text: in

text: the

text: proposed

text: coupled

text: model.

text: Both

text: of

text: the

text: constitutive

text: approaches

text: have

text: one

text: parameter

text: that

text: needs

text: to

text: be

text: determined.

text: For

text: the

text: wake

text: model

text: this

text: is

text: the

text: wake

text: expansion

text: coeﬃcient

text: k

text: and

text: for

text: the

text: “top-down”

text: model

text: this

text: is

text: the

text: eﬀective

text: spanwise

text: spacing

text: s

text: In

text: the

text: CWBL

text: model,

text: w

text: ye

text: the

text: eﬀective

text: spanwise

text: spacing

text: is

text: obtained

text: from

text: the

text: wake

text: model

text: and

text: is

text: then

text: used

text: in

text: the

text: “top-down”

text: model.

text: These

text: results

text: are

text: then

text: coupled

text: through

text: an

text: iterative

text: procedure

text: to

text: obtain

text: the

text: wake

text: expansion

text: coeﬃcient

text: k

text: that

text: ensures

text: that

text: w

text: the

text: turbine

text: velocity

text: is

text: matched

text: in

text: both

text: models.

text: A

text: detailed

text: comparison

text: with

text: LES

text: results

text: for

text: a

text: variety

text: of

text: cases

text: reveals

text: that

text: the

text: model

text: represents

text: the

text: LES

text: data

text: quite

text: well

text: for

text: both

text: the

text: fully

text: developed

text: region

text: and

text: the

text: entrance

text: region

text: of

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 21 [7] I. Kati´c, J. Højstrup, and N. O. Jensen, A simple model for cluster eﬃciency, European Wind Energy Association Con- ference and Exhibition, 7-9 October 1986, Rome, Italy 407 (1986). [8] J. Choi and M. Shan, Advancement of Jensen (PARK) wake model, EWEA Conference, Vienna, February 2013 (2013). [9] A. Pen˜a and O. Rathmann, Atmospheric stability-dependent inﬁnite wind-farm models and the wake-decay coeﬃcient, Wind Energy 17, 1269 (2014). [10] A. Pen˜a, P.-E. R´ethor´e, and O. Rathmann, Modeling large oﬀshore wind farms under diﬀerent atmospheric stability regimes with the Park wake model, Renewable Energy 70, 164 (2014). [11] M. Bastankhah and F. Port´e-Agel, A new analytical model for wind-turbine wakes, Renewable Energy 70, 116 (2014). [12] N. G. Nygaard, Wakes in very large wind farms and the eﬀect of neighbouring wind farms, J. Phys. Conf. Ser. 524, 012162 (2014). [13] B. G. Newman, The spacing of wind turbines in large arrays, Energy conversion 16, 169 (1977). [14] N. O. Jensen, Change of surface roughness and the planetary boundary layer, Quart. J. R. Met. Soc. 104, 351 (1978). [15] S. Frandsen, On the wind speed reduction in the center of large clusters of wind turbines, J. Wind Eng. and Ind. Aerodyn. 39, 251 (1992). [16] S. Emeis and S. Frandsen, Reduction of horizontal wind speed in a boundary layer with obstacles, Bound-Lay. Meteorol. 64, 297 (1993). [17] S. Frandsen, R. Barthelmie, S. Pryor, O. Rathmann, S. Larsen, J. Højstrup, and M. Thøgersen, Analytical modelling of wind speed deﬁcit in large oﬀshore wind farms, Wind Energy 9, 39 (2006). [18] M. Calaf, C. Meneveau, and J. Meyers, Large eddy simulations of fully developed wind-turbine array boundary layers, Phys. Fluids 22, 015110 (2010). [19] C. Meneveau, The top-down model of wind farm boundary layers and its applications, J. of Turbulence 13, 1 (2012). [20] R. J. A. M. Stevens, Dependence of optimal wind-turbine spacing on wind-farm length, Submitted to Wind Energy (2014). [21] J. Meyers and C. Meneveau, Optimal turbine spacing in fully developed wind farm boundary layers, Wind Energy 15, 305 (2012). [22] A. Sescu and C. Meneveau, Large Eddy Simulation and single column modeling of thermally stratiﬁed wind-turbine arrays for fully developed, stationary atmospheric conditions, J. Atmospheric and Oceanic Technology submitted (2014). [23] J. F. Herbert-Acero, O. Probst, P.-E. R´ethor´e, G. C. Larsen, and K. K. Castillo-Villar, A Review of Methodological Approaches for the Design and Optimization of Wind Farms, Energies 7, 6930 (2014). [24] E. Son, S. Lee, B. Hwang, and S. Lee, Characteristics of turbine spacing in a wind farm using an optimal design process, Renewable Energy 65, 245 (2014). [25] G. Hassan, Theory manual 4.0: GH WindFarmer - wind farm design software:, Technical report from Garrad Hassan and Partners, Bristol, England (2009). [26] W. Schlez and A. Neubert, New developments in large wind farm modelling, Proc. European Wind Energy Conf. Marseilles, France, European Wind Energy Association, PO.167 (2009). [27] P. Beaucage, N. Robinson, M. Brower, and C. Alonge, Overview of six commercial and research wake models for large oﬀshore wind farms, Proceedings of the European Wind Energy Association Conference, Copenhagen 95 (2012). [28] M. C. Brower and N. M. Robinson, The openwind deep-array wake model: Development and validation, AWS Truepower (2012). [29] X. Yang, S. Kang, and F. Sotiropoulos, Computational study and modeling of turbine spacing eﬀects in inﬁnite aligned wind farms, Phys. Fluids 24, 115107 (2012). [30] A. Pen˜a, P.-E. R´ethor´e, and O. Rathmann, Modeling large oﬀshore wind farms under diﬀerent atmospheric stability regimes with the Park wake model, International Conference on Aerodynamics of Oﬀshore Wind Energy Systems and Wakes (ICOWES2013) 26 (2013). [31] S. Emeis, A simple analytical wind PARK model considering atmospheric stability, Wind Energy 459-469, 13 (2010). [32] A. Pen˜a, P.-E. R´ethor´e, C. B. Hasager, and K. S. Hansen, Results of wake simulations at the Horns Rev I and Lillgrund wind farms using the modiﬁed Park model, DTU Wind Energy, E-report-0026, (2013). [33] G. Marmidis, S. Lazarou, and E. Pyrgioti, Optimal placement of wind turbines in a wind park using Monte Carlo simulation, Renewable Energy 33, 1455 (2008). [34] A. Emami and P. Noghreh, New approach on optimization in placement of wind turbines within wind farm by genetic algorithms, Renewable Energy 35, 1559 (2010). [35] A. Kusiak and Z. Song, Design of wind farm layout for maximum wind energy capture, Renewable Energy 35, 685 (2010). [36] B. Saavedra-Moreno, S. Salcedo-Sanz, A. Paniagua-Tineo, L. Prieto, and A. Portilla-Figueras, Seeding evolutionary algo- rithms with heuristics for optimal wind turbines positioning in wind farms, Renewable Energy 36, 2838 (2011). [37] J. F. Ainslie, Calculating the ﬂow ﬁeld in the wake of wind turbines, J. Wind Eng. and Ind. Aerodyn. 27, 213 (1988). [38] S. P. van der Pijl and J. G. Schepers, Improvements of the wakefarm wake model,n Workshop on wake modelling and benchmarking of models, Annex XXIII: Oﬀshore wind energy technology and deployment, Billund (2006). [39] J. G. Schepers and S. P. van der Pijl, Improved modelling of wake aerodynamics and assessment of new farm control strategies, J. Phys. Conf. Ser. 75, 012039 (2007). [40] P. J. Eecen and E. T. G. Bot, Improvements to the ECN wind farm optimisation software armFlow , European Wind Energy Conference & Exhibition 2010, 20 - 23 April 2010, Warsaw, Poland (2010). [41] J. G. Schepers, Ph.D. thesis, Delft University, 2012. [42] H. O¨zdemir, M. Versteeg, and A. Brand, Improvements in ECN Wake Model, International Conference on Aerodynamics of Oﬀshore Wind Energy Systems and Wakes (ICOWES2013) 1 (2013).

label: paragraph

label: reference

text: I.

text: Kati´c,

text: J.

text: Højstrup,

text: and

text: N.

text: O.

text: Jensen,

text: A

text: simple

text: model

text: for

text: cluster

text: eﬃciency,

text: European

text: Wind

text: Energy

text: Association

text: Con-

text: ference

text: and

text: Exhibition,

text: October

text: Rome,

text: Italy

text: J.

text: Choi

text: and

text: M.

text: Shan,

text: Advancement

text: of

text: Jensen

text: (PARK)

text: wake

text: model,

text: EWEA

text: Conference,

text: Vienna,

text: February

text: A.

text: Pen˜a

text: and

text: O.

text: Rathmann,

text: Atmospheric

text: stability-dependent

text: inﬁnite

text: wind-farm

text: models

text: and

text: the

text: wake-decay

text: coeﬃcient,

text: Wind

text: Energy

text: A.

text: Pen˜a,

text: P.-E.

text: R´ethor´e,

text: and

text: O.

text: Rathmann,

text: Modeling

text: large

text: oﬀshore

text: wind

text: farms

text: under

text: diﬀerent

text: atmospheric

text: stability

text: regimes

text: with

text: the

text: Park

text: wake

text: model,

text: Renewable

text: Energy

text: M.

text: Bastankhah

text: and

text: F.

text: Port´e-Agel,

text: A

text: new

text: analytical

text: model

text: for

text: wind-turbine

text: wakes,

text: Renewable

text: Energy

text: N.

text: G.

text: Nygaard,

text: Wakes

text: in

text: very

text: large

text: wind

text: farms

text: and

text: the

text: eﬀect

text: of

text: neighbouring

text: wind

text: farms,

text: J.

text: Phys.

text: Conf.

text: Ser.

text: B.

text: G.

text: Newman,

text: The

text: spacing

text: of

text: wind

text: turbines

text: in

text: large

text: arrays,

text: Energy

text: conversion

text: N.

text: O.

text: Jensen,

text: Change

text: of

text: surface

text: roughness

text: and

text: the

text: planetary

text: boundary

text: layer,

text: Quart.

text: J.

text: R.

text: Met.

text: Soc.

text: S.

text: Frandsen,

text: On

text: the

text: wind

text: speed

text: reduction

text: in

text: the

text: center

text: of

text: large

text: clusters

text: of

text: wind

text: turbines,

text: J.

text: Wind

text: Eng.

text: and

text: Ind.

text: Aerodyn.

text: S.

text: Emeis

text: and

text: S.

text: Frandsen,

text: Reduction

text: of

text: horizontal

text: wind

text: speed

text: in

text: a

text: boundary

text: layer

text: with

text: obstacles,

text: Bound-Lay.

text: Meteorol.

text: S.

text: Frandsen,

text: R.

text: Barthelmie,

text: S.

text: Pryor,

text: O.

text: Rathmann,

text: S.

text: Larsen,

text: J.

text: Højstrup,

text: and

text: M.

text: Thøgersen,

text: Analytical

text: modelling

text: of

text: wind

text: speed

text: deﬁcit

text: in

text: large

text: oﬀshore

text: wind

text: farms,

text: Wind

text: Energy

text: M.

text: Calaf,

text: C.

text: Meneveau,

text: and

text: J.

text: Meyers,

text: Large

text: eddy

text: simulations

text: of

text: fully

text: developed

text: wind-turbine

text: array

text: boundary

text: layers,

text: Phys.

text: Fluids

text: C.

text: Meneveau,

text: The

text: top-down

text: model

text: of

text: wind

text: farm

text: boundary

text: layers

text: and

text: its

text: applications,

text: J.

text: of

text: Turbulence

text: R.

text: J.

text: A.

text: M.

text: Stevens,

text: Dependence

text: of

text: optimal

text: wind-turbine

text: spacing

text: on

text: wind-farm

text: length,

text: Submitted

text: to

text: Wind

text: Energy

text: J.

text: Meyers

text: and

text: C.

text: Meneveau,

text: Optimal

text: turbine

text: spacing

text: in

text: fully

text: developed

text: wind

text: farm

text: boundary

text: layers,

text: Wind

text: Energy

text: A.

text: Sescu

text: and

text: C.

text: Meneveau,

text: Large

text: Eddy

text: Simulation

text: and

text: single

text: column

text: modeling

text: of

text: thermally

text: stratiﬁed

text: wind-turbine

text: arrays

text: for

text: fully

text: developed,

text: stationary

text: atmospheric

text: conditions,

text: J.

text: Atmospheric

text: and

text: Oceanic

text: Technology

text: submitted

text: J.

text: F.

text: Herbert-Acero,

text: O.

text: Probst,

text: P.-E.

text: R´ethor´e,

text: G.

text: C.

text: Larsen,

text: and

text: K.

text: K.

text: Castillo-Villar,

text: A

text: Review

text: of

text: Methodological

text: Approaches

text: for

text: the

text: Design

text: and

text: Optimization

text: of

text: Wind

text: Farms,

text: Energies

text: E.

text: Son,

text: S.

text: Lee,

text: B.

text: Hwang,

text: and

text: S.

text: Lee,

text: Characteristics

text: of

text: turbine

text: spacing

text: in

text: a

text: wind

text: farm

text: using

text: an

text: optimal

text: design

text: process,

text: Renewable

text: Energy

text: G.

text: Hassan,

text: Theory

text: manual

text: GH

text: WindFarmer

text: wind

text: farm

text: design

text: software:,

text: Technical

text: report

text: from

text: Garrad

text: Hassan

text: and

text: Partners,

text: Bristol,

text: England

text: W.

text: Schlez

text: and

text: A.

text: Neubert,

text: New

text: developments

text: in

text: large

text: wind

text: farm

text: modelling,

text: Proc.

text: European

text: Wind

text: Energy

text: Conf.

text: Marseilles,

text: France,

text: European

text: Wind

text: Energy

text: Association,

text: P.

text: Beaucage,

text: N.

text: Robinson,

text: M.

text: Brower,

text: and

text: C.

text: Alonge,

text: Overview

text: of

text: six

text: commercial

text: and

text: research

text: wake

text: models

text: for

text: large

text: oﬀshore

text: wind

text: farms,

text: Proceedings

text: of

text: the

text: European

text: Wind

text: Energy

text: Association

text: Conference,

text: Copenhagen

text: M.

text: C.

text: Brower

text: and

text: N.

text: M.

text: Robinson,

text: The

text: openwind

text: deep-array

text: wake

text: model:

text: Development

text: and

text: validation,

text: AWS

text: Truepower

text: X.

text: Yang,

text: S.

text: Kang,

text: and

text: F.

text: Sotiropoulos,

text: Computational

text: study

text: and

text: modeling

text: of

text: turbine

text: spacing

text: eﬀects

text: in

text: inﬁnite

text: aligned

text: wind

text: farms,

text: Phys.

text: Fluids

text: A.

text: Pen˜a,

text: P.-E.

text: R´ethor´e,

text: and

text: O.

text: Rathmann,

text: Modeling

text: large

text: oﬀshore

text: wind

text: farms

text: under

text: diﬀerent

text: atmospheric

text: stability

text: regimes

text: with

text: the

text: Park

text: wake

text: model,

text: International

text: Conference

text: on

text: Aerodynamics

text: of

text: Oﬀshore

text: Wind

text: Energy

text: Systems

text: and

text: Wakes

text: S.

text: Emeis,

text: A

text: simple

text: analytical

text: wind

text: PARK

text: model

text: considering

text: atmospheric

text: stability,

text: Wind

text: Energy

text: A.

text: Pen˜a,

text: P.-E.

text: R´ethor´e,

text: C.

text: B.

text: Hasager,

text: and

text: K.

text: S.

text: Hansen,

text: Results

text: of

text: wake

text: simulations

text: at

text: the

text: Horns

text: Rev

text: I

text: and

text: Lillgrund

text: wind

text: farms

text: using

text: the

text: modiﬁed

text: Park

text: model,

text: DTU

text: Wind

text: Energy,

text: G.

text: Marmidis,

text: S.

text: Lazarou,

text: and

text: E.

text: Pyrgioti,

text: Optimal

text: placement

text: of

text: wind

text: turbines

text: in

text: a

text: wind

text: park

text: using

text: Monte

text: Carlo

text: simulation,

text: Renewable

text: Energy

text: A.

text: Emami

text: and

text: P.

text: Noghreh,

text: New

text: approach

text: on

text: optimization

text: in

text: placement

text: of

text: wind

text: turbines

text: within

text: wind

text: farm

text: by

text: genetic

text: algorithms,

text: Renewable

text: Energy

text: A.

text: Kusiak

text: and

text: Z.

text: Song,

text: Design

text: of

text: wind

text: farm

text: layout

text: for

text: maximum

text: wind

text: energy

text: capture,

text: Renewable

text: Energy

text: B.

text: Saavedra-Moreno,

text: S.

text: Salcedo-Sanz,

text: A.

text: Paniagua-Tineo,

text: L.

text: Prieto,

text: and

text: A.

text: Portilla-Figueras,

text: Seeding

text: evolutionary

text: algo-

text: rithms

text: with

text: heuristics

text: for

text: optimal

text: wind

text: turbines

text: positioning

text: in

text: wind

text: farms,

text: Renewable

text: Energy

text: J.

text: F.

text: Ainslie,

text: Calculating

text: the

text: ﬂow

text: ﬁeld

text: in

text: the

text: wake

text: of

text: wind

text: turbines,

text: J.

text: Wind

text: Eng.

text: and

text: Ind.

text: Aerodyn.

text: S.

text: P.

text: van

text: der

text: Pijl

text: and

text: J.

text: G.

text: Schepers,

text: Improvements

text: of

text: the

text: wakefarm

text: wake

text: model,n

text: Workshop

text: on

text: wake

text: modelling

text: and

text: benchmarking

text: of

text: models,

text: Annex

text: XXIII:

text: Oﬀshore

text: wind

text: energy

text: technology

text: and

text: deployment,

text: Billund

text: J.

text: G.

text: Schepers

text: and

text: S.

text: P.

text: van

text: der

text: Pijl,

text: Improved

text: modelling

text: of

text: wake

text: aerodynamics

text: and

text: assessment

text: of

text: new

text: farm

text: control

text: strategies,

text: J.

text: Phys.

text: Conf.

text: Ser.

text: P.

text: J.

text: Eecen

text: and

text: E.

text: T.

text: G.

text: Bot,

text: Improvements

text: to

text: the

text: ECN

text: wind

text: farm

text: optimisation

text: software

text: armFlow

text: European

text: Wind

text: Energy

text: Conference

text: Exhibition

text: April

text: Warsaw,

text: Poland

text: J.

text: G.

text: Schepers,

text: Ph.D.

text: thesis,

text: Delft

text: University,

text: H.

text: O¨zdemir,

text: M.

text: Versteeg,

text: and

text: A.

text: Brand,

text: Improvements

text: in

text: ECN

text: Wake

text: Model,

text: International

text: Conference

text: on

text: Aerodynamics

text: of

text: Oﬀshore

text: Wind

text: Energy

text: Systems

text: and

text: Wakes


text: 6 1 s =5.24, s =3.49 x y s =7.85, s =3.49 1.6 x y 0.8 s =3.49, s =5.24 x y s =5.24, s =5.24 a 1.4 x y P1 0.6 P,∞ sx=7.85, sy=5.24 P / ∞ 0.4 / ,s∞ 1.2 sx=3.49, sy=7.85 P s =5.24, s =7.85 x y s =7.85, s =7.85 0.2 1 x y s =3.49 y (a) Aligned (b) Staggered (c) Ratio s =5.24 0 0.8 y 2 4 8 12 20 2 4 8 12 20 2 4 8 12 20 s =7.85 y s = √s s s = √s s s = √s s x y x y x y FIG. 3: Power output ratio P∞/P in the fully developed regime √according to the wake model (equations (5)-(7)) with 1 k = 0.0579 as function of the geometric mean turbine spacing s = s s . P∞ is the power output per turbine in the fully w x y developed regime while P is the reference power output of a single turbine without eﬀects from other turbines. The panels show 1 the model results (lines) for the (a) aligned and (b) staggered conﬁguration compared to LES results (symbols) [3, 53, 54]. Panel (c) shows the ratio between the staggered and aligned results. The dashed lines indicate the sum approximations (equations (26)-(31)) for the wake model given in appendix 1. Later on, Figures 5 and 9 compare results from the “top-down” and CWBL model with LES data. As will be seen, only the proposed CWBL model captures the trend for both the aligned and staggered cases. the wake expansion coeﬃcient in the fully developed regime of the wind-farm based on parameters obtained from the “top-down” model. The turbine velocity and power output of the turbines in the fully developed region of the wind-farm can be obtained by applying the wake model to predict the streamwise velocity ﬁeld u(x) at all points on a three-dimensional mesh. For the calculation presented here we use a resolution of ∆x = ∆y = ∆z = 6 meters. To determine the velocity ﬁeld in the fully developed regime we consider the eﬀect of a very large number of upstream rows. Speciﬁcally, we consider 100 upstream rows with up to 4 columns of turbines on the left and the right side including the corresponding “ghost” turbines. These parameters can be shown to lead to fully converged results for the wake model. That is to say, adding more turbines upstream or to the sides does not make any diﬀerence in the results. In fact, for most cases only a fraction of the turbines used in this study are necessary to reach convergence. Note that since the model is analytical the full solution only has to be calculated for visualization purposes, see ﬁgures 2, and ﬁgures 14-17. To determinate the turbine power outputs the velocity only needs to be calculated at the turbine locations. In appendix 1 we present some practically relevant simpliﬁcations that can be used for calculating the velocity at wind-turbine locations more eﬃciently in the aligned and staggered conﬁgurations. Both the presented LES and model results assume that all turbines operate in region II, where the thrust coeﬃcient C is constant as function of the wind-speed. Note that the experimental data to which we compare the model results T ± in section are obtained for a wind-speed of 8 0.5m/s, which corresponds to the turbines operating in region II. At the end of section we explain how the CWBL model can be applied to the cases where the C coeﬃcient of the T turbines in the wind-farm changes in the entrance region of the wind-farm. In addition, both the LES and model neglect the variation of the power coeﬃcient C with wind-speed. For the comparison with the ﬁeld experiments P presented here this is a reasonable assumption since the data have been obtained for a very narrow range of wind speeds and hence C cancels out when relative turbine power outputs are considered. P Figure 2(a) gives a three-dimensional representation of the predicted mean velocity in a fully developed staggered wind-farm for a dimensionless streamwise spacing (in units of rotor diam√eter D) of s = 7.85 and a spanwise spacing x of s = 5.24. The geometric average of the spacing is deﬁned as s = s s and is s = 6.41 in this case. In ﬁgure y x y 2(b) the results from a corresponding LES run, averaged in time, are also shown for a qualitative comparison. The parameters for both the LES and the wake model used here are: D = 100m and z = 100m. The surface roughness h height in the LES was z = 0.1m and C = 0.75. 0,lo T Next, in order to highlight some advantages and drawbacks of the wake model, it is applied (without coupling with the “top-down” model) to predict wind-turbine power output for various wind-farm conﬁgurations consisting of diﬀerent streamwise and spanwise turbine spacings. In ﬁgure 3 the wake model results are compared with the (cid:104) (cid:105) (cid:104) (cid:105) LES results. For the LES the power ratio P∞/P is determined by measuring u3∞ / u3 , where u∞ is the velocity 1 1 averaged over a turbine disk for turbines at the end of the wind-farm and u is the velocity averaged over the disk 1 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: s

text: s

text: x

text: y

text: s

text: s

text: x

text: y

text: s

text: s

text: x

text: y

text: s

text: s

text: a

text: x

text: y

text: P,∞

text: P

text: ,s∞

text: P

text: s

text: s

text: x

text: y

text: s

text: s

text: x

text: y

text: s

text: y

text: (a)

text: Aligned

text: (b)

text: Staggered

text: (c)

text: Ratio

text: s

text: y

text: s

text: y

text: s

text: √s

text: s

text: s

text: √s

text: s

text: s

text: √s

text: s

text: x

text: y

text: x

text: y

text: x

text: y

text: FIG.

text: Power

text: output

text: ratio

text: P∞/P

text: in

text: the

text: fully

text: developed

text: regime

text: √according

text: to

text: the

text: wake

text: model

text: (equations

text: with

text: k

text: as

text: function

text: of

text: the

text: geometric

text: mean

text: turbine

text: spacing

text: s

text: s

text: s

text: P∞

text: is

text: the

text: power

text: output

text: per

text: turbine

text: in

text: the

text: fully

text: w

text: x

text: y

text: developed

text: regime

text: while

text: P

text: is

text: the

text: reference

text: power

text: output

text: of

text: a

text: single

text: turbine

text: without

text: eﬀects

text: from

text: other

text: turbines.

text: The

text: panels

text: show

text: the

text: model

text: results

text: (lines)

text: for

text: the

text: (a)

text: aligned

text: and

text: (b)

text: staggered

text: conﬁguration

text: compared

text: to

text: LES

text: results

text: (symbols)

text: Panel

text: (c)

text: shows

text: the

text: ratio

text: between

text: the

text: staggered

text: and

text: aligned

text: results.

text: The

text: dashed

text: lines

text: indicate

text: the

text: sum

text: approximations

text: (equations

text: for

text: the

text: wake

text: model

text: given

text: in

text: appendix

text: Later

text: on,

text: Figures

text: and

text: compare

text: results

text: from

text: the

text: “top-down”

text: and

text: CWBL

text: model

text: with

text: LES

text: data.

text: As

text: will

text: be

text: seen,

text: only

text: the

text: proposed

text: CWBL

text: model

text: captures

text: the

text: trend

text: for

text: both

text: the

text: aligned

text: and

text: staggered

text: cases.

text: the

text: wake

text: expansion

text: coeﬃcient

text: in

text: the

text: fully

text: developed

text: regime

text: of

text: the

text: wind-farm

text: based

text: on

text: parameters

text: obtained

text: from

text: the

text: “top-down”

text: model.

text: The

text: turbine

text: velocity

text: and

text: power

text: output

text: of

text: the

text: turbines

text: in

text: the

text: fully

text: developed

text: region

text: of

text: the

text: wind-farm

text: can

text: be

text: obtained

text: by

text: applying

text: the

text: wake

text: model

text: to

text: predict

text: the

text: streamwise

text: velocity

text: ﬁeld

text: u(x)

text: at

text: all

text: points

text: on

text: a

text: three-dimensional

text: mesh.

text: For

text: the

text: calculation

text: presented

text: here

text: we

text: use

text: a

text: resolution

text: of

text: ∆x

text: ∆y

text: ∆z

text: meters.

text: To

text: determine

text: the

text: velocity

text: ﬁeld

text: in

text: the

text: fully

text: developed

text: regime

text: we

text: consider

text: the

text: eﬀect

text: of

text: a

text: very

text: large

text: number

text: of

text: upstream

text: rows.

text: Speciﬁcally,

text: we

text: consider

text: upstream

text: rows

text: with

text: up

text: to

text: columns

text: of

text: turbines

text: on

text: the

text: left

text: and

text: the

text: right

text: side

text: including

text: the

text: corresponding

text: “ghost”

text: turbines.

text: These

text: parameters

text: can

text: be

text: shown

text: to

text: lead

text: to

text: fully

text: converged

text: results

text: for

text: the

text: wake

text: model.

text: That

text: is

text: to

text: say,

text: adding

text: more

text: turbines

text: upstream

text: or

text: to

text: the

text: sides

text: does

text: not

text: make

text: any

text: diﬀerence

text: in

text: the

text: results.

text: In

text: fact,

text: for

text: most

text: cases

text: only

text: a

text: fraction

text: of

text: the

text: turbines

text: used

text: in

text: this

text: study

text: are

text: necessary

text: to

text: reach

text: convergence.

text: Note

text: that

text: since

text: the

text: model

text: is

text: analytical

text: the

text: full

text: solution

text: only

text: has

text: to

text: be

text: calculated

text: for

text: visualization

text: purposes,

text: see

text: ﬁgures

text: and

text: ﬁgures

text: To

text: determinate

text: the

text: turbine

text: power

text: outputs

text: the

text: velocity

text: only

text: needs

text: to

text: be

text: calculated

text: at

text: the

text: turbine

text: locations.

text: In

text: appendix

text: we

text: present

text: some

text: practically

text: relevant

text: simpliﬁcations

text: that

text: can

text: be

text: used

text: for

text: calculating

text: the

text: velocity

text: at

text: wind-turbine

text: locations

text: more

text: eﬃciently

text: in

text: the

text: aligned

text: and

text: staggered

text: conﬁgurations.

text: Both

text: the

text: presented

text: LES

text: and

text: model

text: results

text: assume

text: that

text: all

text: turbines

text: operate

text: in

text: region

text: II,

text: where

text: the

text: thrust

text: coeﬃcient

text: C

text: is

text: constant

text: as

text: function

text: of

text: the

text: wind-speed.

text: Note

text: that

text: the

text: experimental

text: data

text: to

text: which

text: we

text: compare

text: the

text: model

text: results

text: T

text: in

text: section

text: are

text: obtained

text: for

text: a

text: wind-speed

text: of

text: which

text: corresponds

text: to

text: the

text: turbines

text: operating

text: in

text: region

text: II.

text: At

text: the

text: end

text: of

text: section

text: we

text: explain

text: how

text: the

text: CWBL

text: model

text: can

text: be

text: applied

text: to

text: the

text: cases

text: where

text: the

text: C

text: coeﬃcient

text: of

text: the

text: T

text: turbines

text: in

text: the

text: wind-farm

text: changes

text: in

text: the

text: entrance

text: region

text: of

text: the

text: wind-farm.

text: In

text: addition,

text: both

text: the

text: LES

text: and

text: model

text: neglect

text: the

text: variation

text: of

text: the

text: power

text: coeﬃcient

text: C

text: with

text: wind-speed.

text: For

text: the

text: comparison

text: with

text: the

text: ﬁeld

text: experiments

text: P

text: presented

text: here

text: this

text: is

text: a

text: reasonable

text: assumption

text: since

text: the

text: data

text: have

text: been

text: obtained

text: for

text: a

text: very

text: narrow

text: range

text: of

text: wind

text: speeds

text: and

text: hence

text: C

text: cancels

text: out

text: when

text: relative

text: turbine

text: power

text: outputs

text: are

text: considered.

text: P

text: Figure

text: gives

text: a

text: three-dimensional

text: representation

text: of

text: the

text: predicted

text: mean

text: velocity

text: in

text: a

text: fully

text: developed

text: staggered

text: wind-farm

text: for

text: a

text: dimensionless

text: streamwise

text: spacing

text: (in

text: units

text: of

text: rotor

text: diam√eter

text: D)

text: of

text: s

text: and

text: a

text: spanwise

text: spacing

text: x

text: of

text: s

text: The

text: geometric

text: average

text: of

text: the

text: spacing

text: is

text: deﬁned

text: as

text: s

text: s

text: s

text: and

text: is

text: s

text: in

text: this

text: case.

text: In

text: ﬁgure

text: y

text: x

text: y

text: the

text: results

text: from

text: a

text: corresponding

text: LES

text: run,

text: averaged

text: in

text: time,

text: are

text: also

text: shown

text: for

text: a

text: qualitative

text: comparison.

text: The

text: parameters

text: for

text: both

text: the

text: LES

text: and

text: the

text: wake

text: model

text: used

text: here

text: are:

text: D

text: and

text: z

text: The

text: surface

text: roughness

text: h

text: height

text: in

text: the

text: LES

text: was

text: z

text: and

text: C

text: T

text: Next,

text: in

text: order

text: to

text: highlight

text: some

text: advantages

text: and

text: drawbacks

text: of

text: the

text: wake

text: model,

text: it

text: is

text: applied

text: (without

text: coupling

text: with

text: the

text: “top-down”

text: model)

text: to

text: predict

text: wind-turbine

text: power

text: output

text: for

text: various

text: wind-farm

text: conﬁgurations

text: consisting

text: of

text: diﬀerent

text: streamwise

text: and

text: spanwise

text: turbine

text: spacings.

text: In

text: ﬁgure

text: the

text: wake

text: model

text: results

text: are

text: compared

text: with

text: the

text: LES

text: results.

text: For

text: the

text: LES

text: the

text: power

text: ratio

text: P∞/P

text: is

text: determined

text: by

text: measuring

text: where

text: u∞

text: is

text: the

text: velocity

text: averaged

text: over

text: a

text: turbine

text: disk

text: for

text: turbines

text: at

text: the

text: end

text: of

text: the

text: wind-farm

text: and

text: u

text: is

text: the

text: velocity

text: averaged

text: over

text: the

text: disk

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: energy and momentum conservation, and is satisﬁed in back-scattering only by multi-GHz phonon modes in most solids. SBS is frequently used for optical gain [3, 13], laser linewidth narrowing [14], optical phase conjugation [15], dynamic gratings [16], and even material characterization and microscopy [17–22]. The applications of SBS in superluminal and slow light experiments have been rec- ognized as well [4–7]. However, unlike the case of electromagnetically induced transparency (EIT) [1, 2], the generation of transparency with SBS has never been demonstrated. This is because in typical Brillouin scattering pump-probe systems the lifetimes of phonons at multi-GHz frequencies are much shorter than the photon lifetimes [3, 23, 24], eﬀectively disabling the coherent inter- ference of probe Stokes scattering and pump anti-Stokes scattering pathways. In a forward-scattering SBS system however, this lifetime relationship can be reversed by coupling the light ﬁelds through a low-frequency long-lived phonon mode as shown recently in the experimental demonstration of forward-SBS las- ing [8] and Brillouin cooling [9]. Nonlinear optical processes such as EIT need to satisfy the energy-momentum conservation, leading to the phase-matching requirement. For EIT in an atomic system, the momentum of the spin wave is set by the relevant optical waves, automatically satisfying the phase matching requirement [1, 2, 26]. In compar- ison, the traveling acoustic wave in Brillouin scattering induced transparency (BSIT) carries a momentum that is intrinsic to the mechanical medium, is dis- cretized by the resonator, and can far exceed the momentum of an optical wave, which leads to special phase matching requirement as will be discussed in detail below. More importantly, this unique property makes BSIT intrinsically non- reciprocal. The BSIT vanishes when either the probe or the control reverses its propagation direction (see Fig 4). Such non-reciprocity [27–29] can be exploited for important applications, including on-chip optical isolators, circulators, and gyroscopes. The generation of BSIT requires a 3-mode system composed of two opti- cal modes and one long-lived propagating acoustic mode. Phase-matching of these modes in both frequency space and momentum space is essential as il- lustrated in Fig. 1a. Notably, the momentum requirement is either automatic or unnecessary in other transparency mechanisms [1, 2, 30–32]. In BSIT, the energies and momenta of these three discrete modes must be matched to sat- − − isfy ω ω = Ω and k k = q simultaneously. Here, (ω , k ) are the 2 1 B 2 1 B 1 1 energy and momentum of the lower energy optical mode, (ω , k ) represent the 2 2 higher energy optical mode, and (Ω , q ) represent the propagating acoustic B B mode (Fig. 1a,b). As described above, the coupling between the three modes is mediated by electrostrictive Brillouin scattering. Previous work has shown that such instances of both forward- and backward-SBS phase matching can occur naturally within a microresonator [8, 9, 33, 34]. When momentum matching is assumed, a 3-level lambda system analogy to EIT can also be made in the case of BSIT (Fig. 2a). The system is excited through the addition of a photon at either (ω , k ) or (ω , k ) to a virtual level. The system then radiatively decays 1 1 2 2 to one of the lower levels with the emission of a Stokes or anti-Stokes photon, along with the creation or annihilation of a phonon. The additional conservation 2

text: energy

text: and

text: momentum

text: conservation,

text: and

text: is

text: satisﬁed

text: in

text: back-scattering

text: only

text: by

text: multi-GHz

text: phonon

text: modes

text: in

text: most

text: solids.

text: SBS

text: is

text: frequently

text: used

text: for

text: optical

text: gain

text: laser

text: linewidth

text: narrowing

text: optical

text: phase

text: conjugation

text: dynamic

text: gratings

text: and

text: even

text: material

text: characterization

text: and

text: microscopy

text: The

text: applications

text: of

text: SBS

text: in

text: superluminal

text: and

text: slow

text: light

text: experiments

text: have

text: been

text: rec-

text: ognized

text: as

text: well

text: However,

text: unlike

text: the

text: case

text: of

text: electromagnetically

text: induced

text: transparency

text: (EIT)

text: the

text: generation

text: of

text: transparency

text: with

text: SBS

text: has

text: never

text: been

text: demonstrated.

text: This

text: is

text: because

text: in

text: typical

text: Brillouin

text: scattering

text: pump-probe

text: systems

text: the

text: lifetimes

text: of

text: phonons

text: at

text: multi-GHz

text: frequencies

text: are

text: much

text: shorter

text: than

text: the

text: photon

text: lifetimes

text: eﬀectively

text: disabling

text: the

text: coherent

text: inter-

text: ference

text: of

text: probe

text: Stokes

text: scattering

text: and

text: pump

text: anti-Stokes

text: scattering

text: pathways.

text: In

text: a

text: forward-scattering

text: SBS

text: system

text: however,

text: this

text: lifetime

text: relationship

text: can

text: be

text: reversed

text: by

text: coupling

text: the

text: light

text: ﬁelds

text: through

text: a

text: low-frequency

text: long-lived

text: phonon

text: mode

text: as

text: shown

text: recently

text: in

text: the

text: experimental

text: demonstration

text: of

text: forward-SBS

text: las-

text: ing

text: and

text: Brillouin

text: cooling

text: Nonlinear

text: optical

text: processes

text: such

text: as

text: EIT

text: need

text: to

text: satisfy

text: the

text: energy-momentum

text: conservation,

text: leading

text: to

text: the

text: phase-matching

text: requirement.

text: For

text: EIT

text: in

text: an

text: atomic

text: system,

text: the

text: momentum

text: of

text: the

text: spin

text: wave

text: is

text: set

text: by

text: the

text: relevant

text: optical

text: waves,

text: automatically

text: satisfying

text: the

text: phase

text: matching

text: requirement

text: In

text: compar-

text: ison,

text: the

text: traveling

text: acoustic

text: wave

text: in

text: Brillouin

text: scattering

text: induced

text: transparency

text: (BSIT)

text: carries

text: a

text: momentum

text: that

text: is

text: intrinsic

text: to

text: the

text: mechanical

text: medium,

text: is

text: dis-

text: cretized

text: by

text: the

text: resonator,

text: and

text: can

text: far

text: exceed

text: the

text: momentum

text: of

text: an

text: optical

text: wave,

text: which

text: leads

text: to

text: special

text: phase

text: matching

text: requirement

text: as

text: will

text: be

text: discussed

text: in

text: detail

text: below.

text: More

text: importantly,

text: this

text: unique

text: property

text: makes

text: BSIT

text: intrinsically

text: non-

text: reciprocal.

text: The

text: BSIT

text: vanishes

text: when

text: either

text: the

text: probe

text: or

text: the

text: control

text: reverses

text: its

text: propagation

text: direction

text: (see

text: Fig

text: Such

text: non-reciprocity

text: can

text: be

text: exploited

text: for

text: important

text: applications,

text: including

text: on-chip

text: optical

text: isolators,

text: circulators,

text: and

text: gyroscopes.

text: The

text: generation

text: of

text: BSIT

text: requires

text: a

text: system

text: composed

text: of

text: two

text: opti-

text: cal

text: modes

text: and

text: one

text: long-lived

text: propagating

text: acoustic

text: mode.

text: Phase-matching

text: of

text: these

text: modes

text: in

text: both

text: frequency

text: space

text: and

text: momentum

text: space

text: is

text: essential

text: as

text: il-

text: lustrated

text: in

text: Fig.

text: Notably,

text: the

text: momentum

text: requirement

text: is

text: either

text: automatic

text: or

text: unnecessary

text: in

text: other

text: transparency

text: mechanisms

text: In

text: BSIT,

text: the

text: energies

text: and

text: momenta

text: of

text: these

text: three

text: discrete

text: modes

text: must

text: be

text: matched

text: to

text: sat-

text: isfy

text: ω

text: ω

text: Ω

text: and

text: k

text: k

text: q

text: simultaneously.

text: Here,

text: (ω

text: k

text: are

text: the

text: B

text: B

text: energy

text: and

text: momentum

text: of

text: the

text: lower

text: energy

text: optical

text: mode,

text: (ω

text: k

text: represent

text: the

text: higher

text: energy

text: optical

text: mode,

text: and

text: (Ω

text: q

text: represent

text: the

text: propagating

text: acoustic

text: B

text: B

text: mode

text: (Fig.

text: As

text: described

text: above,

text: the

text: coupling

text: between

text: the

text: three

text: modes

text: is

text: mediated

text: by

text: electrostrictive

text: Brillouin

text: scattering.

text: Previous

text: work

text: has

text: shown

text: that

text: such

text: instances

text: of

text: both

text: forward-

text: and

text: backward-SBS

text: phase

text: matching

text: can

text: occur

text: naturally

text: within

text: a

text: microresonator

text: When

text: momentum

text: matching

text: is

text: assumed,

text: a

text: lambda

text: system

text: analogy

text: to

text: EIT

text: can

text: also

text: be

text: made

text: in

text: the

text: case

text: of

text: BSIT

text: (Fig.

text: The

text: system

text: is

text: excited

text: through

text: the

text: addition

text: of

text: a

text: photon

text: at

text: either

text: (ω

text: k

text: or

text: (ω

text: k

text: to

text: a

text: virtual

text: level.

text: The

text: system

text: then

text: radiatively

text: decays

text: to

text: one

text: of

text: the

text: lower

text: levels

text: with

text: the

text: emission

text: of

text: a

text: Stokes

text: or

text: anti-Stokes

text: photon,

text: along

text: with

text: the

text: creation

text: or

text: annihilation

text: of

text: a

text: phonon.

text: The

text: additional

text: conservation


text: [4] K. Y. Song, M. G. Herr´aez, and L. Th´evenaz, “Observation of pulse delay- ing and advancement in optical ﬁbers using stimulated Brillouin scatter- ing,” Opt. Express, vol. 13, no. 1, pp. 82–88, 2005. [5] Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gau- thier, R. Boyd, and A. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Physical Review Letters, vol. 94, no. 15, p. 153902, Apr. 2005. [6] L. Th´evenaz, “Slow and fast light in optical ﬁbres,” Nature Photonics, vol. 2, no. 8, pp. 474–481, 2008. [7] R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science, vol. 326, no. 5956, pp. 1074–1077, 2009. [8] G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optome- chanical excitation of surface acoustic waves in a microdevice,” Nature Communications, vol. 2, p. 403, Jul. 2011. [9] G. Bahl, M. Tomes, F. Marquardt, and T. Carmon, “Observation of spon- taneous Brillouin cooling,” Nature Physics, vol. 8, no. 3, pp. 203–207, Mar. 2012. [10] Y. R. Shen and N. Bloembergen, “Theory of Stimulated Brillouin and Ra- man Scattering,” Phys. Rev., vol. 137, no. 6A, pp. A1787–A1805, Mar 1965. [11] A. Yariv, “Quantum theory for parametric interactions of light and hyper- sound,” Quantum Electronics, IEEE Journal of, vol. 1, no. 1, pp. 28–36, 1965. [12] R. W. Boyd, Nonlinear Optics, 3rd ed. Elsevier, 2008, Chapter 9. [13] J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon.” Optics Express, vol. 20, no. 18, pp. 20 170–20 180, Aug. 2012. [14] A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin ﬁber ring lasers,” JOSA B, vol. 18, no. 4, p. 556, 2001. [15] B. Y. Zel’dovich, V. I. Popocivhec, R. V, V, and F. S. Faisullov, “Connec- tion between the wave fronts of the reﬂected and exciting light in stumu- lated Mandelshtam-Brillouin scattering,” JETP Letters, vol. 15, no. 109, Aug. 1972. [16] R. Pant, E. Li, C. G. Poulton, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Observation of Brillouin dynamic grating in a photonic chip,” Optics Letters, vol. 38, no. 3, pp. 305–307, 2013. 13

text: K.

text: Y.

text: Song,

text: M.

text: G.

text: Herr´aez,

text: and

text: L.

text: Th´evenaz,

text: “Observation

text: of

text: pulse

text: delay-

text: ing

text: and

text: advancement

text: in

text: optical

text: ﬁbers

text: using

text: stimulated

text: Brillouin

text: scatter-

text: ing,”

text: Opt.

text: Express,

text: vol.

text: no.

text: pp.

text: Y.

text: Okawachi,

text: M.

text: Bigelow,

text: J.

text: Sharping,

text: Z.

text: Zhu,

text: A.

text: Schweinsberg,

text: D.

text: Gau-

text: thier,

text: R.

text: Boyd,

text: and

text: A.

text: Gaeta,

text: “Tunable

text: All-Optical

text: Delays

text: via

text: Brillouin

text: Slow

text: Light

text: in

text: an

text: Optical

text: Fiber,”

text: Physical

text: Review

text: Letters,

text: vol.

text: no.

text: p.

text: Apr.

text: L.

text: Th´evenaz,

text: “Slow

text: and

text: fast

text: light

text: in

text: optical

text: ﬁbres,”

text: Nature

text: Photonics,

text: vol.

text: no.

text: pp.

text: R.

text: W.

text: Boyd

text: and

text: D.

text: J.

text: Gauthier,

text: “Controlling

text: the

text: velocity

text: of

text: light

text: pulses,”

text: Science,

text: vol.

text: no.

text: pp.

text: G.

text: Bahl,

text: J.

text: Zehnpfennig,

text: M.

text: Tomes,

text: and

text: T.

text: Carmon,

text: “Stimulated

text: optome-

text: chanical

text: excitation

text: of

text: surface

text: acoustic

text: waves

text: in

text: a

text: microdevice,”

text: Nature

text: Communications,

text: vol.

text: p.

text: Jul.

text: G.

text: Bahl,

text: M.

text: Tomes,

text: F.

text: Marquardt,

text: and

text: T.

text: Carmon,

text: “Observation

text: of

text: spon-

text: taneous

text: Brillouin

text: cooling,”

text: Nature

text: Physics,

text: vol.

text: no.

text: pp.

text: Mar.

text: Y.

text: R.

text: Shen

text: and

text: N.

text: Bloembergen,

text: “Theory

text: of

text: Stimulated

text: Brillouin

text: and

text: Ra-

text: man

text: Scattering,”

text: Phys.

text: Rev.,

text: vol.

text: no.

text: pp.

text: Mar

text: A.

text: Yariv,

text: “Quantum

text: theory

text: for

text: parametric

text: interactions

text: of

text: light

text: and

text: hyper-

text: sound,”

text: Quantum

text: Electronics,

text: IEEE

text: Journal

text: of,

text: vol.

text: no.

text: pp.

text: R.

text: W.

text: Boyd,

text: Nonlinear

text: Optics,

text: ed.

text: Elsevier,

text: Chapter

text: J.

text: Li,

text: H.

text: Lee,

text: T.

text: Chen,

text: and

text: K.

text: J.

text: Vahala,

text: “Characterization

text: of

text: a

text: high

text: coherence,

text: Brillouin

text: microcavity

text: laser

text: on

text: silicon.”

text: Optics

text: Express,

text: vol.

text: no.

text: pp.

text: Aug.

text: A.

text: Debut,

text: S.

text: Randoux,

text: and

text: J.

text: Zemmouri,

text: “Experimental

text: and

text: theoretical

text: study

text: of

text: linewidth

text: narrowing

text: in

text: Brillouin

text: ﬁber

text: ring

text: lasers,”

text: JOSA

text: B,

text: vol.

text: no.

text: p.

text: B.

text: Y.

text: Zel’dovich,

text: V.

text: I.

text: Popocivhec,

text: R.

text: V,

text: V,

text: and

text: F.

text: S.

text: Faisullov,

text: “Connec-

text: tion

text: between

text: the

text: wave

text: fronts

text: of

text: the

text: reﬂected

text: and

text: exciting

text: light

text: in

text: stumu-

text: lated

text: Mandelshtam-Brillouin

text: scattering,”

text: JETP

text: Letters,

text: vol.

text: no.

text: Aug.

text: R.

text: Pant,

text: E.

text: Li,

text: C.

text: G.

text: Poulton,

text: D.-Y.

text: Choi,

text: S.

text: Madden,

text: B.

text: Luther-Davies,

text: and

text: B.

text: J.

text: Eggleton,

text: “Observation

text: of

text: Brillouin

text: dynamic

text: grating

text: in

text: a

text: photonic

text: chip,”

text: Optics

text: Letters,

text: vol.

text: no.

text: pp.


text: a Phonon dispersion Optical dispersion Ω ω (ω , k ) Optical phonons 2 2 THz (ω , k ) ω - ω =Ω (Ω , q ) 1 1 2 1 B GHz A c o u s t i c p h o n o n s B B k2- k1=qB q k b Control ω Tapered coupler c BSIT measured in transmission Probe ω p Resonator Op(cid:3)cal signals in (ω , k ) 1 1 in (ω , k ) 2 2 Microscope image Acoustic signal in (Ω , q ) q k B B B k 1 2 Forward SBS interaction c Stokes gain n n gai ΓB ωc + ΩB ω ui ω - Ω ω p o rill c B c ΓB B An(cid:3)-Stokes loss Induced transparency n eo dsi guimis κ1 κ2 es Wavtran (ω1, k1) (ω2, k2) ω ω p c Figure 1: Theoretical description of BSIT. a. Brillouin phase matching requires two optical modes of a system that are separated by the acoustic parameters in ω-k space. b. In a BSIT experiment, control and probe lasers are coupled to phase matched optical whispering gallery modes of a resonator by means of a tapered optical ﬁber. A forward-SBS opto-acoustic interaction takes place within the resonator. c. The control laser pumps the lower frequency optical mode to allow anti-Stokes scattering while the resonator suppresses Stokes scattering [9]. (Bottom) A probe laser scans through the optical resonances, visible as a drop in waveguide transmission [25]. (Top) From traditional SBS theory [12] anti-Stokes absorption is predicted for a probe signal at ω = ω + Ω , p c B which intuitively should increase the probe loss. However, when (Ω , q ) represent a long lived phonon mode, B B a transparency is observed due to interference between Stokes and anti-Stokes scattering pathways. κ represent i the respective dissipation rates of the two optical modes. 3 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: a

text: Phonon

text: dispersion

text: Optical

text: dispersion

text: Ω

text: ω

text: (ω

text: k

text: Optical

text: phonons

text: THz

text: (ω

text: k

text: ω

text: ω

text: =Ω

text: (Ω

text: q

text: B

text: GHz

text: A

text: c

text: o

text: u

text: s

text: t

text: i

text: c

text: p

text: h

text: o

text: n

text: o

text: n

text: s

text: B

text: B

text: q

text: k

text: b

text: Control

text: ω

text: Tapered

text: coupler

text: c

text: BSIT

text: measured

text: in

text: transmission

text: Probe

text: ω

text: p

text: Resonator

text: signals

text: in

text: (ω

text: k

text: in

text: (ω

text: k

text: Microscope

text: image

text: Acoustic

text: signal

text: in

text: (Ω

text: q

text: q

text: k

text: B

text: B

text: B

text: k

text: Forward

text: SBS

text: interaction

text: c

text: Stokes

text: gain

text: n

text: n

text: gai

text: ΓB

text: ωc

text: ΩB

text: ω

text: ui

text: ω

text: Ω

text: ω

text: p

text: o

text: rill

text: c

text: B

text: c

text: ΓB

text: B

text: loss

text: Induced

text: transparency

text: n

text: eo

text: dsi

text: guimis

text: κ1

text: κ2

text: es

text: Wavtran

text: ω

text: ω

text: p

text: c

text: Figure

text: Theoretical

text: description

text: of

text: BSIT.

text: a.

text: Brillouin

text: phase

text: matching

text: requires

text: two

text: optical

text: modes

text: of

text: a

text: system

text: that

text: are

text: separated

text: by

text: the

text: acoustic

text: parameters

text: in

text: ω-k

text: space.

text: b.

text: In

text: a

text: BSIT

text: experiment,

text: control

text: and

text: probe

text: lasers

text: are

text: coupled

text: to

text: phase

text: matched

text: optical

text: whispering

text: gallery

text: modes

text: of

text: a

text: resonator

text: by

text: means

text: of

text: a

text: tapered

text: optical

text: ﬁber.

text: A

text: forward-SBS

text: opto-acoustic

text: interaction

text: takes

text: place

text: within

text: the

text: resonator.

text: c.

text: The

text: control

text: laser

text: pumps

text: the

text: lower

text: frequency

text: optical

text: mode

text: to

text: allow

text: anti-Stokes

text: scattering

text: while

text: the

text: resonator

text: suppresses

text: Stokes

text: scattering

text: (Bottom)

text: A

text: probe

text: laser

text: scans

text: through

text: the

text: optical

text: resonances,

text: visible

text: as

text: a

text: drop

text: in

text: waveguide

text: transmission

text: (Top)

text: From

text: traditional

text: SBS

text: theory

text: anti-Stokes

text: absorption

text: is

text: predicted

text: for

text: a

text: probe

text: signal

text: at

text: ω

text: ω

text: Ω

text: p

text: c

text: B

text: which

text: intuitively

text: should

text: increase

text: the

text: probe

text: loss.

text: However,

text: when

text: (Ω

text: q

text: represent

text: a

text: long

text: lived

text: phonon

text: mode,

text: B

text: B

text: a

text: transparency

text: is

text: observed

text: due

text: to

text: interference

text: between

text: Stokes

text: and

text: anti-Stokes

text: scattering

text: pathways.

text: κ

text: represent

text: i

text: the

text: respective

text: dissipation

text: rates

text: of

text: the

text: two

text: optical

text: modes.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Forward probe Backward probe Optical mode Optical mode ) u. a. ( n o si s 118888 192 196 200 118888 192 196 200 mi s n a e tr BSIT No transparency b observed o r P 192.6 193 193.4 193.8 194.2 192.6 193 193.4 193.8 194.2 Frequency offset from control laser (MHz) Frequency offset from control laser (MHz) Figure 4: Demonstration of non-reciprocity in BSIT. (Left) Transparency is induced in the anti-Stokes optical mode for a forward propagating probe signal in a Brillouin coupled system. (Right) There is no trans- parency observed when the optical probe signal is switched to the backward direction. The background noise and asymmetries are created due to the Fabry-Perot resonances in the diﬀering forward and backward ﬁber propagation pathways, and due to the transfer functions of the two diﬀerent electro-optic modulators used for the experiment. 9 ##LTFigure## ##LTFigure## ##LTFigure##

text: Forward

text: probe

text: Backward

text: probe

text: Optical

text: mode

text: Optical

text: mode

text: u.

text: a.

text: n

text: o

text: si

text: s

text: mi

text: s

text: n

text: a

text: e

text: tr

text: BSIT

text: No

text: transparency

text: b

text: observed

text: o

text: r

text: P

text: Frequency

text: offset

text: from

text: control

text: laser

text: (MHz)

text: Frequency

text: offset

text: from

text: control

text: laser

text: (MHz)

text: Figure

text: Demonstration

text: of

text: non-reciprocity

text: in

text: BSIT.

text: (Left)

text: Transparency

text: is

text: induced

text: in

text: the

text: anti-Stokes

text: optical

text: mode

text: for

text: a

text: forward

text: propagating

text: probe

text: signal

text: in

text: a

text: Brillouin

text: coupled

text: system.

text: (Right)

text: There

text: is

text: no

text: trans-

text: parency

text: observed

text: when

text: the

text: optical

text: probe

text: signal

text: is

text: switched

text: to

text: the

text: backward

text: direction.

text: The

text: background

text: noise

text: and

text: asymmetries

text: are

text: created

text: due

text: to

text: the

text: Fabry-Perot

text: resonances

text: in

text: the

text: diﬀering

text: forward

text: and

text: backward

text: ﬁber

text: propagation

text: pathways,

text: and

text: due

text: to

text: the

text: transfer

text: functions

text: of

text: the

text: two

text: diﬀerent

text: electro-optic

text: modulators

text: used

text: for

text: the

text: experiment.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 5 seems highly unrealistic to expect that nonlinear eﬀects this correction is small when the stopping force is due can make the deceleration correction insigniﬁcant. For a to the plasma electrons, but can be signiﬁcant when the He2+-projectile, the left-hand side of the inequality (??) stopping force is due to the ions. (cid:39) (cid:39) is 0.3, while δF/F 0.24, which is still a noticeable 0 correction. Acknowledgments To conclude, we have rigorously addressed the eﬀect of the projectile deceleration on the stopping force. We have provided a general expression that allows investigating The authors thank Mikhail Pustylnik for useful dis- this eﬀect for any speciﬁed dielectric function [Eq. (??)]. cussions. The work received funding from the European For a cold plasma, we have derived a simple expression Research Council under the European Union’s Seventh for the correction to the stopping force [Eq. (??)]. As- Framework Programme (FP7/2007—2013)/ERC Grant suming that the projectile is an ion, we have found that agreement 267499. [] T. Peter and J. Meyer-Ter-Vehn, Phys. Rev. A 43, 1998 [] O. Ishihara, J. Phys. D: Appl. Phys. 40, 121 (2007). (1991). [] W. J. Miloch, M. Kroll, and D. Block, Phys. Plasmas 17, [] D. H. H. Hoﬀmann, R. Bock, A. Y. Faenov, U. Funk, 103703 (2010). M. Geissel, U. Neuner, T. A. Pikuz, F. Rosmej, M. Roth, [] A. V. Ivlev, S. K. Zhdanov, S. A. Khrapak, and G. E. W. Su¨ß, et al., Nucl. Instrum. Meth. B 161, 9 (2000). Morﬁll, Phys. Rev. E 71, 016405 (2005). [] S. Kawata, K. Horioka, M. Murakami, Y. Oguri, [] N. Bohr, Philos. Mag. 30, 581 (1915). J. Hasegawa, K. Takayama, H. Yoneda, K. Miyazawa, [] T. Peter, J. Plasma Phys. 44, 269 (1990). T. Someya, A. I. Ogoyski, et al., Nucl. Instrum. Meth. A [] R. Kompaneets, S. V. Vladimirov, A. V. Ivlev, and 577, 21 (2007). G. Morﬁll, New J. Phys. 10, 063018 (2008). [] M. Lampe, G. Joyce, G. Ganguli, and V. Gavrishchaka, [] E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics Phys. Plasmas 7, 3851 (2000). (Pergamon, Oxford, 1981). [] S. A. Khrapak, A. V. Ivlev, G. E. Morﬁll, and H. M. [] A. F. Alexandrov, L. S. Bogdankevich, and A. A. Thomas, Phys. Rev. E 66, 046414 (2002). Rukhadze, Principles of Plasma Electrodynamics [] V. Yaroshenko, S. Ratynskaia, S. Khrapak, M. H. (Springer, Berlin, 1984). Thoma, M. Kretschmer, H. Ho¨fner, G. E. Morﬁll, A. Zob- [] H.-D. Betz, R. Ho¨ppler, R. Schramm, and W. Oswald, nin, A. Usachev, O. Petrov, et al., Phys. Plasmas 12, Nucl. Instrum. Meth. B 33, 185 (1988). 093503 (2005). ##LTLine## ##LTLine## ##LTLine##

text: seems

text: highly

text: unrealistic

text: to

text: expect

text: that

text: nonlinear

text: eﬀects

text: this

text: correction

text: is

text: small

text: when

text: the

text: stopping

text: force

text: is

text: due

text: can

text: make

text: the

text: deceleration

text: correction

text: insigniﬁcant.

text: For

text: a

text: to

text: the

text: plasma

text: electrons,

text: but

text: can

text: be

text: signiﬁcant

text: when

text: the

text: the

text: left-hand

text: side

text: of

text: the

text: inequality

text: stopping

text: force

text: is

text: due

text: to

text: the

text: ions.

text: is

text: while

text: δF/F

text: which

text: is

text: still

text: a

text: noticeable

text: correction.

text: Acknowledgments

text: To

text: conclude,

text: we

text: have

text: rigorously

text: addressed

text: the

text: eﬀect

text: of

text: the

text: projectile

text: deceleration

text: on

text: the

text: stopping

text: force.

text: We

text: have

text: provided

text: a

text: general

text: expression

text: that

text: allows

text: investigating

text: The

text: authors

text: thank

text: Mikhail

text: Pustylnik

text: for

text: useful

text: dis-

text: this

text: eﬀect

text: for

text: any

text: speciﬁed

text: dielectric

text: function

text: [Eq.

text: cussions.

text: The

text: work

text: received

text: funding

text: from

text: the

text: European

text: For

text: a

text: cold

text: plasma,

text: we

text: have

text: derived

text: a

text: simple

text: expression

text: Research

text: Council

text: under

text: the

text: European

text: Union’s

text: Seventh

text: for

text: the

text: correction

text: to

text: the

text: stopping

text: force

text: [Eq.

text: As-

text: Framework

text: Programme

text: Grant

text: suming

text: that

text: the

text: projectile

text: is

text: an

text: ion,

text: we

text: have

text: found

text: that

text: agreement

text: T.

text: Peter

text: and

text: J.

text: Meyer-Ter-Vehn,

text: Phys.

text: Rev.

text: A

text: O.

text: Ishihara,

text: J.

text: Phys.

text: D:

text: Appl.

text: Phys.

text: W.

text: J.

text: Miloch,

text: M.

text: Kroll,

text: and

text: D.

text: Block,

text: Phys.

text: Plasmas

text: D.

text: H.

text: H.

text: Hoﬀmann,

text: R.

text: Bock,

text: A.

text: Y.

text: Faenov,

text: U.

text: Funk,

text: M.

text: Geissel,

text: U.

text: Neuner,

text: T.

text: A.

text: Pikuz,

text: F.

text: Rosmej,

text: M.

text: Roth,

text: A.

text: V.

text: Ivlev,

text: S.

text: K.

text: Zhdanov,

text: S.

text: A.

text: Khrapak,

text: and

text: G.

text: E.

text: W.

text: Su¨ß,

text: et

text: al.,

text: Nucl.

text: Instrum.

text: Meth.

text: B

text: Morﬁll,

text: Phys.

text: Rev.

text: E

text: S.

text: Kawata,

text: K.

text: Horioka,

text: M.

text: Murakami,

text: Y.

text: Oguri,

text: N.

text: Bohr,

text: Philos.

text: Mag.

text: J.

text: Hasegawa,

text: K.

text: Takayama,

text: H.

text: Yoneda,

text: K.

text: Miyazawa,

text: T.

text: Peter,

text: J.

text: Plasma

text: Phys.

text: T.

text: Someya,

text: A.

text: I.

text: Ogoyski,

text: et

text: al.,

text: Nucl.

text: Instrum.

text: Meth.

text: A

text: R.

text: Kompaneets,

text: S.

text: V.

text: Vladimirov,

text: A.

text: V.

text: Ivlev,

text: and

text: G.

text: Morﬁll,

text: New

text: J.

text: Phys.

text: M.

text: Lampe,

text: G.

text: Joyce,

text: G.

text: Ganguli,

text: and

text: V.

text: Gavrishchaka,

text: E.

text: M.

text: Lifshitz

text: and

text: L.

text: P.

text: Pitaevskii,

text: Physical

text: Kinetics

text: Phys.

text: Plasmas

text: (Pergamon,

text: Oxford,

text: S.

text: A.

text: Khrapak,

text: A.

text: V.

text: Ivlev,

text: G.

text: E.

text: Morﬁll,

text: and

text: H.

text: M.

text: A.

text: F.

text: Alexandrov,

text: L.

text: S.

text: Bogdankevich,

text: and

text: A.

text: A.

text: Thomas,

text: Phys.

text: Rev.

text: E

text: Rukhadze,

text: Principles

text: of

text: Plasma

text: Electrodynamics

text: V.

text: Yaroshenko,

text: S.

text: Ratynskaia,

text: S.

text: Khrapak,

text: M.

text: H.

text: (Springer,

text: Berlin,

text: Thoma,

text: M.

text: Kretschmer,

text: H.

text: Ho¨fner,

text: G.

text: E.

text: Morﬁll,

text: A.

text: Zob-

text: H.-D.

text: Betz,

text: R.

text: Ho¨ppler,

text: R.

text: Schramm,

text: and

text: W.

text: Oswald,

text: nin,

text: A.

text: Usachev,

text: O.

text: Petrov,

text: et

text: al.,

text: Phys.

text: Plasmas

text: Nucl.

text: Instrum.

text: Meth.

text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Super-surveys modes and cosmic shear 3 (cid:10) (cid:11) (cid:80) most conveniently the dimensionless local average density δ¯ = δ˜ /V , this curvature is on average nothing else than the Fisher − 0 · information content ∂α22 ln p of the ﬁeld on(cid:88)this parameter. Since δ(xi) (cid:88)= δ¯ + k(cid:54)=0 δ˜k eik xi follows ∂δ(xi)/∂δ¯ = 1 and thus ∂ ln p d ∂ ln p ∂2 ln p d ∂2 ln p δ = δ and δ = δ (5) ∂δ¯ i=1 ∂δi(cid:42)(cid:32) ∂δ¯2(cid:33) i(cid:43),j=1 ∂δi∂δj (cid:88) Therefore in full generality holds − d ∂2 Fδ¯δ¯ = ∂δ ∂δ ln pδ . (6) (cid:42)(cid:32)i,j=1 i j(cid:33) (cid:43) (cid:88) Likewise the degeneracy to a model parameter is described by − d ∂2 Fαδ¯ = ∂δ ∂α ln pδ . (7) i i=1 2.2 Background modes in lognormal ﬁelds We now state the results for the background mode extraction in lognormal ﬁelds upon which a fair amount of our subsequent considerations is based. For full generality, we consider a number N of jointly lognormal ﬁelds δ (x ) with bin n i zero ensemble mean (in a tomographic analysis of a weak-lensing survey these ﬁelds represent the rescaled convergence ﬁelds | | δ (x ) = κ (x )/ κ where κ is the total(cid:68)weight of th(cid:69)e nth lensing kernel, see section later for deﬁnitions and more n i n i 0,n 0,n × details). The ﬁelds are described statistically by N N hermitian spectral matrices Pnm(k) bin bin δ ∗ (cid:48) δ˜ (k)δ˜ (k ) = V δ (cid:48)Pnm(k) (8) n m kk δ × Assuming periodic boundary conditions, these are Fourier transforms of the d d covariance matrices ξnm. The log-densities − δ − A (x ) = ln(1 + δ (x )) form jointly Gaussian ﬁelds with covariance matrices ξnm(x x ) = ln(1 + ξnm(x x )), and n i n i(cid:104) (cid:105) − A i j δ i j ensemble mean vector A = σ2 /2. The Fourier transforms of ξ are the spectral matrices P (k), which are the most n A,n A A natural variables to describe the lognormal ﬁ(cid:34)eld. There are(cid:88)Nbi(cid:2)n backgr(cid:3)ound modes δ¯n(cid:18). (cid:19) (cid:35) Appendix details the derivation of the following results. The Fisher information matrix on the background modes is − Fδ¯nδ¯m = eσA2 neσA2 m δnm d eσA2 n + PA−1(k) nm Pδnm(k) + PAV(0) 1 . (9) k nm The last term is the Gaussian limit where the modes are statistically independent. It only depends on the survey geometry. The ﬁrst terms are due to mode coupling and are directly proportional to the number d of modes available. For a given survey volume the zeroth mode can thus be reconstructed meaningfully from the others provided the resolution k is large enough ∼ max for the ﬁrst terms to be dominant. This happens roughly at the critical value n¯P (0) 1, where n¯ = d/V is the number A density of grid points, in one to one corr(cid:34)e(cid:88)spon(cid:18)dence to kmax. (cid:19)The deg(cid:88)ene(cid:18)racy to a(cid:19)mod(cid:18)el parame(cid:19)te(cid:35)r is described by the Fisher matrix element − Fδ¯nα = eσA2 ,n PA−1(k)PA∂(αk) + Nbin PAV(0) 1 12 ∂σ∂A2α,m . (10) k nn m=1 nm In stark contrast to a Gaussian ﬁeld (for which Fδ¯α = 0, see (A2)), we ﬁnd that all modes correlate equally with δ¯, and Fαδ¯ is typically proportional to the n(cid:88)umbe(cid:20)r of modes present. Finally, the in(cid:21)form(cid:18)ation m(cid:19)at(cid:18)rix for m(cid:19)odel(cid:18)parame(cid:19)ters in the lognormal ﬁeld is identical to the well known formula for Gaussian ﬁelds − 1 − ∂P (k) − ∂P (k) 1 ∂σ2 t P (0) 1 1 ∂σ2 Fαβ = 2 Tr PA 1(k) ∂Aα PA 1(k) ∂Aβ + 2 ∂αA AV 2 ∂βA (11) k − This last equation follows directly from the fact that the log-density ﬁelds A are jointly Gaussian with mean vector 1σ2 . 2 A Before proceeding, a word of caution. Eqs. (10) and (11) hold assuming that κ is independent of the parameters of interest. 0 This holds throughout this paper as we will be interested in σ and the spectral index n , κ being purely geometric. The 8 s 0 appendix provides the complete expressions including the derivatives of κ for completeness. 0 3 TESTS AND IMPLICATIONS We test the background mode reconstruction and the above results in a large ensemble of simulations of lognormal ﬁelds. We simulate a square survey of L = 10 degrees on the side with d = N2 points on the ﬂat sky. To generate each map as realistically as possible we use the circulant embedding method (for details see C(cid:32)ar(cid:33)ron et al. 2014). The spectrum assigned to (cid:88) the discrete Fourier modes is deﬁned through the following equation V − · L i −N · · · N − Pδd,iAscrete((cid:96)) = d ξδ,A(xi)e i(cid:96) xi with grid points xi = N j , i, j = 2 , , 2 1 (12) xi (cid:13) c 2014 RAS, MNRAS 000, 1–?? ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Super-surveys

text: modes

text: and

text: cosmic

text: shear

text: most

text: conveniently

text: the

text: dimensionless

text: local

text: average

text: density

text: δ¯

text: δ˜

text: /V

text: this

text: curvature

text: is

text: on

text: average

text: nothing

text: else

text: than

text: the

text: Fisher

text: information

text: content

text: ∂α22

text: ln

text: p

text: of

text: the

text: ﬁeld

text: parameter.

text: Since

text: δ(xi)

text: δ¯

text: δ˜k

text: eik

text: xi

text: follows

text: ∂δ(xi)/∂δ¯

text: and

text: thus

text: ln

text: p

text: d

text: ln

text: p

text: ∂2

text: ln

text: p

text: d

text: ∂2

text: ln

text: p

text: δ

text: δ

text: and

text: δ

text: δ

text: ∂δ¯

text: ∂δi∂δj

text: Therefore

text: in

text: full

text: generality

text: holds

text: d

text: ∂2

text: Fδ¯δ¯

text: ∂δ

text: ∂δ

text: ln

text: pδ

text: i

text: Likewise

text: the

text: degeneracy

text: to

text: a

text: model

text: parameter

text: is

text: described

text: by

text: d

text: ∂2

text: Fαδ¯

text: ∂δ

text: ∂α

text: ln

text: pδ

text: i

text: Background

text: modes

text: in

text: lognormal

text: ﬁelds

text: We

text: now

text: state

text: the

text: results

text: for

text: the

text: background

text: mode

text: extraction

text: in

text: lognormal

text: ﬁelds

text: upon

text: which

text: a

text: fair

text: amount

text: of

text: our

text: subsequent

text: considerations

text: is

text: based.

text: For

text: full

text: generality,

text: we

text: consider

text: a

text: number

text: N

text: of

text: jointly

text: lognormal

text: ﬁelds

text: δ

text: (x

text: with

text: bin

text: n

text: i

text: zero

text: ensemble

text: mean

text: (in

text: a

text: tomographic

text: analysis

text: of

text: a

text: weak-lensing

text: survey

text: these

text: ﬁelds

text: represent

text: the

text: rescaled

text: convergence

text: ﬁelds

text: δ

text: (x

text: κ

text: (x

text: κ

text: where

text: κ

text: is

text: the

text: of

text: nth

text: lensing

text: kernel,

text: see

text: section

text: later

text: for

text: deﬁnitions

text: and

text: more

text: n

text: i

text: n

text: i

text: details).

text: The

text: ﬁelds

text: are

text: described

text: statistically

text: by

text: N

text: N

text: hermitian

text: spectral

text: matrices

text: Pnm(k)

text: bin

text: bin

text: δ

text: δ˜

text: (k)δ˜

text: (k

text: V

text: δ

text: n

text: m

text: kk

text: δ

text: Assuming

text: periodic

text: boundary

text: conditions,

text: these

text: are

text: Fourier

text: transforms

text: of

text: the

text: d

text: d

text: covariance

text: matrices

text: ξnm.

text: The

text: log-densities

text: δ

text: A

text: (x

text: δ

text: (x

text: form

text: jointly

text: Gaussian

text: ﬁelds

text: with

text: covariance

text: matrices

text: ξnm(x

text: x

text: ξnm(x

text: x

text: and

text: n

text: i

text: n

text: A

text: i

text: j

text: δ

text: i

text: j

text: ensemble

text: mean

text: vector

text: A

text: σ2

text: The

text: Fourier

text: transforms

text: of

text: ξ

text: are

text: the

text: spectral

text: matrices

text: P

text: (k),

text: which

text: are

text: the

text: most

text: n

text: A,n

text: A

text: A

text: natural

text: variables

text: to

text: describe

text: the

text: lognormal

text: There

text: modes

text: Appendix

text: details

text: the

text: derivation

text: of

text: the

text: following

text: results.

text: The

text: Fisher

text: information

text: matrix

text: on

text: the

text: background

text: modes

text: is

text: Fδ¯nδ¯m

text: m

text: δnm

text: d

text: n

text: nm

text: Pδnm(k)

text: k

text: nm

text: The

text: last

text: term

text: is

text: the

text: Gaussian

text: limit

text: where

text: the

text: modes

text: are

text: statistically

text: independent.

text: It

text: only

text: depends

text: on

text: the

text: survey

text: geometry.

text: The

text: ﬁrst

text: terms

text: are

text: due

text: to

text: mode

text: coupling

text: and

text: are

text: directly

text: proportional

text: to

text: the

text: number

text: d

text: of

text: modes

text: available.

text: For

text: a

text: given

text: survey

text: volume

text: the

text: zeroth

text: mode

text: can

text: thus

text: be

text: reconstructed

text: meaningfully

text: from

text: the

text: others

text: provided

text: the

text: resolution

text: k

text: is

text: large

text: enough

text: max

text: for

text: the

text: ﬁrst

text: terms

text: to

text: be

text: dominant.

text: This

text: happens

text: roughly

text: at

text: the

text: critical

text: value

text: n¯P

text: where

text: n¯

text: d/V

text: is

text: the

text: number

text: A

text: density

text: of

text: grid

text: points,

text: in

text: one

text: to

text: one

text: to

text: kmax.

text: to

text: is

text: described

text: by

text: the

text: Fisher

text: matrix

text: element

text: Fδ¯nα

text: ,n

text: Nbin

text: ∂σ∂A2α,m

text: k

text: nn

text: nm

text: In

text: stark

text: contrast

text: to

text: a

text: Gaussian

text: ﬁeld

text: (for

text: which

text: Fδ¯α

text: see

text: we

text: ﬁnd

text: that

text: all

text: modes

text: correlate

text: equally

text: with

text: δ¯,

text: and

text: Fαδ¯

text: is

text: typically

text: proportional

text: to

text: the

text: of

text: modes

text: present.

text: Finally,

text: the

text: for

text: in

text: the

text: lognormal

text: ﬁeld

text: is

text: identical

text: to

text: the

text: well

text: known

text: formula

text: for

text: Gaussian

text: ﬁelds

text: ∂P

text: (k)

text: ∂P

text: (k)

text: ∂σ2

text: t

text: P

text: ∂σ2

text: Fαβ

text: Tr

text: PA

text: ∂Aα

text: PA

text: ∂Aβ

text: ∂αA

text: AV

text: ∂βA

text: k

text: This

text: last

text: equation

text: follows

text: directly

text: from

text: the

text: fact

text: that

text: the

text: log-density

text: ﬁelds

text: A

text: are

text: jointly

text: Gaussian

text: with

text: mean

text: vector

text: A

text: Before

text: proceeding,

text: a

text: word

text: of

text: caution.

text: Eqs.

text: and

text: hold

text: assuming

text: that

text: κ

text: is

text: independent

text: of

text: the

text: parameters

text: of

text: interest.

text: This

text: holds

text: throughout

text: this

text: paper

text: as

text: we

text: will

text: be

text: interested

text: in

text: σ

text: and

text: the

text: spectral

text: index

text: n

text: κ

text: being

text: purely

text: geometric.

text: The

text: s

text: appendix

text: provides

text: the

text: complete

text: expressions

text: including

text: the

text: derivatives

text: of

text: κ

text: for

text: completeness.

text: TESTS

text: AND

text: IMPLICATIONS

text: We

text: test

text: the

text: background

text: mode

text: reconstruction

text: and

text: the

text: above

text: results

text: in

text: a

text: large

text: ensemble

text: of

text: simulations

text: of

text: lognormal

text: ﬁelds.

text: We

text: simulate

text: a

text: square

text: survey

text: of

text: L

text: degrees

text: on

text: the

text: side

text: with

text: d

text: points

text: on

text: the

text: ﬂat

text: sky.

text: To

text: generate

text: each

text: map

text: as

text: realistically

text: as

text: possible

text: we

text: use

text: the

text: circulant

text: embedding

text: method

text: (for

text: details

text: see

text: et

text: al.

text: The

text: spectrum

text: assigned

text: to

text: the

text: discrete

text: Fourier

text: modes

text: is

text: deﬁned

text: through

text: the

text: following

text: equation

text: V

text: L

text: i

text: −N

text: N

text: d

text: ξδ,A(xi)e

text: xi

text: with

text: grid

text: points

text: xi

text: N

text: j

text: i,

text: j

text: xi

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: d Xn 1 1 d Xn 2 2 d Xn K K d Xn K+1 K+1 n + d (n) max Y n+dmax(n) D Fig. 1: Gaussian time asynchronous multiple access relay channel (TA-MARC), with delays d , · · · , d . 1 K+1 Xn denotes the length-n MARC input codeword (X [0], · · · , X [n−1]) of the (cid:96)th transmitter, and based (cid:96) (cid:96) (cid:96) on this, we also denote (X [a], · · · , X [b]) by Xb . The n-length discrete Fourier transforms (DFT) of (cid:96) (cid:96) (cid:96),a (cid:44) the n-length codeword Xn is denoted by Xˆn = DFT(Xn). Furthermore, let [1, K] {1, · · · , K}, for (cid:96) (cid:96) (cid:96) N ∀K ∈ . ∞ Consider K ﬁnite alphabet sources {(U [i], U [i], · · · , U [i])} as correlated random variables drawn 1 2 K i=0 according to a distribution p(u , u , · · · , u ). The sources are memoryless, i.e., (U [i], U [i], · · · , U [i])’s 1 2 K 1 2 K are independent and identically distributed (i.i.d) for i = 1, 2, · · · . The indices 1, · · · , K, represent the transmitter nodes and the index K + 1 represents the relay transmitter. All of the sources are to be transmitted to a destination by the help of a relay through a continuous alphabet, discrete-time memoryless multiple-access relay channel (MARC) with time asynchronism between different transmitters and the relay. Speciﬁcally, as depicted in Fig. ??, the encoders use different time references and thus we assume that the encoders start transmitting with offsets of 0 ≤ d ≤ d (n), (cid:96) = 1, · · · , K + 1, (1) (cid:96) max symbols with respect to a ﬁxed time reference, where d is the offset for the relay transmitter with K+1 respect to the time reference. 4 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: d

text: Xn

text: d

text: Xn

text: d

text: Xn

text: K

text: K

text: d

text: Xn

text: n

text: d

text: (n)

text: max

text: Y

text: n+dmax(n)

text: D

text: Fig.

text: Gaussian

text: time

text: asynchronous

text: multiple

text: access

text: relay

text: channel

text: (TA-MARC),

text: with

text: delays

text: d

text: d

text: Xn

text: denotes

text: the

text: length-n

text: MARC

text: input

text: codeword

text: (X

text: X

text: of

text: the

text: transmitter,

text: and

text: based

text: on

text: this,

text: we

text: also

text: denote

text: (X

text: [a],

text: X

text: [b])

text: by

text: Xb

text: The

text: n-length

text: discrete

text: Fourier

text: transforms

text: (DFT)

text: of

text: the

text: n-length

text: codeword

text: Xn

text: is

text: denoted

text: by

text: Xˆn

text: DFT(Xn).

text: Furthermore,

text: let

text: K]

text: K},

text: for

text: N

text: ∀K

text: Consider

text: K

text: ﬁnite

text: alphabet

text: sources

text: {(U

text: [i],

text: U

text: [i],

text: U

text: [i])}

text: as

text: correlated

text: random

text: variables

text: drawn

text: K

text: according

text: to

text: a

text: distribution

text: p(u

text: u

text: u

text: The

text: sources

text: are

text: memoryless,

text: i.e.,

text: (U

text: [i],

text: U

text: [i],

text: U

text: [i])’s

text: K

text: K

text: are

text: independent

text: and

text: identically

text: distributed

text: (i.i.d)

text: for

text: i

text: The

text: indices

text: K,

text: represent

text: the

text: transmitter

text: nodes

text: and

text: the

text: index

text: K

text: represents

text: the

text: relay

text: transmitter.

text: All

text: of

text: the

text: sources

text: are

text: to

text: be

text: transmitted

text: to

text: a

text: destination

text: by

text: the

text: help

text: of

text: a

text: relay

text: through

text: a

text: continuous

text: alphabet,

text: discrete-time

text: memoryless

text: multiple-access

text: relay

text: channel

text: (MARC)

text: with

text: time

text: asynchronism

text: between

text: different

text: transmitters

text: and

text: the

text: relay.

text: Speciﬁcally,

text: as

text: depicted

text: in

text: Fig.

text: the

text: encoders

text: use

text: different

text: time

text: references

text: and

text: thus

text: we

text: assume

text: that

text: the

text: encoders

text: start

text: transmitting

text: with

text: offsets

text: of

text: d

text: d

text: (n),

text: K

text: max

text: symbols

text: with

text: respect

text: to

text: a

text: ﬁxed

text: time

text: reference,

text: where

text: d

text: is

text: the

text: offset

text: for

text: the

text: relay

text: transmitter

text: with

text: respect

text: to

text: the

text: time

text: reference.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##