text: 18 ther modify the ﬂow by allowing a signiﬁcant Maxwell stress planet Hill sphere, and the enhanced magnetic coupling also to develop near the disk midplane through the combined ac- led to the launching of a magnetocentrifugally accelerated jet tion of ﬁeld winding and Hall drift (Kunz 2008; Kunz & Lesur from the circumplanetary disk. It remains unclear whether 2013; Lesur et al. 2014; Bai 2014), but does not lead to tur- or not these effects will arise when ambipolar diffusion is in- bulence. We are then left with a picture of PPDs in which the cluded, given the low density of the gap region. innermost regions sustain vigorous MRI-induced turbulence because of collisional ionization of alkali metals, the interme- diate regions between ∼ 0.5 − 20 au are laminar with angular We thank Martin Pessah, Anna Tenerani and Marco Velli momentum transport occurring through a disk wind and per- for useful advice on plasma instabilities, and the referee for haps through Maxwell stresses near the midplane (an effect a helpful report. The research was carried out in part at the that only arises if Ω · B > 0 and when including the Hall Jet Propulsion Laboratory, California Institute of Technology, term), and an outer region where the Hall effect and Ohmic under a contract with the National Aeronautics and Space resistivity are sub-dominant and weak turbulence arises (Si- Administration and with the support of the NASA Origins mon et al. 2013; Bai 2015). of Solar Systems program via grant 13-OSS13-0114. The The picture painted above has important implications for research leading to these results has received funding from dust evolution and planet formation. In principle, dust coagu- the Danish Council for Independent Research (DFF) and lation and settling in the laminar region should occur rapidly, FP7 Marie Curie Actions – COFUND under the grant-ID: and in the absence of a source of stirring the loss of small DFF – 1325-00111, and from the People Programme (Marie grains should affect the spectral energy distribution (SED). Curie Actions) of the European Union’s Seventh Framework This was examined by Dullemond & Dominik (2005), who Programme (FP7/2007-2013) under REA grant agreement showed that this rapid evolution of the SED is not in agree- 327995. This work used the NIRVANA-III code developed ment with observations, and for many years this important by Udo Ziegler at the Leibniz Institute for Astrophysics result has provided indirect evidence for the existence of tur- (AIP). We acknowledge that the results of this research have bulent stirring in the planet forming regions of protoplanetary been partly achieved using the PRACE-3IP project (FP7 RI- disks. In the absence of a magnetic origin for local turbulence 312763) resource Fionn based in Ireland at the Irish Centre this suggests that either another source of turbulence exists, for High-End Computing (ICHEC). Computations were also such as the vertical shear instability (e.g. Nelson, Gressel, & performed on the astro2 node at the Danish Center for Su- Umurhan 2013), or that small grains are continuously deliv- percomputing (DCSC), and on the U.K. STFC-funded DiRAC ered to the laminar region from the outer and/or inner turbu- facility. lent regions. The results in this paper are consistent with the remanent REFERENCES magnetic ﬁeld measurements in chondrules by Fu et al. (2014) Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2009, ApJ, 700, 1502 in that the minimum midplane ﬁeld strength for a wind driven Bai, X.-N. 2011, ApJ, 739, 50 disk at the presumed chondrule forming location (the mid- —. 2013, ApJ, 772, 96 plane at 2 to 3 au) we ﬁnd is smaller than the upper bound —. 2014, ApJ, 791, 137 on the ambient ﬁeld strength in the chondrule cooling region —. 2015, ApJ, 798, 84 found in that work. This allows the chondrules studied by Fu Bai, X.-N., & Goodman, J. 2009, ApJ, 701, 737 Bai, X.-N., & Stone, J. M. 2013, ApJ, 769, 76 et al. (2014) to have formed in the midplane of a wind driven Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214 laminar disk, while also allowing for the chondrule formation Balbus, S. A., & Terquem, C. 2001, ApJ, 552, 235 process to locally concentrate the magnetic ﬁeld. Bans, A., & Ko¨nigl, A. 2012, ApJ, 758, 100 The inﬂuence of the traditional dead-zone picture for proto- Baruteau, C., Fromang, S., Nelson, R. P., & Masset, F. 2011, A&A, 533, A84 planetary disks during various stages in planet formation has Beckwith, K., Armitage, P. J., & Simon, J. B. 2011, MNRAS, 416, 361 Bethell, T. J., & Bergin, E. A. 2011, ApJ, 739, 78 been examined in some detail. For example, Gressel, Nelson, Blaes, O. M., & Balbus, S. A. 1994, ApJ, 421, 163 & Turner (2011, 2012) showed that the stirring of planetesi- Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883 mals by the propagation of waves from the active region into Boyd, J. P. 2000, Chebyshev and Fourier Spectral Methods, 2nd edn. (Dover, (cid:46) the dead zone can stir up small planetesimals ( 10 km) and New York) cause them to undergo collisional destruction, or prevent their Chen, Q., Otto, A., & Lee, L. C. 1997, J. Geophys. Res., 102, 151 Choi, E., Kim, J., & Wiita, P. J. 2009, ApJS, 181, 413 runaway growth. In recent work (Nelson et al. 2014, in prep.), Cleeves, L. I., Adams, F. C., & Bergin, E. A. 2013, ApJ, 772, 5 it has furthermore been shown that low mass planets orbiting Davidson, J. A., Li, Z.-Y., Hull, C. L. H., et al. 2014, The Astrophysical in disk models with traditional dead zones are likely to un- Journal, 797, 74 dergo very rapid inward migration because the weak stresses Dullemond, C. P., & Dominik, C. 2005, A&A, 434, 971 there are unable to prevent saturation of the corotation torque. Dzyurkevich, N., Flock, M., Turner, N. J., Klahr, H., & Henning, T. 2010, A&A, 515, A70 In principle, the new emerging picture of PPDs – that is, as- Ercolano, B., & Glassgold, A. E. 2013, MNRAS, 436, 3446 suming that Maxwell stresses operate near the midplane be- Evans, C. R., & Hawley, J. F. 1988, ApJ, 332, 659 cause of the Hall effect – can ameliorate these problems by Faure, J., Fromang, S., & Latter, H. 2014, A&A, 564, A22 removing the turbulent stirring, and by providing a signiﬁcant Fleming, T., & Stone, J. 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This created a turbulent and chaotic gas ﬂow into the —. 2012, MNRAS, 422, 1140

text: ther

text: modify

text: the

text: ﬂow

text: by

text: allowing

text: a

text: signiﬁcant

text: Maxwell

text: stress

text: planet

text: Hill

text: sphere,

text: and

text: the

text: enhanced

text: magnetic

text: coupling

text: also

text: to

text: develop

text: near

text: the

text: disk

text: midplane

text: through

text: the

text: combined

text: ac-

text: led

text: to

text: the

text: launching

text: of

text: a

text: magnetocentrifugally

text: accelerated

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text: tion

text: of

text: ﬁeld

text: winding

text: and

text: Hall

text: drift

text: (Kunz

text: Kunz

text: Lesur

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text: the

text: circumplanetary

text: disk.

text: It

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text: unclear

text: whether

text: Lesur

text: et

text: al.

text: Bai

text: but

text: does

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text: to

text: tur-

text: or

text: not

text: these

text: effects

text: will

text: arise

text: when

text: ambipolar

text: diffusion

text: is

text: in-

text: bulence.

text: We

text: are

text: then

text: left

text: with

text: a

text: picture

text: of

text: PPDs

text: in

text: which

text: the

text: cluded,

text: given

text: the

text: low

text: density

text: of

text: the

text: gap

text: region.

text: innermost

text: regions

text: sustain

text: vigorous

text: MRI-induced

text: turbulence

text: because

text: of

text: collisional

text: ionization

text: of

text: alkali

text: metals,

text: the

text: interme-

text: diate

text: regions

text: between

text: au

text: are

text: laminar

text: with

text: angular

text: We

text: thank

text: Martin

text: Pessah,

text: Anna

text: Tenerani

text: and

text: Marco

text: Velli

text: momentum

text: transport

text: occurring

text: through

text: a

text: disk

text: wind

text: and

text: per-

text: for

text: useful

text: advice

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text: plasma

text: instabilities,

text: and

text: the

text: referee

text: for

text: haps

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text: Maxwell

text: stresses

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text: the

text: midplane

text: (an

text: effect

text: a

text: helpful

text: report.

text: The

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text: carried

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text: the

text: that

text: only

text: arises

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text: Ω

text: B

text: and

text: when

text: including

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text: Hall

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text: Propulsion

text: Laboratory,

text: California

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text: of

text: Technology,

text: term),

text: and

text: an

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text: where

text: the

text: Hall

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text: and

text: Ohmic

text: under

text: a

text: contract

text: with

text: the

text: National

text: Aeronautics

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text: sub-dominant

text: and

text: weak

text: turbulence

text: arises

text: (Si-

text: Administration

text: and

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text: the

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text: the

text: NASA

text: Origins

text: mon

text: et

text: al.

text: Bai

text: of

text: Solar

text: Systems

text: program

text: via

text: grant

text: The

text: The

text: picture

text: painted

text: above

text: has

text: important

text: implications

text: for

text: research

text: leading

text: to

text: these

text: results

text: has

text: received

text: funding

text: from

text: dust

text: evolution

text: and

text: planet

text: formation.

text: In

text: principle,

text: dust

text: coagu-

text: the

text: Danish

text: Council

text: for

text: Independent

text: Research

text: (DFF)

text: and

text: lation

text: and

text: settling

text: in

text: the

text: laminar

text: region

text: should

text: occur

text: rapidly,

text: Marie

text: Curie

text: Actions

text: COFUND

text: under

text: the

text: grant-ID:

text: and

text: in

text: the

text: absence

text: of

text: a

text: source

text: of

text: stirring

text: the

text: loss

text: of

text: small

text: DFF

text: and

text: from

text: the

text: People

text: Programme

text: (Marie

text: grains

text: should

text: affect

text: the

text: spectral

text: energy

text: distribution

text: (SED).

text: Curie

text: Actions)

text: of

text: the

text: European

text: Union’s

text: Seventh

text: Framework

text: This

text: was

text: examined

text: by

text: Dullemond

text: Dominik

text: who

text: Programme

text: under

text: REA

text: grant

text: agreement

text: showed

text: that

text: this

text: rapid

text: evolution

text: of

text: the

text: SED

text: is

text: not

text: in

text: agree-

text: This

text: work

text: used

text: the

text: NIRVANA-III

text: code

text: developed

text: ment

text: with

text: observations,

text: and

text: for

text: many

text: years

text: this

text: important

text: by

text: Udo

text: Ziegler

text: at

text: the

text: Leibniz

text: Institute

text: for

text: Astrophysics

text: result

text: has

text: provided

text: indirect

text: evidence

text: for

text: the

text: existence

text: of

text: tur-

text: (AIP).

text: We

text: acknowledge

text: that

text: the

text: results

text: of

text: this

text: research

text: have

text: bulent

text: stirring

text: in

text: the

text: planet

text: forming

text: regions

text: of

text: protoplanetary

text: been

text: partly

text: achieved

text: using

text: the

text: project

text: RI-

text: disks.

text: In

text: the

text: absence

text: of

text: a

text: magnetic

text: origin

text: for

text: local

text: turbulence

text: resource

text: Fionn

text: based

text: in

text: Ireland

text: at

text: the

text: Irish

text: Centre

text: this

text: suggests

text: that

text: either

text: another

text: source

text: of

text: turbulence

text: exists,

text: for

text: High-End

text: Computing

text: (ICHEC).

text: Computations

text: were

text: also

text: such

text: as

text: the

text: vertical

text: shear

text: instability

text: (e.g.

text: Nelson,

text: Gressel,

text: performed

text: on

text: the

text: node

text: at

text: the

text: Danish

text: Center

text: for

text: Su-

text: Umurhan

text: or

text: that

text: small

text: grains

text: are

text: continuously

text: deliv-

text: percomputing

text: (DCSC),

text: and

text: on

text: the

text: U.K.

text: STFC-funded

text: DiRAC

text: ered

text: to

text: the

text: laminar

text: region

text: from

text: the

text: outer

text: and/or

text: inner

text: turbu-

text: facility.

text: lent

text: regions.

text: The

text: results

text: in

text: this

text: paper

text: are

text: consistent

text: with

text: the

text: remanent

text: REFERENCES

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text: ﬁeld

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text: D.

text: J.,

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text: disks

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text: various

text: stages

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text: has

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text: K.,

text: Armitage,

text: P.

text: J.,

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text: MNRAS,

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text: M.,

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text: planetesi-

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text: stir

text: up

text: small

text: planetesimals

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text: and

text: New

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text: cause

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text: undergo

text: collisional

text: destruction,

text: or

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text: L.

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text: Ercolano,

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text: E.

text: MNRAS,

text: suming

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text: Maxwell

text: stresses

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text: midplane

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text: R.,

text: Hawley,

text: J.

text: F.

text: ApJ,

text: cause

text: of

text: the

text: Hall

text: effect

text: can

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text: problems

text: by

text: Faure,

text: J.,

text: Fromang,

text: S.,

text: Latter,

text: H.

text: A&A,

text: removing

text: the

text: turbulent

text: stirring,

text: and

text: by

text: providing

text: a

text: signiﬁcant

text: Fleming,

text: T.,

text: Stone,

text: J.

text: M.

text: ApJ,

text: stress

text: that

text: can

text: prevent

text: corotation

text: torque

text: saturation.

text: Full

text: sim-

text: Flock,

text: M.,

text: Dzyurkevich,

text: N.,

text: Klahr,

text: H.,

text: Turner,

text: N.,

text: Henning,

text: T.

text: ApJ,

text: ulations

text: are

text: clearly

text: required

text: to

text: examine

text: whether

text: or

text: not

text: the

text: Fromang,

text: S.,

text: Latter,

text: H.,

text: Lesur,

text: G.,

text: Ogilvie,

text: G.

text: I.

text: A&A,

text: latter

text: effect

text: can

text: be

text: realized

text: during

text: full

text: nonlinear

text: evolution.

text: Fu,

text: R.

text: S.,

text: Weiss,

text: B.

text: P.,

text: Lima,

text: E.

text: A.,

text: et

text: al.

text: Science,

text: Finally,

text: Gressel

text: et

text: al.

text: examined

text: gap

text: formation

text: and

text: gas

text: Gammie,

text: C.

text: F.

text: ApJ,

text: accretion

text: onto

text: a

text: giant

text: planet

text: within

text: a

text: disk

text: with

text: a

text: traditional

text: Gardiner,

text: T.

text: A.,

text: Stone,

text: J.

text: M.

text: JCoPh,

text: Garmire,

text: G.,

text: Feigelson,

text: E.

text: D.,

text: Broos,

text: P.,

text: et

text: al.

text: AJ,

text: dead

text: zone,

text: and

text: observed

text: that

text: the

text: gas

text: in

text: the

text: gap

text: region

text: became

text: Girart,

text: J.

text: M.,

text: Rao,

text: R.,

text: Marrone,

text: D.

text: P.

text: Science,

text: enlivened

text: because

text: of

text: enhanced

text: penetration

text: of

text: X-rays

text: and

text: cos-

text: Gressel,

text: O.,

text: Nelson,

text: R.

text: P.,

text: Turner,

text: N.

text: J.

text: MNRAS,

text: mic

text: rays.

text: This

text: created

text: a

text: turbulent

text: and

text: chaotic

text: gas

text: ﬂow

text: into

text: the

text: MNRAS,


text: 2 QCALT˙CCALT printed on May 27, 2020 The presence of a calorimeter in the quadrupoles region is needed to intercept photons coming from K decays and lost in the beam pipe. In- L → creased eﬃciency for these photons implies a strong reduction of the K0 → L 3π0 background in CP violating events such as K π0π0. Moreover, high L granularity in the calorimeter will help on reducing accidental contribution from machine background [2]. For the KLOE-2 upgrade [3, 4, 5] we have designed and built two new calorimeters, named QCALT [6]. The detector requirements can be sum- marised as: a realization of a high granularity calorimeters with suﬃcient X , fast timing response with a time resolution better than < 1 ns plus 0 some boundary conditions arising from the location of the existing detector: maximum length below 1 meter, maximum height 5 cm and photosensors working in B ﬁeld. 1.1. QCALT design and components QCALT consists of two dodecagonal sampling calorimeters covering the quadrupoles region close to the IP. Each module, 1/12 of one calorimeter, is composed of16 towers with 5 layers of 3.5 mm absorber and 5 scintillator tiles 5 mm thick (See Fig.1). The sampling absorber is made with a 90% tungsten and 10% copper alloy. Scintillator tiles are made using EJ-200 plastic scintillator produced by Eljen and painted using BC-620 refractive paint. Light from scintillator is routed outside via multi-cladding BCF-92 1mm round ﬁbers inserted in a circular groove ensuring a fast emission time (5 ns/pe), long attenuation length and a large light yield with respect to standard single cladding ﬁbers. At one side of the calorimeter, ﬁbers are collected and glued into a plastic holder polished with 0.05 mm planarity and coupled to an aluminum dioxide PCB where 80 SMD SiPM are bounded. Each tile is independently coupled to a photodetector ensuring 0.05 mm clearance in each direction. 1.2. QCALT photosensors and FEE In the QCALT we use a custom SMD circular SiPM produced by FBK of 1.2 mm diameter, 25 µm pixel size and PDE (photon detection eﬃciency) peaked at 480 nm. The 80 photodetectors for each module are bonded on an Alluminum PCB and covered with optical resin. The bias voltage required is between 30 and 35 V. Each PCB is then connected to 4 boards, 20 channels/each, including × 20 trans-impedance preampliﬁers, HV regulators (0.1% precision, 0.01% stability) and single threshold discriminators with diﬀerential output. Each module requires a power consumption of 2 W. Boards are inserted in a brass Faraday cage with air cooling. Calorimeter reconstruction is based only on timing using multi-hit TDC with 0.5 ns resolution. 1.3. QCALT commissioning

label: paragraph

text: QCALT˙CCALT

text: printed

text: on

text: May

text: The

text: presence

text: of

text: a

text: calorimeter

text: in

text: the

text: quadrupoles

text: region

text: is

text: needed

text: to

text: intercept

text: photons

text: coming

text: from

text: K

text: decays

text: and

text: lost

text: in

text: the

text: beam

text: pipe.

text: In-

text: L

text: creased

text: eﬃciency

text: for

text: these

text: photons

text: implies

text: a

text: strong

text: reduction

text: of

text: the

text: L

text: background

text: in

text: CP

text: violating

text: events

text: such

text: as

text: K

text: Moreover,

text: high

text: L

text: granularity

text: in

text: the

text: calorimeter

text: will

text: help

text: on

text: reducing

text: accidental

text: contribution

text: from

text: machine

text: background

text: For

text: the

text: upgrade

text: we

text: have

text: designed

text: and

text: built

text: two

text: new

text: calorimeters,

text: named

text: QCALT

text: The

text: detector

text: requirements

text: can

text: be

text: sum-

text: marised

text: as:

text: a

text: realization

text: of

text: a

text: high

text: granularity

text: calorimeters

text: with

text: suﬃcient

text: X

text: fast

text: timing

text: response

text: with

text: a

text: time

text: resolution

text: better

text: than

text: ns

text: plus

text: some

text: boundary

text: conditions

text: arising

text: from

text: the

text: location

text: of

text: the

text: existing

text: detector:

text: maximum

text: length

text: below

text: meter,

text: maximum

text: height

text: cm

text: and

text: photosensors

text: working

text: in

text: B

text: ﬁeld.

text: QCALT

text: design

text: and

text: components

text: QCALT

text: consists

text: of

text: two

text: dodecagonal

text: sampling

text: calorimeters

text: covering

text: the

text: quadrupoles

text: region

text: close

text: to

text: the

text: IP.

text: Each

text: module,

text: of

text: one

text: calorimeter,

text: is

text: composed

text: towers

text: with

text: layers

text: of

text: mm

text: absorber

text: and

text: scintillator

text: tiles

text: mm

text: thick

text: (See

text: The

text: sampling

text: absorber

text: is

text: made

text: with

text: a

text: tungsten

text: and

text: copper

text: alloy.

text: Scintillator

text: tiles

text: are

text: made

text: using

text: plastic

text: scintillator

text: produced

text: by

text: Eljen

text: and

text: painted

text: using

text: refractive

text: paint.

text: Light

text: from

text: scintillator

text: is

text: routed

text: outside

text: via

text: multi-cladding

text: round

text: ﬁbers

text: inserted

text: in

text: a

text: circular

text: groove

text: ensuring

text: a

text: fast

text: emission

text: time

text: ns/pe),

text: long

text: attenuation

text: length

text: and

text: a

text: large

text: light

text: yield

text: with

text: respect

text: to

text: standard

text: single

text: cladding

text: ﬁbers.

text: At

text: one

text: side

text: of

text: the

text: calorimeter,

text: ﬁbers

text: are

text: collected

text: and

text: glued

text: into

text: a

text: plastic

text: holder

text: polished

text: with

text: mm

text: planarity

text: and

text: coupled

text: to

text: an

text: aluminum

text: dioxide

text: PCB

text: where

text: SMD

text: SiPM

text: are

text: bounded.

text: Each

text: tile

text: is

text: independently

text: coupled

text: to

text: a

text: photodetector

text: ensuring

text: mm

text: clearance

text: in

text: each

text: direction.

text: QCALT

text: photosensors

text: and

text: FEE

text: In

text: the

text: QCALT

text: we

text: use

text: a

text: custom

text: SMD

text: circular

text: SiPM

text: produced

text: by

text: FBK

text: of

text: mm

text: diameter,

text: µm

text: pixel

text: size

text: and

text: PDE

text: (photon

text: detection

text: eﬃciency)

text: peaked

text: at

text: nm.

text: The

text: photodetectors

text: for

text: each

text: module

text: are

text: bonded

text: on

text: an

text: Alluminum

text: PCB

text: and

text: covered

text: with

text: optical

text: resin.

text: The

text: bias

text: voltage

text: required

text: is

text: between

text: and

text: V.

text: Each

text: PCB

text: is

text: then

text: connected

text: to

text: boards,

text: channels/each,

text: including

text: trans-impedance

text: preampliﬁers,

text: HV

text: regulators

text: precision,

text: stability)

text: and

text: single

text: threshold

text: discriminators

text: with

text: diﬀerential

text: output.

text: Each

text: module

text: requires

text: a

text: power

text: consumption

text: of

text: W.

text: Boards

text: are

text: inserted

text: in

text: a

text: brass

text: Faraday

text: cage

text: with

text: air

text: cooling.

text: Calorimeter

text: reconstruction

text: is

text: based

text: only

text: on

text: timing

text: using

text: multi-hit

text: TDC

text: with

text: ns

text: resolution.

text: QCALT

text: commissioning


text: QCALT˙CCALT printed on May 27, 2020 3 Fig. 1. CAD drawings of the QCALT calorimeter. Details of the tile ﬁber arrange- ment are shown as well s the FEE readout boxes. The QCALT calorimeters have been installed on May 2013 while the FEE boards have been tested during each installation steps. At the end of the beam pipe insertion, we had less than 10 dead channels over about 1800 readout channels. During ﬁrst beam insertion, we have experienced heating problems due to insuﬃcient cooling of the electronics. This problem has been solved in October 2013 with the installation of a dedicated air compressed line for both IT and QCALT cooling which decreases the SiPM temperature from ##LTFigure## ##LTFigure##

text: QCALT˙CCALT

text: printed

text: on

text: May

text: Fig.

text: CAD

label: caption

text: drawings

text: of

text: the

text: QCALT

text: calorimeter.

text: Details

text: of

text: the

text: tile

text: ﬁber

text: arrange-

text: ment

text: are

text: shown

text: as

text: well

text: s

text: the

text: FEE

text: readout

text: boxes.

text: The

text: QCALT

text: calorimeters

text: have

text: been

text: installed

text: on

text: May

text: while

text: the

text: FEE

text: boards

text: have

text: been

text: tested

text: during

text: each

text: installation

text: steps.

text: At

text: the

text: end

text: of

text: the

text: beam

text: pipe

text: insertion,

text: we

text: had

text: less

text: than

text: dead

text: channels

text: over

text: about

text: readout

text: channels.

text: During

text: ﬁrst

text: beam

text: insertion,

text: we

text: have

text: experienced

text: heating

text: problems

text: due

text: to

text: insuﬃcient

text: cooling

text: of

text: the

text: electronics.

text: This

text: problem

text: has

text: been

text: solved

text: in

text: October

text: with

text: the

text: installation

text: of

text: a

text: dedicated

text: air

text: compressed

text: line

text: for

text: both

text: IT

text: and

text: QCALT

text: cooling

text: which

text: decreases

text: the

text: SiPM

text: temperature

text: from

text: ##LTFigure##

text: ##LTFigure##


text: 2 FIG. 1: Explosion in a fully chaotic billiard. (a) Cardioid billiard, whose boundary in polar coordinates is r(φ) = 1 + cos(φ) ∈ − with φ [ π, π] [9]. The gain region (gray, marked by g) is a disc of radius 0.5 in the middle of the billiard. (b) Collision time distribution τ((cid:126)x) in the cardioid billiard (velocity modulus is unity). Birkhoﬀ coordinates (cid:126)x = (s, p = sin θ) are used, where s is the arc length along the boundary and θ is the collision angle. (c) Reﬂection coeﬃcient distribution: R((cid:126)x) = egτg with g = 1 and τ given by the length of the intersection of the ray with the gain region. (d) Time-independent density ρ in the phase g ≈ c space. (e) Time-dependence of the intensity integrated over the phase space. The explosion rate κ 0.215 is the slope of the curve (note the log-scale). Inset shows the non-exponential behavior for short times. We are interested in the density ρ((cid:126)x, t) (i.e., the col- times τ, and stretching of phase space volume D . The f lective intensity of an ensemble of trajectories) at time t rate κ follows from the constraint that the compensated in (cid:126)x. Here we consider the class of (ergodic and chaotic) intensity neither increases nor decreases for large n [3]. maps f((cid:126)x ) for which we show that for any smooth initial Comparing with our previous results [4, 6], we see that n ρ((cid:126)x, t = 0) one observes for long times (3) is an extension to cases with R > 1 of the Frobenius- Perron operator considered previously only in systems with absorption (R ≤ 1 for any (cid:126)x). The explosion rate κ ρ((cid:126)x, t) ∼ ρc((cid:126)x)eκt, (2) plays the role of a negative escape rate and the attracting limit distribution ρ is the conditionally invariant den- c sity [6] (also known as the steady probability distribu- where κ is the temporal rate of the total energy change tion in the optics community [12, 13]). By integrating, (cid:48) (an explosion rate for κ > 0) and is independent of for n → ∞, (3) over all (cid:126)x , the left hand side is unity due ρ((cid:126)x, t = 0). to normalization of ρ ((cid:126)x), while the right hand side is an n We can expect that ρc((cid:126)x) of (2) is the attracting den- average tak−en with respect to ρc. This yields (see also sity of an iteration scheme for ρ((cid:126)x) o−f the extended Ref. [4]) (cid:104)e κτR(cid:105)c = 1. For chaotic explosions−this means map (1). With a compensation factor e κτ((cid:126)x) per itera- that typically R > 1, but its reduced value e κτR taken tion, this scheme evolves a density ρn((cid:126)x) at discrete time with the proper explosion rate averages out to unity if n into ρn+1((cid:126)x) at the next intersection with the Poincar´e the average is taken with the ρc associated to κ. surface of section as (cid:88) III. GENERAL FEATURES It is remarkable that the formalism of transient (cid:48) − R((cid:126)x)ρ ((cid:126)x) ρn+1((cid:126)x ) = (cid:126)x∈f−1((cid:126)x(cid:48)) e κτ((cid:126)x) | Df((cid:126)xn) | , (3) cdheascorsib[2e,c3h]aoctainc ebxeplaopspiolinesd. Bweitlhowslwigehtusme othdiisﬁcfoartmioanlsistmo to explore the most interesting eﬀects of gain (R > 1). where D represents the Jacobian of the Poincar´e map A. Fractality f f. In the special case of invertible area-preserving dy- namics (as in the billiard of Fig. 1), D ((cid:126)x) = 1 and there We ﬁrst derive general relations between the fractality f is no sum in (3) (map f has a single preimage). of ρ – the eigenfunctions of Eq. (3) – and the distribu- c In any extended map, there exists one κ – the one ap- tions (τ((cid:126)x), R((cid:126)x)) characterizing the extended map f . ext pearing in Eq. (2) – for which ρ ((cid:126)x) arises as the limiting Let us consider the map f in (1) to be invertible and two- c (n → ∞) distribution of ρ ((cid:126)x) iterated by scheme (3). dimensional. We are interested in the dynamics within a n The right-hand-side of Eq. (3), with the proper κ, is an region of interest Γ only, which has a size of order unity operator with largest eigenvalue unity and ρ as the cor- in appropriately chosen units. Considering the n-th im- c responding eigenfunction. The second-largest eigenvalue age and preimage of Γ taken with the map f, we ﬁnd for controls the convergence of a smooth initial density ρ n (cid:29) 1 a set of narrow “columns” in the unstable direc- 0 to ρ . In agreement with the physical picture, we see tion and narrow “strips” along the stable directions, as c that for extended maps f there are three diﬀerent fac- illustrated in Fig. 2. They are good approximants of the ext tors contributing to the density ρ: reﬂectivity R, return unstable and stable foliation of the chaotic set (a chaotic ##LTFigure## ##LTFigure## ##LTLine##

text: FIG.

text: Explosion

text: in

text: a

text: fully

text: chaotic

text: billiard.

text: (a)

text: Cardioid

text: billiard,

text: whose

text: boundary

text: in

text: polar

text: coordinates

text: is

text: r(φ)

text: cos(φ)

text: with

text: φ

text: π,

text: π]

text: The

text: gain

text: region

text: (gray,

text: marked

text: by

text: g)

text: is

text: a

text: disc

text: of

text: radius

text: in

text: the

text: middle

text: of

text: the

text: billiard.

text: (b)

text: Collision

text: time

text: distribution

text: in

text: the

text: cardioid

text: billiard

text: (velocity

text: modulus

text: is

text: unity).

text: Birkhoﬀ

text: coordinates

text: (s,

text: p

text: sin

text: θ)

text: are

text: used,

text: where

text: s

text: is

text: the

text: arc

text: length

text: along

text: the

text: boundary

text: and

text: θ

text: is

text: the

text: collision

text: angle.

text: (c)

text: Reﬂection

text: coeﬃcient

text: distribution:

text: egτg

text: with

text: g

text: and

text: τ

text: given

text: by

text: the

text: length

text: of

text: the

text: intersection

text: of

text: the

text: ray

text: with

text: the

text: gain

text: region.

text: (d)

text: Time-independent

text: density

text: ρ

text: in

text: the

text: phase

text: g

text: c

text: space.

text: (e)

text: Time-dependence

text: of

text: the

text: intensity

text: integrated

text: over

text: the

text: phase

text: space.

text: The

text: explosion

text: rate

text: κ

text: is

text: the

text: slope

text: of

text: the

text: curve

text: (note

text: the

text: log-scale).

text: Inset

text: shows

text: the

text: non-exponential

text: behavior

text: for

text: short

text: times.

text: We

text: are

text: interested

text: in

text: the

text: density

text: t)

text: (i.e.,

text: the

text: col-

text: times

text: τ,

text: and

text: stretching

text: of

text: phase

text: space

text: volume

text: D

text: The

text: f

text: lective

text: intensity

text: of

text: an

text: ensemble

text: of

text: trajectories)

text: at

text: time

text: t

text: rate

text: κ

text: follows

text: from

text: the

text: constraint

text: that

text: the

text: compensated

text: in

text: Here

text: we

text: consider

text: the

text: class

text: of

text: (ergodic

text: and

text: chaotic)

text: intensity

text: neither

text: increases

text: nor

text: decreases

text: for

text: large

text: n

text: maps

text: for

text: which

text: we

text: show

text: that

text: for

text: any

text: smooth

text: initial

text: Comparing

text: with

text: our

text: previous

text: results

text: we

text: see

text: that

text: n

text: t

text: one

text: observes

text: for

text: long

text: times

text: is

text: an

text: extension

text: to

text: cases

text: with

text: R

text: of

text: the

text: Frobenius-

text: Perron

text: operator

text: considered

text: previously

text: only

text: in

text: systems

text: with

text: absorption

text: (R

text: for

text: any

text: The

text: explosion

text: rate

text: κ

text: t)

text: plays

text: the

text: role

text: of

text: a

text: negative

text: escape

text: rate

text: and

text: the

text: attracting

text: limit

text: distribution

text: ρ

text: is

text: the

text: conditionally

text: invariant

text: den-

text: c

text: sity

text: (also

text: known

text: as

text: the

text: steady

text: probability

text: distribu-

text: where

text: κ

text: is

text: the

text: temporal

text: rate

text: of

text: the

text: total

text: energy

text: change

text: tion

text: in

text: the

text: optics

text: community

text: By

text: integrating,

text: (an

text: explosion

text: rate

text: for

text: κ

text: and

text: is

text: independent

text: of

text: for

text: n

text: over

text: all

text: the

text: left

text: hand

text: side

text: is

text: unity

text: due

text: t

text: to

text: normalization

text: of

text: ρ

text: while

text: the

text: right

text: hand

text: side

text: is

text: an

text: n

text: We

text: can

text: expect

text: that

text: of

text: is

text: the

text: attracting

text: den-

text: average

text: tak−en

text: with

text: respect

text: to

text: ρc.

text: This

text: yields

text: (see

text: also

text: sity

text: of

text: an

text: iteration

text: scheme

text: for

text: o−f

text: the

text: extended

text: Ref.

text: For

text: chaotic

text: explosions−this

text: means

text: map

text: With

text: a

text: compensation

text: factor

text: e

text: per

text: itera-

text: that

text: typically

text: R

text: but

text: its

text: reduced

text: value

text: e

text: κτR

text: taken

text: tion,

text: this

text: scheme

text: evolves

text: a

text: density

text: at

text: discrete

text: time

text: with

text: the

text: proper

text: explosion

text: rate

text: averages

text: out

text: to

text: unity

text: if

text: n

text: into

text: at

text: the

text: next

text: intersection

text: with

text: the

text: Poincar´e

text: the

text: average

text: is

text: taken

text: with

text: the

text: ρc

text: associated

text: to

text: κ.

text: surface

text: of

text: section

text: as

text: III.

text: GENERAL

text: FEATURES

text: It

text: is

text: remarkable

text: that

text: the

text: formalism

text: of

text: transient

text: e

text: ebxeplaopspiolinesd.

text: Bweitlhowslwigehtusme

text: othdiisﬁcfoartmioanlsistmo

text: to

text: explore

text: the

text: most

text: interesting

text: eﬀects

text: of

text: gain

text: (R

text: where

text: D

text: represents

text: the

text: Jacobian

text: of

text: the

text: Poincar´e

text: map

text: A.

text: Fractality

text: f

text: f.

text: In

text: the

text: special

text: case

text: of

text: invertible

text: area-preserving

text: dy-

text: namics

text: (as

text: in

text: the

text: billiard

text: of

text: Fig.

text: D

text: and

text: there

text: We

text: ﬁrst

text: derive

text: general

text: relations

text: between

text: the

text: fractality

text: f

text: is

text: no

text: sum

text: in

text: (map

text: f

text: has

text: a

text: single

text: preimage).

text: of

text: ρ

text: the

text: eigenfunctions

text: of

text: Eq.

text: and

text: the

text: distribu-

text: c

text: In

text: any

text: extended

text: map,

text: there

text: exists

text: one

text: κ

text: the

text: one

text: ap-

text: tions

text: characterizing

text: the

text: extended

text: map

text: f

text: ext

text: pearing

text: in

text: Eq.

text: for

text: which

text: ρ

text: arises

text: as

text: the

text: limiting

text: Let

text: us

text: consider

text: the

text: map

text: f

text: in

text: to

text: be

text: invertible

text: and

text: two-

text: c

text: (n

text: distribution

text: of

text: ρ

text: iterated

text: by

text: scheme

text: dimensional.

text: We

text: are

text: interested

text: in

text: the

text: dynamics

text: within

text: a

text: n

text: The

text: right-hand-side

text: of

text: Eq.

text: with

text: the

text: proper

text: κ,

text: is

text: an

text: region

text: of

text: interest

text: Γ

text: only,

text: which

text: has

text: a

text: size

text: of

text: order

text: unity

text: operator

text: with

text: largest

text: eigenvalue

text: unity

text: and

text: ρ

text: as

text: the

text: cor-

text: in

text: appropriately

text: chosen

text: units.

text: Considering

text: the

text: n-th

text: im-

text: c

text: responding

text: eigenfunction.

text: The

text: second-largest

text: eigenvalue

text: age

text: and

text: preimage

text: of

text: Γ

text: taken

text: with

text: the

text: map

text: f,

text: we

text: ﬁnd

text: for

text: controls

text: the

text: convergence

text: of

text: a

text: smooth

text: initial

text: density

text: ρ

text: n

text: a

text: set

text: of

text: narrow

text: “columns”

text: in

text: the

text: unstable

text: direc-

text: to

text: ρ

text: In

text: agreement

text: with

text: the

text: physical

text: picture,

text: we

text: see

text: tion

text: and

text: narrow

text: “strips”

text: along

text: the

text: stable

text: directions,

text: as

text: c

text: that

text: for

text: extended

text: maps

text: f

text: there

text: are

text: three

text: diﬀerent

text: fac-

text: illustrated

text: in

text: Fig.

text: They

text: are

text: good

text: approximants

text: of

text: the

text: ext

text: tors

text: contributing

text: to

text: the

text: density

text: ρ:

text: reﬂectivity

text: R,

text: return

text: unstable

text: and

text: stable

text: foliation

text: of

text: the

text: chaotic

text: set

text: (a

text: chaotic

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 6 rather R + R = 2. 1 2 As a ﬁnal illustration of the signiﬁcance of our general results, we perform numerical simulation of rays in an op- tical cavity with gain and absorption. Imagine that the cardioid billiard with localized gain introduced in Fig. 1 is composed of a dielectric material (with refraction in- dex η). Figure 5 reports results for transverse-electric polarized light in such a conﬁguration. Whether explo- sion occurs, depends on the gain parameter g (κ depends smoothly on g, see panel a). At the critical value g = g∗ gain and loss cancel each other (κ = 0) and an energetic steady state sets in. The density ρ is fractal for any g, ∗ c also at g = g (see lower inset of panel a). The trans- mitted rays can be detected outside the billiard as an emission in a given direction (represented by the far-ﬁeld angle Φ, see upper inset in Fig. 5a). We obtain that the (observable) far-ﬁeld emission (as a function of angle Φ) is modiﬁed by the gain (compare panels b-d). We thus conclude that (non-uniform[? ]) gain has to be included in ray simulations. V. CONCLUSIONS We considered chaotic systems in which the intensities of trajectories may grow in time (e.g., due to gain or a reﬂection coeﬃcient R > 1). We extended the formal- FIG. 5: Ray emission from an optical cardioid billiard with ism of systems with absorption (R < 1) to show how the gain and transmission. (a) Asymptotic rate κ as a function of theory of (open) chaotic systems can be used in this new the gain strength g. The steady state κ = 0 is achieved at g = ∗ ≈ (cid:82) g 1.0367, the lower inset shows the invariant density ρ for class of systems. For instance, we derived a formula – (cid:29) c this case. (b-d) Long-time (t 1) emission E(Φ) of intensity Eq. (7) – that relates the invariant properties (e.g., frac- J in the far-ﬁeld angle Φ (normalized as E(Φ)dΦ = 1) for tal dimensions and the explosion/escape rates) of systems ∗ (b) g = 0, no gain and decaying intensity: κ < 0; (c) g = g , with gain/absorption, a generalization of important re- steady state, κ = 0; (d) g = 2, explosion, κ > 0. The intensity sults in the theory of open systems (in which R is ei- of rays grows with rate g at the gain region (gray circle, see ther 0 or 1) [2, 3]. Despite the unifying formalism, our also Fig. 1) and is split between reﬂected and transmitted results reveal an intricate relationship between systems rays for small collision angles θ. This leads to a partial optical − | |≡| with gain and with absorption. For instance, the inverse reﬂection R (θ) = [sin(θ θ)/ sin(θ + θ)]2 < 1 for p | o T T of a system with gain is a system with absorption, but – sin θ < 1/η, where we used η = 3 and the transmitted angle in contrast to usual dynamical systems – their invariant θ is given by Snell’s law as η sin θ = sin θ [12, 13]. T T properties are not trivially related to each other. For applications in optical microcavities, our results nates (κ > 0). For α < α = 1/3 absorption dominates show how gain can be incorporated in the usual ray sim- c (κ < 0). At the critical value α = α , κ = 0, which is ulations. Whenever the gain is not uniform in the cavity, c the steady state condition[? ], R ≡ R (α = α ) = 4/3, we ﬁnd that the emission is modiﬁed and can be described 1c 1 c R ≡ R (α = α ) = 2/3, P = 2/3, P = 1/3, the through our formalism of chaotic explosions. These re- 2c 2 c 1c 2c average of ln R is 5/3 ln 2 − ln 3, and thus Eq. (7) yields sults can be tested experimentally by constructing optical again fractality D(2) = 1 − ln R/ ln 2 = 0.919 for any cavities with diﬀerent localized gain regions [7] and com- 1 paring the emission to ray simulations with and without τ , τ . 1 2 gain. Even though the return times τ are irrelevant at the i steady state, the steady state is not achieved at R R = VI. ACKNOWLEDGMENTS 1 2 1. In our example, R (α)R (α) = 8α2 = 1 for α = √ 1 2 1/ 8 = 0.354 > α = 1/3 (explosion). This remains We thank M. Sch¨onwetter for the careful reading of the c valid in maps (τ = 1) for any R , R : introducing R = manuscript. This work has been supported by OTKA i 1 2 1 1/R in Eq. (12) leads to κτ = ln ((R + 1/R )/2) ≥ 0. grant No. NK100296 and by the Alexander von Hum- 2 1 1 The stead-state condition is thus not R R = 1, but boldt Foundation. 1 2 [1] Ott E, Chaos in dynamical systems, Cambrdige Univ. [2] Gaspard P, Chaos, Scattering and Statistical Mechanics, Press, 1993. ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: rather

text: R

text: R

text: As

text: a

text: ﬁnal

text: illustration

text: of

text: the

text: signiﬁcance

text: of

text: our

text: general

text: results,

text: we

text: perform

text: numerical

text: simulation

text: of

text: rays

text: in

text: an

text: op-

text: tical

text: cavity

text: with

text: gain

text: and

text: absorption.

text: Imagine

text: that

text: the

text: cardioid

text: billiard

text: with

text: localized

text: gain

text: introduced

text: in

text: Fig.

text: is

text: composed

text: of

text: a

text: dielectric

text: material

text: (with

text: refraction

text: in-

text: dex

text: η).

text: Figure

text: reports

text: results

text: for

text: transverse-electric

text: polarized

text: light

text: in

text: such

text: a

text: conﬁguration.

text: Whether

text: explo-

text: sion

text: occurs,

text: depends

text: on

text: the

text: gain

text: parameter

text: g

text: (κ

text: depends

text: smoothly

text: on

text: g,

text: see

text: panel

text: a).

text: At

text: the

text: critical

text: value

text: g

text: g∗

text: gain

text: and

text: loss

text: cancel

text: each

text: other

text: (κ

text: and

text: an

text: energetic

text: steady

text: state

text: sets

text: in.

text: The

text: density

text: ρ

text: is

text: fractal

text: for

text: any

text: g,

text: c

text: also

text: at

text: g

text: g

text: (see

text: lower

text: inset

text: of

text: panel

text: a).

text: The

text: trans-

text: mitted

text: rays

text: can

text: be

text: detected

text: outside

text: the

text: billiard

text: as

text: an

text: emission

text: in

text: a

text: given

text: direction

text: (represented

text: by

text: the

text: far-ﬁeld

text: angle

text: Φ,

text: see

text: upper

text: inset

text: in

text: Fig.

text: We

text: obtain

text: that

text: the

text: (observable)

text: far-ﬁeld

text: emission

text: (as

text: a

text: function

text: of

text: angle

text: Φ)

text: is

text: modiﬁed

text: by

text: the

text: gain

text: (compare

text: panels

text: b-d).

text: We

text: thus

text: conclude

text: that

text: (non-uniform[?

text: gain

text: has

text: to

text: be

text: included

text: in

text: ray

text: simulations.

text: V.

text: CONCLUSIONS

text: We

text: considered

text: chaotic

text: systems

text: in

text: which

text: the

text: intensities

text: of

text: trajectories

text: may

text: grow

text: in

text: time

text: (e.g.,

text: due

text: to

text: gain

text: or

text: a

text: reﬂection

text: coeﬃcient

text: R

text: We

text: extended

text: the

text: formal-

text: FIG.

text: Ray

text: emission

text: from

text: an

text: optical

text: cardioid

text: billiard

text: with

text: ism

text: of

text: systems

text: with

text: absorption

text: (R

text: to

text: show

text: how

text: the

text: gain

text: and

text: transmission.

text: (a)

text: Asymptotic

text: rate

text: κ

text: as

text: a

text: function

text: of

text: theory

text: of

text: (open)

text: chaotic

text: systems

text: can

text: be

text: used

text: in

text: this

text: new

text: the

text: gain

text: strength

text: g.

text: The

text: steady

text: state

text: κ

text: is

text: achieved

text: at

text: g

text: g

text: the

text: lower

text: inset

text: shows

text: the

text: invariant

text: density

text: ρ

text: for

text: class

text: of

text: systems.

text: For

text: instance,

text: we

text: derived

text: a

text: formula

text: c

text: this

text: case.

text: (b-d)

text: Long-time

text: (t

text: emission

text: E(Φ)

text: of

text: intensity

text: Eq.

text: that

text: relates

text: the

text: invariant

text: properties

text: (e.g.,

text: frac-

text: J

text: in

text: the

text: far-ﬁeld

text: angle

text: Φ

text: (normalized

text: as

text: E(Φ)dΦ

text: for

text: tal

text: dimensions

text: and

text: the

text: explosion/escape

text: rates)

text: of

text: systems

text: (b)

text: g

text: no

text: gain

text: and

text: decaying

text: intensity:

text: κ

text: (c)

text: g

text: g

text: with

text: gain/absorption,

text: a

text: generalization

text: of

text: important

text: re-

text: steady

text: state,

text: κ

text: (d)

text: g

text: explosion,

text: κ

text: The

text: intensity

text: sults

text: in

text: the

text: theory

text: of

text: open

text: systems

text: (in

text: which

text: R

text: is

text: ei-

text: of

text: rays

text: grows

text: with

text: rate

text: g

text: at

text: the

text: gain

text: region

text: (gray

text: circle,

text: see

text: ther

text: or

text: Despite

text: the

text: unifying

text: formalism,

text: our

text: also

text: Fig.

text: and

text: is

text: split

text: between

text: reﬂected

text: and

text: transmitted

text: results

text: reveal

text: an

text: intricate

text: relationship

text: between

text: systems

text: rays

text: for

text: small

text: collision

text: angles

text: θ.

text: This

text: leads

text: to

text: a

text: partial

text: optical

text: with

text: gain

text: and

text: with

text: absorption.

text: For

text: instance,

text: the

text: inverse

text: reﬂection

text: R

text: (θ)

text: [sin(θ

text: θ)/

text: sin(θ

text: for

text: p

text: o

text: T

text: T

text: of

text: a

text: system

text: with

text: gain

text: is

text: a

text: system

text: with

text: absorption,

text: but

text: sin

text: θ

text: where

text: we

text: used

text: η

text: and

text: the

text: transmitted

text: angle

text: in

text: contrast

text: to

text: usual

text: dynamical

text: systems

text: their

text: invariant

text: θ

text: is

text: given

text: by

text: Snell’s

text: law

text: as

text: η

text: sin

text: θ

text: sin

text: θ

text: T

text: T

text: properties

text: are

text: not

text: trivially

text: related

text: to

text: each

text: other.

text: For

text: applications

text: in

text: optical

text: microcavities,

text: our

text: results

text: nates

text: (κ

text: For

text: α

text: α

text: absorption

text: dominates

text: show

text: how

text: gain

text: can

text: be

text: incorporated

text: in

text: the

text: usual

text: ray

text: sim-

text: c

text: (κ

text: At

text: the

text: critical

text: value

text: α

text: α

text: κ

text: which

text: is

text: ulations.

text: Whenever

text: the

text: gain

text: is

text: not

text: uniform

text: in

text: the

text: cavity,

text: c

text: the

text: steady

text: state

text: condition[?

text: R

text: R

text: (α

text: α

text: we

text: ﬁnd

text: that

text: the

text: emission

text: is

text: modiﬁed

text: and

text: can

text: be

text: described

text: c

text: R

text: R

text: (α

text: α

text: P

text: P

text: the

text: through

text: our

text: formalism

text: of

text: chaotic

text: explosions.

text: These

text: re-

text: c

text: average

text: of

text: ln

text: R

text: is

text: ln

text: ln

text: and

text: thus

text: Eq.

text: yields

text: sults

text: can

text: be

text: tested

text: experimentally

text: by

text: constructing

text: optical

text: again

text: fractality

text: ln

text: R/

text: ln

text: for

text: any

text: cavities

text: with

text: diﬀerent

text: localized

text: gain

text: regions

text: and

text: com-

text: paring

text: the

text: emission

text: to

text: ray

text: simulations

text: with

text: and

text: without

text: τ

text: τ

text: gain.

text: Even

text: though

text: the

text: return

text: times

text: τ

text: are

text: irrelevant

text: at

text: the

text: i

text: steady

text: state,

text: the

text: steady

text: state

text: is

text: not

text: achieved

text: at

text: R

text: R

text: VI.

text: ACKNOWLEDGMENTS

text: In

text: our

text: example,

text: R

text: (α)R

text: (α)

text: for

text: α

text: α

text: (explosion).

text: This

text: remains

text: We

text: thank

text: M.

text: Sch¨onwetter

text: for

text: the

text: careful

text: reading

text: of

text: the

text: c

text: valid

text: in

text: maps

text: (τ

text: for

text: any

text: R

text: R

text: introducing

text: R

text: manuscript.

text: This

text: work

text: has

text: been

text: supported

text: by

text: OTKA

text: i

text: in

text: Eq.

text: leads

text: to

text: κτ

text: ln

text: ((R

text: grant

text: No.

text: and

text: by

text: the

text: Alexander

text: von

text: Hum-

text: The

text: stead-state

text: condition

text: is

text: thus

text: not

text: R

text: R

text: but

text: boldt

text: Foundation.

text: Ott

text: E,

text: Chaos

text: in

text: dynamical

text: systems,

text: Cambrdige

text: Univ.

text: Gaspard

text: P,

text: Chaos,

text: Scattering

text: and

text: Statistical

text: Mechanics,

text: Press,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##