text: Hydrogen-loss rates of protoplanetary cores 5 Figure 1. Hydrogen mass loss rates as a function of orbital distance for rocky protoplanetary cores with 1M⊕ (a), 2M⊕ (b), 3M⊕ (c) and 5M⊕ (d) calculated for the stellar EUV ﬂux 100 times higher compared to the present solar value in various orbit locations between 0.1 - 1 AU. The planetary radii related to assumed hydrogen envelope fractions are given in Table 1. Dash-dotted lines correspond to the hydrogen loss rate L of the hydrodynamic model by neglecting ionization, dissociation and recombination; dash-dotted-dotted-dotted lines correspond to loss rates Ln,i if ionization, dissociation and recombination processes are not neglected; dashed lines correspond to the loss rates Li of ionized hydrogen atoms only; solid lines correspond to the loss rates Ln of neutral hydrogen atoms only; the upper dotted lines correspond to the energy-limited los∗s formula multiplied by a heating eﬃciency η of 15 %, Len according to eq. (1) and the lower dotted lines correspond to the loss rates L according to eq. (3). en limited formula, we show also mass loss rates, L , obtained Figs 1., eq. (3) tends to underestimate the mass loss rates en by this approach, but multiplied with a heating eﬃciency η because of the assumption that the EUV ﬂux is absorbed of 15 %. Depending on the planetary parameters, the orbital close to the planetary radius, whereas it is actually absorbed distance, corresponding EUV ﬂux and eﬀective temperature, at larger altitudes. For the small planets considered here, ∼ − − the hydrogen mass loss rates are between 108 g s 1 at 1 R may be located at 2 3 R , much higher than for more ∼ − EUV 0 AU and 1010 g s 1 at 0.1 AU. One can also see that L massive giant planets (Erkaev et al. 2007; Murray-Clay et n,i yields negligible diﬀerences that are less than a factor of two al. 2009). Therefore, the application of the energy-limited for all test planet loss rates between the hydrodynamic up- formulae given in eqs. (1) and (3) are of limited use for low- per atmosphere model and the energy-limited formula of eq. mass planets. (1) at orbital distances of 1 AU. For closer orbits such as Moreover, the discrepancy between the mass loss rates 0.5 AU or 0.1 AU, depending on planetary parameters, the calculated with the hydrodynamic code and the energy lim- diﬀerences between the energy-limited formula given in eq. ≈ ited escape formula arises because eq. (1) yields the max- (1) and the results obtained for L are factors of 1.5–3.5 ≈ n,i imum EUV-driven mass-loss rate that a planet can have, and 3.0–9.0, respectively. Therefore, ionization, dissocia- even if multiplied by the heating eﬃciency η. The numer- tion and recombination can not be neglected for close orbital ator represents the integrated EUV heating rate provided distances or highly active young host stars. to the planet, i.e. the total stellar EUV ﬂux absorbed at Fig. 1 shows the hydrogen mass loss rates for the four R multiplied by the heating eﬃciency, i.e., the fraction EUV planetary masses as a function of orbital distance. One can of incident energy converted to heating. Since the denomi- see that the energy-limited formula underestimates the mass nator represents the potential energy of the planet, eq. (1) ≈ ≈ loss rates when one assumes R R R (eq. 3) assumes that the total absorbed EUV energy is used to lift EUV 0 pl and overestimates the mass loss rates when one uses the the planetary atmosphere out of the planet’s gravitational more accurate formula of eq. (1). As it is obvious also from well (Lammer et al. 2016). 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text: 10 N. V. Erkaev et al. (cid:16) (cid:17) (cid:16) (cid:17) For computational convenience we introduce dimension- less quantities ∂ n ∂ n vR2 H+ 1 H+ − 2 + 2 = ν n α n n .(A12) ∂t R2 ∂R H2 H2 H2 e H+2 (cid:112) ρ˜ = ρ/ρ , ρ = N m , (A19) 0 0 0 H2 The electron density is determined for quasi-neutrality r = R/R0, U˜ = mHU/(kT0), (A20) conditions V˜ = V/V , V = kT /m , (A21) T0 T0 0 H T˜ = T/T , P˜ = P/(ρ V 2 ), (A22) 0 0 T0 ne = nH+ + nH2+ (A13) X = mHnH/ρ, X+ = mH+nH+/ρ, (A23) Y = m n /ρ, Y + = m n /ρ, (A24) H2 H2 H+ H+ and the total hydrogen number density 2 2 Q˜ = ησ φ R /(m V 3 ), (A25) EUV E·UV− EUV 0 H2 T0 Q˜ = 7.5 10 19N R /(m V 3 ), (A26) n = n + n + n + n . (A14) Lα 0 0 H2 T0 H H+ H2 H+ 2 ν˜ = ν R /V , ν˜ = ν R /V , (A27) H H 0 T0 H2 H2 0 T0 α˜ = α N R /V , α˜ = α N R /V , (A28) α is the recombination rate related to the reac- H H 0 0 T0 H2 H2 0 0 T0 H → × − − tion H+ + e H of 4 10 12(300→/T)0.64 cm3 s 1, αH×2 ν˜Hcol = νHcolN0R0/VT0, ν˜diss = νdissN0R0/VT0, (A29) is −the dissociation ra−te of H+2 +e H + H: αH2=2.3 γ˜H = γHN02R0/VT0, χ˜ = χT0/(ρ0VT30R0). (A30) 10 8(300/T)0.4 cm3 s 1, ν is the thermal dissociation → · dis−s − rate of H H + H: 1.5 10 9 exp( 49000/T), γ is the 2 → · − H rate of reaction H + H H : γ = 8.0 10 33 (300/T)0.6 Subscript “0” denotes lower boundary values. The nor- 2 H (cid:0) (cid:1) (Yelle, 2004). malized equations can be written as follows ν is the hydrogen ionization rate, and ν is the ioniza- H H2 tion rate of molecular hydrogen (Storey and Hummer 1995; (cid:0) (cid:1) ∂ρ˜ ∂ r2ρ˜V˜ Murray-Clay et al. 2009), + = 0, (A31) ∂t r2∂r · − − · − − ∂ρ˜X ∂ r2ρ˜XV˜ ν = 5.9 10 8φ s 1, ν = 3.3 10 8φ s 1, (A15) + = H EUV H2 EUV ∂t r2∂r − − ν˜ Xρ˜ ν˜ T˜1/2ρ˜2X(2X+ + Y +) + H Hcol ν a=nd5ν.H9·c1o0l −is11thTe1/c2oellxispi(o−na1l5i7o8n0i9za/tTio)n. rate (Black, 1981), α˜Hρ˜2X+(2X+ + Y +) + HcolφEUV is(cid:90)the function describing the EUV ﬂux absorp- α˜H2ρ˜2Y +(2X+ + Y +)−+ νdissρ˜2Y(cid:0)(1 + X + (cid:1)X+) tion in the atmosphere γ˜HHρ˜3(1 + X + X+)X2. (A32) 1 π/2+arccos(1/r) ∂ρ˜X+ ∂ r2ρ˜X+V˜ φ = J (r, θ)2π sin(θ)dθ. (A16) + = EUV 4π EUV ∂t r2∂r 0 ν˜ Xρ˜+ ν˜ T˜1/2ρ˜2X(cid:0)(2X+ +(cid:1)Y +) H Hcol Here, JEUV(r, θ) is the function of spherical coordinates −α˜ ρ˜2X+(2X+ + Y +), (A33) that describes the spatial variation of the EUV ﬂux due H to the atmospheric absorption (Erkaev et al. 2015), r cor- ∂ρ˜Y + ∂ r2ρ˜Y +V˜ +(cid:2) (cid:3) = responds to the radial distance from the planetary center ∂t r2∂r − noramalized to R0 . ν˜ ρ˜Y α˜ ρ˜2Y +(2X+ + Y +), (cid:16) (cid:17) H2 H2 In the hydrodynamic equations, the mass density, ρ, and the pressure, P, can then be written as ∂ρ˜V˜ ∂ r2(ρ˜V˜2 + P˜) + = ∂t r2∂r (cid:16) (cid:17) (cid:2) (cid:3) ρ = mH (nH + nH+) + mH2 nH2 + nH+2 , (A17) −ρ˜∂U˜ + 2P˜ , (A34) ∂r r (cid:2) (cid:3) ∂ 1ρV˜2 + E˜ + ρ˜U˜ P = n + n + n + n + n kT, (A18) 2 H H+ H2 H2+ e ∂t (cid:18) (cid:19) where T is the upper atmosphere temperature and k is ∂V˜r2 1ρV˜2 + E˜ + P˜ + ρ˜U˜ the Boltzmann constant, and mH and mH2 are the masses + 2 r2∂r = of the hydrogen atoms and molecules, respectively. − ∂ ∂T˜ For atmospheres that are in long-term radiative equi- Q˜EUV Q˜Lα + r2∂r r2χ˜ ∂r , (A35) librium, the temperature T near the lower boundary of 0 the simulation domain is quite close to the planetary eﬀec- P˜ = ρ˜T˜(1 + X + 3X+ + Y +)/2.0, (A36) ≈ tive and equilibrium temperatures Teﬀ Teq. The hydrody- E˜ = ρ˜V˜2/2 + ρ˜T˜(5 + X + 7X+ + 3Y +)/4.0. (A37) namic model is only applicable as long as enough collisions occur, which is the case if the Knudsen number is < 0.1. We set the upper boundary conditions in the supersonic region The obtained equations (A31-A37) make a self- assuming the radial derivatives of the density, temperature consistent closed system with respect to 6 unknown quanti- and velocity are zero. ties ρ, X, X+, Y +, T, V . The seventh quantity Y (ratio of the (cid:13) c 2015 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##

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text: r

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text: set

text: the

text: upper

text: boundary

text: conditions

text: in

text: the

text: supersonic

text: region

text: The

text: obtained

text: equations

text: make

text: a

text: self-

text: assuming

text: the

text: radial

text: derivatives

text: of

text: the

text: density,

text: temperature

text: consistent

text: closed

text: system

text: with

text: respect

text: to

text: unknown

text: quanti-

text: and

text: velocity

text: are

text: zero.

text: ties

text: ρ,

text: X,

text: X+,

text: Y

text: T,

text: V

text: The

text: seventh

text: quantity

text: Y

text: (ratio

text: of

text: the

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

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text: 3 (a) (b) FIG. 2. Pulse spectral evolution during propagation for two diﬀerent input pulse chirps exempliﬁed with ﬁber 1 ((a): 0fs2; FIG. 3. Peak power of compressed pulse in ﬁber 1 as a func- (b): 305 fs2). N=2.25, 12 fs hyperbolic secant pulses were tion of pre-chirp applied to the 12 fs input pulse. used. The horizontal line indicates the maximum spectral broadening. Figure taken from [17]. erably further along the ﬁber. In addition the extent of the spectral broadening has slightly decreased. at the compression point with the amount critically de- pendent on the compression ratio. As the pulse compresses and broadens, its peak power As we will see in the simulations and experiments, the will rise to a maximum and then decrease. As we see, compression ratio will be greater for a small positive chirp the RR is alomst entirely generated where the pulse com- compared to zero chirp and then reduce as the chirp be- presses, i.e. spectrally broadens, and assumes the maxi- comes larger. mum peak power. Hence it is instructive to ﬁnd out how eﬃcient the pulse compression is as the pulse chirp is IV. SIMULATIONS OF CHIRPED PULSE PROPAGATION varied. Figure 3 shows the maximum, compressed peak power as a function of the input pulse chirp. The peak power assumes a maximum for small positive chirps ex- We perform numerical simulations of pulse propaga- ceeding the power for zero chirp. For larger positive tion under the GNLSE. We use a standard split-step chirps the maximum peak power decreases, indicating Fourier tool [19], which takes into account the full dis- that the pulse is not compressing as much. persion proﬁle as well as nonlinear and Raman eﬀects. For both ﬁbers we carried out the same simulations for We use dispersion proﬁles determined from RR measure- N values up to N = 5 and a pulse lengths from 12 fs to ments for the two ﬁbers used in the experiments [23]. 200 fs. In all cases we found an optimum chirp for pulse Fiber 1 (NL-1.5-590, NKT Photonics, Ltd.) has a zero- compression. In Fig.4 the variation in the value of this dispersion wavelength (ZDW) of 685 nm and ﬁber 2 (NL- optimum chirp with soliton order and pulse length can 1.6-615, NKT Photonics, Ltd.) a ZDW of 697 nm. Both be seen. In Fig.4(a) and in Fig.4(b) a 12 fs pulse length is ﬁbers have anomalous dispersion at longer wavelengths. used in each case and in Fig.4(c) and Fig.4(d) N = 2.25 Propagations of hyperbolic secant pulses with varying is used for each pulse length. The ﬁgure shows that a linear chirp are simulated. positive initial chirp leads to optimal pulse compression. The main factors for the generation of RR are the evo- The obtained compressed peak powers increase between lution of the pulse spectrum and the pulse peak power, 10 and 75 percent. The chirp plays an important role for stimulating and driving RR production, respectively. a fundamental soliton, although the pulse compression is The simulation parameters are matched to the experi- small. For higher energies the pulse peak power increses mental parameters. We used a 12 fs hyperbolic secant less, but due to the strong compression RR is generated. pulse with a power corresponding to a soliton order of The RR amplitude and eﬃciency critically depend on N = 2.25 and a center wavelength of 800 nm in the the pulse peak power and so we expect a systematic and anomalous dispersion region. We also extended the in- strong dependence of RR on pulse chirp. The peak power vestigation to diﬀerent soliton orders and pulse lengths. enhancement applies to a large range of pulse lengths. Detailed results of the ﬁber 1 simulations can be found We also carried out simulations without self-steepening in [17]. Here we brieﬂy review the results and conclusions and Raman scattering. Whilst they had an impact on the relevant to RR generation. Two example spectral evolu- position of compression in the ﬁber they did not signiﬁ- tions for this ﬁber are shown in ﬁgure 2 for an unchirped cantly aﬀect the optimum prechirp. (a) and a chirped pulse (b). For zero input chirp the input V. EXPERIMENT pulse spectrum expands to almost three octaves within the ﬁrst 2 mm of ﬁber and then contracts again rapidly. RR is generated when the spectrum expands and reaches The simulations demonstrate how the input pulse chirp the RR wavelength between 400 and 500 nm. Similar re- can be used to optimise pulse compression. In the experi- sults are found for ﬁber 2. For the large input chirp in ments we investigate the impact of this on the generation Fig. 2 (b) the behavior is similar, however, the point of eﬃciency of RR. We can additionally use the chirp vary- temporal compression and spectral expansion is consid- ing technique described in [17] to qualitatively investigate ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: (b)

text: FIG.

text: Pulse

label: caption

text: spectral

text: evolution

text: during

text: propagation

text: for

text: two

text: diﬀerent

text: input

text: pulse

text: chirps

text: exempliﬁed

text: with

text: ﬁber

text: ((a):

text: FIG.

text: Peak

text: power

text: of

text: compressed

text: pulse

text: in

text: ﬁber

text: as

text: a

text: func-

text: (b):

text: fs

text: hyperbolic

text: secant

text: pulses

text: were

text: tion

text: of

text: pre-chirp

text: applied

text: to

text: the

text: fs

text: input

text: pulse.

text: used.

text: The

text: horizontal

text: line

text: indicates

text: the

text: maximum

text: spectral

text: broadening.

text: Figure

text: taken

text: from

text: erably

text: further

text: along

text: the

text: ﬁber.

text: In

text: addition

text: the

text: extent

text: of

text: the

text: spectral

text: broadening

text: has

text: slightly

text: decreased.

text: at

text: the

text: compression

text: point

text: with

text: the

text: amount

text: critically

text: de-

text: pendent

text: on

text: the

text: compression

text: ratio.

text: As

text: the

text: pulse

text: compresses

text: and

text: broadens,

text: its

text: peak

text: power

text: As

text: we

text: will

text: see

text: in

text: the

text: simulations

text: and

text: experiments,

text: the

text: will

text: rise

text: to

text: a

text: maximum

text: and

text: then

text: decrease.

text: As

text: we

text: see,

text: compression

text: ratio

text: will

text: be

text: greater

text: for

text: a

text: small

text: positive

text: chirp

text: the

text: RR

text: is

text: alomst

text: entirely

text: generated

text: where

text: the

text: pulse

text: com-

text: compared

text: to

text: zero

text: chirp

text: and

text: then

text: reduce

text: as

text: the

text: chirp

text: be-

text: presses,

text: i.e.

text: spectrally

text: broadens,

text: and

text: assumes

text: the

text: maxi-

text: comes

text: larger.

text: mum

text: peak

text: power.

text: Hence

text: it

text: is

text: instructive

text: to

text: ﬁnd

text: out

text: how

text: eﬃcient

text: the

text: pulse

text: compression

text: is

text: as

text: the

text: pulse

text: chirp

text: is

text: IV.

text: SIMULATIONS

text: OF

text: CHIRPED

text: PULSE

text: PROPAGATION

text: varied.

text: Figure

text: shows

text: the

text: maximum,

text: compressed

text: peak

text: power

text: as

text: a

text: function

text: of

text: the

text: input

text: pulse

text: chirp.

text: The

text: peak

text: power

text: assumes

text: a

text: maximum

text: for

text: small

text: positive

text: chirps

text: ex-

text: We

text: perform

text: numerical

text: simulations

text: of

text: pulse

text: propaga-

text: ceeding

text: the

text: power

text: for

text: zero

text: chirp.

text: For

text: larger

text: positive

text: tion

text: under

text: the

text: GNLSE.

text: We

text: use

text: a

text: standard

text: split-step

text: chirps

text: the

text: maximum

text: peak

text: power

text: decreases,

text: indicating

text: Fourier

text: tool

text: which

text: takes

text: into

text: account

text: the

text: full

text: dis-

text: that

text: the

text: pulse

text: is

text: not

text: compressing

text: as

text: much.

text: persion

text: proﬁle

text: as

text: well

text: as

text: nonlinear

text: and

text: Raman

text: eﬀects.

text: For

text: both

text: ﬁbers

text: we

text: carried

text: out

text: the

text: same

text: simulations

text: for

text: We

text: use

text: dispersion

text: proﬁles

text: determined

text: from

text: RR

text: measure-

text: N

text: values

text: up

text: to

text: N

text: and

text: a

text: pulse

text: lengths

text: from

text: fs

text: to

text: ments

text: for

text: the

text: two

text: ﬁbers

text: used

text: in

text: the

text: experiments

text: fs.

text: In

text: all

text: cases

text: we

text: found

text: an

text: optimum

text: chirp

text: for

text: pulse

text: Fiber

text: NKT

text: Photonics,

text: Ltd.)

text: has

text: a

text: zero-

text: compression.

text: In

text: the

text: variation

text: in

text: the

text: value

text: of

text: this

text: dispersion

text: wavelength

text: (ZDW)

text: of

text: nm

text: and

text: ﬁber

text: (NL-

text: optimum

text: chirp

text: with

text: soliton

text: order

text: and

text: pulse

text: length

text: can

text: NKT

text: Photonics,

text: Ltd.)

text: a

text: ZDW

text: of

text: nm.

text: Both

text: be

text: seen.

text: In

text: and

text: in

text: a

text: fs

text: pulse

text: length

text: is

text: ﬁbers

text: have

text: anomalous

text: dispersion

text: at

text: longer

text: wavelengths.

text: used

text: in

text: each

text: case

text: and

text: in

text: and

text: N

text: Propagations

text: of

text: hyperbolic

text: secant

text: pulses

text: with

text: varying

text: is

text: used

text: for

text: each

text: pulse

text: length.

text: The

text: ﬁgure

text: shows

text: that

text: a

text: linear

text: chirp

text: are

text: simulated.

text: positive

text: initial

text: chirp

text: leads

text: to

text: optimal

text: pulse

text: compression.

text: The

text: main

text: factors

text: for

text: the

text: generation

text: of

text: RR

text: are

text: the

text: evo-

text: The

text: obtained

text: compressed

text: peak

text: powers

text: increase

text: between

text: lution

text: of

text: the

text: pulse

text: spectrum

text: and

text: the

text: pulse

text: peak

text: power,

text: and

text: percent.

text: The

text: chirp

text: plays

text: an

text: important

text: role

text: for

text: stimulating

text: and

text: driving

text: RR

text: production,

text: respectively.

text: a

text: fundamental

text: soliton,

text: although

text: the

text: pulse

text: compression

text: is

text: The

text: simulation

text: parameters

text: are

text: matched

text: to

text: the

text: experi-

text: small.

text: For

text: higher

text: energies

text: the

text: pulse

text: peak

text: power

text: increses

text: mental

text: parameters.

text: We

text: used

text: a

text: fs

text: hyperbolic

text: secant

text: less,

text: but

text: due

text: to

text: the

text: strong

text: compression

text: RR

text: is

text: generated.

text: pulse

text: with

text: a

text: power

text: corresponding

text: to

text: a

text: soliton

text: order

text: of

text: The

text: RR

text: amplitude

text: and

text: eﬃciency

text: critically

text: depend

text: on

text: N

text: and

text: a

text: center

text: wavelength

text: of

text: nm

text: in

text: the

text: the

text: pulse

text: peak

text: power

text: and

text: so

text: we

text: expect

text: a

text: systematic

text: and

text: anomalous

text: dispersion

text: region.

text: We

text: also

text: extended

text: the

text: in-

text: strong

text: dependence

text: of

text: RR

text: on

text: pulse

text: chirp.

text: The

text: peak

text: power

text: vestigation

text: to

text: diﬀerent

text: soliton

text: orders

text: and

text: pulse

text: lengths.

text: enhancement

text: applies

text: to

text: a

text: large

text: range

text: of

text: pulse

text: lengths.

text: Detailed

text: results

text: of

text: the

text: ﬁber

text: simulations

text: can

text: be

text: found

text: We

text: also

text: carried

text: out

text: simulations

text: without

text: self-steepening

text: in

text: Here

text: we

text: brieﬂy

text: review

text: the

text: results

text: and

text: conclusions

text: and

text: Raman

text: scattering.

text: Whilst

text: they

text: had

text: an

text: impact

text: on

text: the

text: relevant

text: to

text: RR

text: generation.

text: Two

text: example

text: spectral

text: evolu-

text: position

text: of

text: compression

text: in

text: the

text: ﬁber

text: they

text: did

text: not

text: signiﬁ-

text: tions

text: for

text: this

text: ﬁber

text: are

text: shown

text: in

text: ﬁgure

text: for

text: an

text: unchirped

text: cantly

text: aﬀect

text: the

text: optimum

text: prechirp.

text: (a)

text: and

text: a

text: chirped

text: pulse

text: (b).

text: For

text: zero

text: input

text: chirp

text: the

text: input

text: V.

text: EXPERIMENT

text: pulse

text: spectrum

text: expands

text: to

text: almost

text: three

text: octaves

text: within

text: the

text: ﬁrst

text: mm

text: of

text: ﬁber

text: and

text: then

text: contracts

text: again

text: rapidly.

text: RR

text: is

text: generated

text: when

text: the

text: spectrum

text: expands

text: and

text: reaches

text: The

text: simulations

text: demonstrate

text: how

text: the

text: input

text: pulse

text: chirp

text: the

text: RR

text: wavelength

text: between

text: and

text: nm.

text: Similar

text: re-

text: can

text: be

text: used

text: to

text: optimise

text: pulse

text: compression.

text: In

text: the

text: experi-

text: sults

text: are

text: found

text: for

text: ﬁber

text: For

text: the

text: large

text: input

text: chirp

text: in

text: ments

text: we

text: investigate

text: the

text: impact

text: of

text: this

text: on

text: the

text: generation

text: Fig.

text: (b)

text: the

text: behavior

text: is

text: similar,

text: however,

text: the

text: point

text: of

text: eﬃciency

text: of

text: RR.

text: We

text: can

text: additionally

text: use

text: the

text: chirp

text: vary-

text: temporal

text: compression

text: and

text: spectral

text: expansion

text: is

text: consid-

text: ing

text: technique

text: described

text: in

text: to

text: qualitatively

text: investigate

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 4 (a) (b) FIG. 4. Simulations of pulse propagation: Optimum chirp as a function of (a,b): soliton order N ,(c,d): input pulse length FIG. 5. Spectral density of the visible and IR spectrum as for (a,c): ﬁber 1, (b,d): ﬁber 2. the input chirp is varied for (a) ﬁber 1 (L=8 mm, N=2.25, τ=12 fs), (b) ﬁber 2 (L=5 mm, N=2.11, τ=7 fs). the evolution of RR. In the experiments we use short pulses centered at 800 nm with a bandwidth of over 200 nm from a mode- locked Ti:Sapphire laser (Rainbow, FemtoSource GmBH). These are coupled into the two photonic crystal ﬁbers (PCFs). The PCFs are typically around L=5 mm in length as the pulses evolve over yet shorter distances. The input pulse lies in the anomalous dispersion region of the PCFs which, in the absence of higher order eﬀects, FIG. 6. Resonant radiation signal strength as a function of allows for soliton formation. The PCF output in the the input pulse chirp. (a) ﬁber 1 (L=8 mm/10 mm (red cir- visible and IR is measured using a CCD spectrometer cles/ blue diamonds), N=2.25, τ=12 fs), (b) ﬁber 2 (L=5 mm, (AvaSpec, Avantes BV). N=2.11, τ=7 fs). The variation in input pulse length and chirp is achieved by inserting glass elements in the beam path. Negative dispersion can be added by increasing the number of re- creasing propagation distance for the RR generated at ﬂections from a pair of double chirped mirrors (DCMs). the compression point. For large pulse chirps the GVD The minimum pulse length (FWHM) used is 7fs for the is unable to compensate the chirp and the pulse does not unchirped pulse. Additional data is taken with a 12fs compress fully in the ﬁber. Hence the output spectrum pulse. The typical average input power was 100mW. is narrow and the RR negligible. As the chirp decreases The coupling into the PCFs is around 25% but varies the pulse is able to fully compress at the end of the ﬁber between ﬁber samples. The pulse chirp C leads to an and we observe the spectral broadening of the pulse and (cid:80) increase in the pulse length and a variation in the in- the initial generation of the RR in the ﬁber output. Fur- stantaneous frequency across the pulse. It is given by ther decrease of the chirp moves the compression point | | C = ( β z ) + D , where (β ) and z are the towards the input end of the ﬁber. The spectral broaden- e 2 e e 2DCM 2 e e GVD and path distances through each dispersive element ing reduces as the pulse has temporally expanded again e before the ﬁber and D is the negative group delay by the time it reaches the output. The distance propa- 2DCM dispersion (GDD) of the DCMs. gated by the RR increases and as in the simulation plots The contour plots in Fig. 5 show how the pulse and we see it forming an isolated peak at a ﬁxed wavelength. RR spectra vary as the input chirp is changed for the two Having seen that we can use the chirp to investigate ﬁbers used in the experiments. RR peaks are measured at spectral evolution we now look at the impact of chirp on about 480nm (ﬁber 1) and 510nm (ﬁber 2). The plots are the generation eﬃciency of RR. We expect the RR to be qualitatively similar to those in Fig. 2 showing spectral generated only during pulse compression and to propa- evolution as a function of propagation distance. How- gate after this with little change in energy. Therefore, ever, in this case an increasing pulse chirp corresponds the total amount of RR generated should not depend on to compression further along the ﬁber and hence a de- where the pulse compresses. However, in the simulations ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: (b)

text: FIG.

text: Simulations

text: of

text: pulse

text: propagation:

text: Optimum

text: chirp

text: as

text: a

text: function

text: of

text: (a,b):

text: soliton

text: order

text: N

text: ,(c,d):

text: input

text: pulse

text: length

text: FIG.

text: Spectral

text: density

text: of

text: the

text: visible

text: and

text: IR

text: spectrum

text: as

text: for

text: (a,c):

text: ﬁber

text: (b,d):

text: ﬁber

text: the

text: input

text: chirp

text: is

text: varied

text: for

text: (a)

text: ﬁber

text: mm,

text: fs),

text: (b)

text: ﬁber

text: mm,

text: fs).

text: the

text: evolution

text: of

text: RR.

text: In

text: the

text: experiments

text: we

text: use

text: short

text: pulses

text: centered

text: at

text: nm

text: with

text: a

text: bandwidth

text: of

text: over

text: nm

text: from

text: a

text: mode-

text: locked

text: Ti:Sapphire

text: laser

text: (Rainbow,

text: FemtoSource

text: GmBH).

text: These

text: are

text: coupled

text: into

text: the

text: two

text: photonic

text: crystal

text: ﬁbers

text: (PCFs).

text: The

text: PCFs

text: are

text: typically

text: around

text: mm

text: in

text: length

text: as

text: the

text: pulses

text: evolve

text: over

text: yet

text: shorter

text: distances.

text: The

text: input

text: pulse

text: lies

text: in

text: the

text: anomalous

text: dispersion

text: region

text: of

text: the

text: PCFs

text: which,

text: in

text: the

text: absence

text: of

text: higher

text: order

text: eﬀects,

text: FIG.

text: Resonant

text: radiation

text: signal

text: strength

text: as

text: a

text: function

text: of

text: allows

text: for

text: soliton

text: formation.

text: The

text: PCF

text: output

text: in

text: the

text: the

text: input

text: pulse

text: chirp.

text: (a)

text: ﬁber

text: mm

text: (red

text: cir-

text: visible

text: and

text: IR

text: is

text: measured

text: using

text: a

text: CCD

text: spectrometer

text: cles/

text: blue

text: diamonds),

text: fs),

text: (b)

text: ﬁber

text: mm,

text: (AvaSpec,

text: Avantes

text: BV).

text: fs).

text: The

text: variation

text: in

text: input

text: pulse

text: length

text: and

text: chirp

text: is

text: achieved

text: by

text: inserting

text: glass

text: elements

text: in

text: the

text: beam

text: path.

text: Negative

text: dispersion

text: can

text: be

text: added

text: by

text: increasing

text: the

text: number

text: of

text: re-

text: creasing

text: propagation

text: distance

text: for

text: the

text: RR

text: generated

text: at

text: ﬂections

text: from

text: a

text: pair

text: of

text: double

text: chirped

text: mirrors

text: (DCMs).

text: the

text: compression

text: point.

text: For

text: large

text: pulse

text: chirps

text: the

text: GVD

text: The

text: minimum

text: pulse

text: length

text: (FWHM)

text: used

text: is

text: for

text: the

text: is

text: unable

text: to

text: compensate

text: the

text: chirp

text: and

text: the

text: pulse

text: does

text: not

text: unchirped

text: pulse.

text: Additional

text: data

text: is

text: taken

text: with

text: a

text: compress

text: fully

text: in

text: the

text: ﬁber.

text: Hence

text: the

text: output

text: spectrum

text: pulse.

text: The

text: typical

text: average

text: input

text: power

text: was

text: is

text: narrow

text: and

text: the

text: RR

text: negligible.

text: As

text: the

text: chirp

text: decreases

text: The

text: coupling

text: into

text: the

text: PCFs

text: is

text: around

text: but

text: varies

text: the

text: pulse

text: is

text: able

text: to

text: fully

text: compress

text: at

text: the

text: end

text: of

text: the

text: ﬁber

text: between

text: ﬁber

text: samples.

text: The

text: pulse

text: chirp

text: C

text: leads

text: to

text: an

text: and

text: we

text: observe

text: the

text: spectral

text: broadening

text: of

text: the

text: pulse

text: and

text: increase

text: in

text: the

text: pulse

text: length

text: and

text: a

text: variation

text: in

text: the

text: in-

text: the

text: initial

text: generation

text: of

text: the

text: RR

text: in

text: the

text: ﬁber

text: output.

text: Fur-

text: stantaneous

text: frequency

text: across

text: the

text: pulse.

text: It

text: is

text: given

text: by

text: ther

text: decrease

text: of

text: the

text: chirp

text: moves

text: the

text: compression

text: point

text: C

text: β

text: z

text: D

text: where

text: (β

text: and

text: z

text: are

text: the

text: towards

text: the

text: input

text: end

text: of

text: the

text: ﬁber.

text: The

text: spectral

text: broaden-

text: e

text: e

text: e

text: e

text: e

text: GVD

text: and

text: path

text: distances

text: through

text: each

text: dispersive

text: element

text: ing

text: reduces

text: as

text: the

text: pulse

text: has

text: temporally

text: expanded

text: again

text: e

text: before

text: the

text: ﬁber

text: and

text: D

text: is

text: the

text: negative

text: group

text: delay

text: by

text: the

text: time

text: it

text: reaches

text: the

text: output.

text: The

text: distance

text: propa-

text: dispersion

text: (GDD)

text: of

text: the

text: DCMs.

text: gated

text: by

text: the

text: RR

text: increases

text: and

text: as

text: in

text: the

text: simulation

text: plots

text: The

text: contour

text: plots

text: in

text: Fig.

text: show

text: how

text: the

text: pulse

text: and

text: we

text: see

text: it

text: forming

text: an

text: isolated

text: peak

text: at

text: a

text: ﬁxed

text: wavelength.

text: RR

text: spectra

text: vary

text: as

text: the

text: input

text: chirp

text: is

text: changed

text: for

text: the

text: two

text: Having

text: seen

text: that

text: we

text: can

text: use

text: the

text: chirp

text: to

text: investigate

text: ﬁbers

text: used

text: in

text: the

text: experiments.

text: RR

text: peaks

text: are

text: measured

text: at

text: spectral

text: evolution

text: we

text: now

text: look

text: at

text: the

text: impact

text: of

text: chirp

text: on

text: about

text: (ﬁber

text: and

text: (ﬁber

text: The

text: plots

text: are

text: the

text: generation

text: eﬃciency

text: of

text: RR.

text: We

text: expect

text: the

text: RR

text: to

text: be

text: qualitatively

text: similar

text: to

text: those

text: in

text: Fig.

text: showing

text: spectral

text: generated

text: only

text: during

text: pulse

text: compression

text: and

text: to

text: propa-

text: evolution

text: as

text: a

text: function

text: of

text: propagation

text: distance.

text: How-

text: gate

text: after

text: this

text: with

text: little

text: change

text: in

text: energy.

text: Therefore,

text: ever,

text: in

text: this

text: case

text: an

text: increasing

text: pulse

text: chirp

text: corresponds

text: the

text: total

text: amount

text: of

text: RR

text: generated

text: should

text: not

text: depend

text: on

text: to

text: compression

text: further

text: along

text: the

text: ﬁber

text: and

text: hence

text: a

text: de-

text: where

text: the

text: pulse

text: compresses.

text: However,

text: in

text: the

text: simulations

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 5 the compressed peak power slightly varied along the ﬁber pulse spectrum will have expanded to overlap with the and therefore the generation eﬃciency of the RR varied, RR hence increasing the signal at that wavelength and too. We expect a maximum in the RR output for the op- producing the second maximum. The diﬀerences in the timum chirp followed by a decline as the chirp increases ﬁber dispersion and the pulse parameters in each case further until the RR becomes negligible when the chirp may explain why the peak is less pronounced for ﬁber is too large to be compensated inside the ﬁber. 2. The second peak, therefore, includes light from the pulse. Otherwise, note how the RR generation follows The graphs in Fig. 6 show the RR peak height as a the development of the pulse peak power in Fig. 3. function of input chirp for the two ﬁbers. In both cases In conclusion, the generation eﬃciency of RR can be the RR has the expected maximum for a small positive maximized by using a small positive chirp. Such con- chirp and then declines. In Fig. 6(a) the RR reaches a siderations are of particular importance when high con- maximum around 50 fs2 for the two ﬁber lengths. This version eﬃciencies are required for practical applications. is in agreement with the simulation in Fig. 4 (a). In Even a small improvement in the pulse compression can Fig. 6(b) the maximum is at around 31 fs2. The RR lead to a large increase of the RR. In addition, we have peak height at the optimum chirp is over 3 times higher shown that the evolution of the pulse and RR spectra than at zero chirp. Note that the optimum chirp is lower can be investigated without cutting the ﬁber by using than expected from Fig. 4 (b,d). The diﬀerence may be the input chirp. If dispersion mircomanaged ﬁbers, such due to inaccuracies in the dispersion proﬁle of ﬁber 2 for as tapered ﬁbers, are used, the chirp will tune this RR IR wavelengths, which weigh much stronger if the group pulse source through the visible range. velocity dispersion is lower due to the larger ZDW. ACKNOWLEDGMENTS Fig. 6 also displays a second eﬃcient chirp corre- sponding to the end of the ﬁber; in Fig. 6(a) at 231 fs2 We would like to acknowledge useful discussions with (L=8 mm) and 256 fs2(L=10 mm), in Fig. 6(b) at 100 fs2. F. Biancalana and S. Kehr. If the compression occurs at the end of the ﬁber, the fxundeﬁned [1]ifx1 fnum [1]1ﬁrstoftwo fx [1]secondoftwo noop Ilaria Cristiani, Riccardo Tediosi, Luca Tartara, and Vit- [0]secondoftworef[1]@startlink1@hrefhref[1]1@endlinkanitize@urltorio Degiorgio. Dispersive wave generation by soli- [0]startlink[1]endlink[0]rl [1]href 1bib@innerbibempty tons in microstructured optical ﬁbers. Opt. Express, Govind P. Agrawal. Nonlinear Fiber Optics. Optics 12(1):124–135, Jan 2004. and Photonics. Academic Press, 3rd edition, 2001. M.Clerici V.Gorini G.Ortenzi L. Rizzi E. Rubino V.G.Sala Karlsson Magnus Akhmediev, Nail. Cherenkov radiation D. Faccio F.Belgiorno, S.L.Cacciatori. Hawking radi- emitted by solitons in optical ﬁbers. Physical Review A, ation from ultrashort laser pulse ﬁlaments. Phys Rev 51(3):2602–2607, 03 1995. Lett, 105(20), November 2010. Guoqing Chang, Li-Jin Chen, and Franz X. K¨artner. Xiquan Fu, Liejia Qian, Shuangchun Wen, and Di- Highly eﬃcient cherenkov radiation in photonic crystal anyuan Fan. Nonlinear chirped pulse propagation and ﬁbers for broadband visible wavelength generation. Opt. supercontinuum generation in microstructured optical Lett., 35(14):2361–2363, Jul 2010. ﬁbre. Journal of Optics A: Pure and Applied Optics, 6(11):1012–, 2004. Guoqing Chang, Chih-Hao Li, A. Glenday, G. Furesz, N. Langellier, Li-Jin Chen, M.W. Webber, Jinkang Lim, J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, Hung-Wen Chen, D.F. Phillips, A. Szentgyorgyi, R.L. D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Rus- Walsworth, and F.X. Kartner. Spectrally ﬂat, broad- sell, and G. Korn. Experimental evid2ence for supercon- band visible-wavelength astro-comb. In Lasers and tinuum generation by ﬁssion of higher-order solitons in Electro-Optics (CLEO), 2012 Conference on, pages 1–2, photonic ﬁbers. Physical Review Letters, 88(17):173901– 2012. , 04 2002. Chunfu Cheng, Youqing Wang, and Qinghua Lv. Eﬀect A V Husakou and J Herrmann. Supercontinuum genera- of initial frequency chirp on the supercontinuum genera- tion of higher-order solitons by ﬁssion in photonic crystal tion in all-normal dispersion photonic crystal ﬁbers. In ﬁbers. Phys Rev Lett, 87(20):203901, Nov 2001. Photonics and Optoelectronics Meetings (POEM) 2011: Anatoly A. Ivanov, Mikhail V. Alﬁmov, Aleksei M. Optical Communication Systems and Networking, vol- Zheltikov, Marcin Szpulak, Waclaw Urbanczyk, and ume 8331, pages 83310O–83310O–8, 2011. Jan W´ojcik. Polarization-controlled vectorial spectral Amol Choudhary and Friedrich K¨onig. Eﬃcient fre- transformations of femtosecond pulses in a birefringent quency shifting of dispersive waves at solitons. Opt. photonic-crystal ﬁber. J. Opt. Soc. Am. B, 23(5):986– Express, 20(5):5538–5546, Feb 2012. 991, May 2006. ##LTLine## ##LTLine## ##LTLine##

text: the

text: compressed

text: peak

text: power

text: slightly

text: varied

text: along

text: the

text: ﬁber

text: pulse

text: spectrum

text: will

text: have

text: expanded

text: to

text: overlap

text: with

text: the

text: and

text: therefore

text: the

text: generation

text: eﬃciency

text: of

text: the

text: RR

text: varied,

text: RR

text: hence

text: increasing

text: the

text: signal

text: at

text: that

text: wavelength

text: and

text: too.

text: We

text: expect

text: a

text: maximum

text: in

text: the

text: RR

text: output

text: for

text: the

text: op-

text: producing

text: the

text: second

text: maximum.

text: The

text: diﬀerences

text: in

text: the

text: timum

text: chirp

text: followed

text: by

text: a

text: decline

text: as

text: the

text: chirp

text: increases

text: ﬁber

text: dispersion

text: and

text: the

text: pulse

text: parameters

text: in

text: each

text: case

text: further

text: until

text: the

text: RR

text: becomes

text: negligible

text: when

text: the

text: chirp

text: may

text: explain

text: why

text: the

text: peak

text: is

text: less

text: pronounced

text: for

text: ﬁber

text: is

text: too

text: large

text: to

text: be

text: compensated

text: inside

text: the

text: ﬁber.

text: The

text: second

text: peak,

text: therefore,

text: includes

text: light

text: from

text: the

text: pulse.

text: Otherwise,

text: note

text: how

text: the

text: RR

text: generation

text: follows

text: The

text: graphs

text: in

text: Fig.

text: show

text: the

text: RR

text: peak

text: height

text: as

text: a

text: the

text: development

text: of

text: the

text: pulse

text: peak

text: power

text: in

text: Fig.

text: function

text: of

text: input

text: chirp

text: for

text: the

text: two

text: ﬁbers.

text: In

text: both

text: cases

text: In

text: conclusion,

text: the

text: generation

text: eﬃciency

text: of

text: RR

text: can

text: be

text: the

text: RR

text: has

text: the

text: expected

text: maximum

text: for

text: a

text: small

text: positive

text: maximized

text: by

text: using

text: a

text: small

text: positive

text: chirp.

text: Such

text: con-

text: chirp

text: and

text: then

text: declines.

text: In

text: Fig.

text: the

text: RR

text: reaches

text: a

text: siderations

text: are

text: of

text: particular

text: importance

text: when

text: high

text: con-

text: maximum

text: around

text: for

text: the

text: two

text: ﬁber

text: lengths.

text: This

text: version

text: eﬃciencies

text: are

text: required

text: for

text: practical

text: applications.

text: is

text: in

text: agreement

text: with

text: the

text: simulation

text: in

text: Fig.

text: (a).

text: In

text: Even

text: a

text: small

text: improvement

text: in

text: the

text: pulse

text: compression

text: can

text: Fig.

text: the

text: maximum

text: is

text: at

text: around

text: The

text: RR

text: lead

text: to

text: a

text: large

text: increase

text: of

text: the

text: RR.

text: In

text: addition,

text: we

text: have

text: peak

text: height

text: at

text: the

text: optimum

text: chirp

text: is

text: over

text: times

text: higher

text: shown

text: that

text: the

text: evolution

text: of

text: the

text: pulse

text: and

text: RR

text: spectra

text: than

text: at

text: zero

text: chirp.

text: Note

text: that

text: the

text: optimum

text: chirp

text: is

text: lower

text: can

text: be

text: investigated

text: without

text: cutting

text: the

text: ﬁber

text: by

text: using

text: than

text: expected

text: from

text: Fig.

text: (b,d).

text: The

text: diﬀerence

text: may

text: be

text: the

text: input

text: chirp.

text: If

text: dispersion

text: mircomanaged

text: ﬁbers,

text: such

text: due

text: to

text: inaccuracies

text: in

text: the

text: dispersion

text: proﬁle

text: of

text: ﬁber

text: for

text: as

text: tapered

text: ﬁbers,

text: are

text: used,

text: the

text: chirp

text: will

text: tune

text: this

text: RR

text: IR

text: wavelengths,

text: which

text: weigh

text: much

text: stronger

text: if

text: the

text: group

text: pulse

text: source

text: through

text: the

text: visible

text: range.

text: velocity

text: dispersion

text: is

text: lower

text: due

text: to

text: the

text: larger

text: ZDW.

text: ACKNOWLEDGMENTS

text: Fig.

text: also

text: displays

text: a

text: second

text: eﬃcient

text: chirp

text: corre-

text: sponding

text: to

text: the

text: end

text: of

text: the

text: ﬁber;

text: in

text: Fig.

text: at

text: We

text: would

text: like

text: to

text: acknowledge

text: useful

text: discussions

text: with

text: mm)

text: and

text: mm),

text: in

text: Fig.

text: at

text: F.

text: Biancalana

text: and

text: S.

text: Kehr.

text: If

text: the

text: compression

text: occurs

text: at

text: the

text: end

text: of

text: the

text: ﬁber,

text: the

text: fxundeﬁned

text: fnum

text: fx

text: noop

text: Ilaria

text: Cristiani,

text: Riccardo

text: Tediosi,

text: Luca

text: Tartara,

text: and

text: Vit-

text: Degiorgio.

text: Dispersive

text: wave

text: generation

text: by

text: soli-

text: tons

text: in

text: microstructured

text: optical

text: ﬁbers.

text: Opt.

text: Express,

text: Govind

text: P.

text: Agrawal.

text: Nonlinear

text: Fiber

text: Optics.

text: Optics

text: Jan

text: and

text: Photonics.

text: Academic

text: Press,

text: edition,

text: M.Clerici

text: V.Gorini

text: G.Ortenzi

text: L.

text: Rizzi

text: E.

text: Rubino

text: V.G.Sala

text: Karlsson

text: Magnus

text: Akhmediev,

text: Nail.

text: Cherenkov

text: radiation

text: D.

text: Faccio

text: F.Belgiorno,

text: S.L.Cacciatori.

text: Hawking

text: radi-

text: emitted

text: by

text: solitons

text: in

text: optical

text: ﬁbers.

text: Physical

text: Review

text: A,

text: ation

text: from

text: ultrashort

text: laser

text: pulse

text: ﬁlaments.

text: Phys

text: Rev

text: Lett,

text: November

text: Guoqing

text: Chang,

text: Li-Jin

text: Chen,

text: and

text: Franz

text: X.

text: K¨artner.

text: Xiquan

text: Fu,

text: Liejia

text: Qian,

text: Shuangchun

text: Wen,

text: and

text: Di-

text: Highly

text: eﬃcient

text: cherenkov

text: radiation

text: in

text: photonic

text: crystal

text: anyuan

text: Fan.

text: Nonlinear

text: chirped

text: pulse

text: propagation

text: and

text: ﬁbers

text: for

text: broadband

text: visible

text: wavelength

text: generation.

text: Opt.

text: supercontinuum

text: generation

text: in

text: microstructured

text: optical

text: Lett.,

text: Jul

text: ﬁbre.

text: Journal

text: of

text: Optics

text: A:

text: Pure

text: and

text: Applied

text: Optics,

text: Guoqing

text: Chang,

text: Chih-Hao

text: Li,

text: A.

text: Glenday,

text: G.

text: Furesz,

text: N.

text: Langellier,

text: Li-Jin

text: Chen,

text: M.W.

text: Webber,

text: Jinkang

text: Lim,

text: J.

text: Herrmann,

text: U.

text: Griebner,

text: N.

text: Zhavoronkov,

text: A.

text: Husakou,

text: Hung-Wen

text: Chen,

text: D.F.

text: Phillips,

text: A.

text: Szentgyorgyi,

text: R.L.

text: D.

text: Nickel,

text: J.

text: C.

text: Knight,

text: W.

text: J.

text: Wadsworth,

text: P.

text: St.

text: J.

text: Rus-

text: Walsworth,

text: and

text: F.X.

text: Kartner.

text: Spectrally

text: ﬂat,

text: broad-

text: sell,

text: and

text: G.

text: Korn.

text: Experimental

text: for

text: supercon-

text: band

text: visible-wavelength

text: astro-comb.

text: In

text: Lasers

text: and

text: tinuum

text: generation

text: by

text: ﬁssion

text: of

text: higher-order

text: solitons

text: in

text: Electro-Optics

text: (CLEO),

text: Conference

text: on,

text: pages

text: photonic

text: ﬁbers.

text: Physical

text: Review

text: Letters,

text: Chunfu

text: Cheng,

text: Youqing

text: Wang,

text: and

text: Qinghua

text: Lv.

text: Eﬀect

text: A

text: V

text: Husakou

text: and

text: J

text: Herrmann.

text: Supercontinuum

text: genera-

text: of

text: initial

text: frequency

text: chirp

text: on

text: the

text: supercontinuum

text: genera-

text: tion

text: of

text: higher-order

text: solitons

text: by

text: ﬁssion

text: in

text: photonic

text: crystal

text: tion

text: in

text: all-normal

text: dispersion

text: photonic

text: crystal

text: ﬁbers.

text: In

text: ﬁbers.

text: Phys

text: Rev

text: Lett,

text: Nov

text: Photonics

text: and

text: Optoelectronics

text: Meetings

text: (POEM)

text: Anatoly

text: A.

text: Ivanov,

text: Mikhail

text: V.

text: Alﬁmov,

text: Aleksei

text: M.

text: Optical

text: Communication

text: Systems

text: and

text: Networking,

text: vol-

text: Zheltikov,

text: Marcin

text: Szpulak,

text: Waclaw

text: Urbanczyk,

text: and

text: ume

text: pages

text: Jan

text: W´ojcik.

text: Polarization-controlled

text: vectorial

text: spectral

text: Amol

text: Choudhary

text: and

text: Friedrich

text: K¨onig.

text: Eﬃcient

text: fre-

text: transformations

text: of

text: femtosecond

text: pulses

text: in

text: a

text: birefringent

text: quency

text: shifting

text: of

text: dispersive

text: waves

text: at

text: solitons.

text: Opt.

text: photonic-crystal

text: ﬁber.

text: J.

text: Opt.

text: Soc.

text: Am.

text: B,

text: Express,

text: Feb

text: May

text: ##LTLine##

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text: Optimal block designs for experiments with responses drawn from a Poisson distribution Stephen Bush (Stephen.Bush@uts.edu.au) Department of Mathematical and Physical Sciences, University of Technology Sydney, Australia Katya Ruggiero (k.ruggiero@auckland.ac.nz) Department of Statistics, University of Auckland, New Zealand May 28, 2020 Abstract Optimal block designs for additive models achieve their eﬃciency by dividing experimental units among relatively homogenous blocks and allocating treatments equally to blocks. Responses in many modern experiments, however, are drawn from distributions such as the one- and two-parameter exponen- tial families, e.g., RNA sequence counts from a negative binomial distribution. These violate additivity. Yet, designs generated by assuming additivity continue to be used, because better approaches are not available, and because the issues are not widely recognised. We solve this problem for single-factor experiments in which treatments, taking categorical values only, are arranged in blocks and responses drawn from a Poisson distribution. We derive expressions for two objective functions, based on D - and A C-optimality, with eﬃcient estimation of linear contrasts of the ﬁxed eﬀects parameters in a Poisson generalised linear mixed model (GLMM) being the objective. These objective functions are shown to be computational eﬃcient, requiring no matrix inversion. Using simulated annealing to generate Poisson GLMM-based locally optimal designs, we show that the replication numbers of treatments in these designs are inversely proportional to the relative magnitudes of the treatments’ expected counts. Importantly, for non-negligible treatment eﬀect sizes, Poisson GLMM-based optimal designs may be substantially more eﬃcient than their classically optimal counterparts. 1 Introduction The introduction of gene expression microarrays (Schena et al., 1995) towards the end of the twentieth century initiated the start of a biotechnology revolution of rapidly evolving instruments capable of proﬁling a wide range of diﬀerent molecular species at the cellular level. Today, high resolution instruments, such as next-generation sequencing (NGS) technologies (Craig et al., 2008), are capable of generating counts of individual copies of, for example, diﬀerent gene transcripts. Experiments using these technologies are relatively expensive, resulting in studies with low numbers of biological replicates of treatments, making the eﬃcient statistical design of such experiment critical. Yet, with few exceptions, classically optimal designs based on the assumption of unit-treatment additivity (i.e. functional independence of variances and covariances on their means) continue to be used. Optimal statistical designs are central to conducting eﬃcient comparative experiments, enabling contrasts of treatment parameters to be estimated without bias and minimum variance while requiring minimum eﬀort, subjects, or other resources. A rich body of literature on optimal designs (John and Williams, 1995; Atkinson et al., 2007) has grown since the creation of this ﬁeld of statistics (Smith, 1918). Until very recently however, the criteria used to deﬁne optimal designs has depended on the assumption of unit-treatment additivity. Under additivity, optimal designs achieve their eﬃciency by dividing the experimental material into relatively homogeneous blocks and allocating treatments equally to blocks. Full eﬃciency, whereby a model’s treatment parameters can be estimated independently of its block parameters, is attained when treatments can be arranged in a complete block design (Fisher, 1926), i.e. each treatment occurs equally frequently, usually once, in each block. However, as already noted, responses in many modern experiments are drawn from distributions such as the one and two parameter exponential families, e.g., RNA sequence counts from a negative binomial distribution (Auer and Doerge, 2010). These violate additivity. Our current work on 1

text: Optimal

label: title

text: block

text: designs

text: for

text: experiments

text: with

text: responses

text: drawn

text: from

text: a

text: Poisson

text: distribution

text: Stephen

text: Bush

text: (Stephen.Bush@uts.edu.au)

text: Department

text: of

text: Mathematical

text: and

text: Physical

text: Sciences,

text: University

text: of

text: Technology

text: Sydney,

text: Australia

text: Katya

text: Ruggiero

text: (k.ruggiero@auckland.ac.nz)

text: Department

text: of

text: Statistics,

text: University

text: of

text: Auckland,

text: New

text: Zealand

text: May

text: Abstract

text: Optimal

label: abstract

text: block

text: designs

text: for

text: additive

text: models

text: achieve

text: their

text: eﬃciency

text: by

text: dividing

text: experimental

text: units

text: among

text: relatively

text: homogenous

text: blocks

text: and

text: allocating

text: treatments

text: equally

text: to

text: blocks.

text: Responses

text: in

text: many

text: modern

text: experiments,

text: however,

text: are

text: drawn

text: from

text: distributions

text: such

text: as

text: the

text: one-

text: and

text: two-parameter

text: exponen-

text: tial

text: families,

text: e.g.,

text: RNA

text: sequence

text: counts

text: from

text: a

text: negative

text: binomial

text: distribution.

text: These

text: violate

text: additivity.

text: Yet,

text: designs

text: generated

text: by

text: assuming

text: additivity

text: continue

text: to

text: be

text: used,

text: because

text: better

text: approaches

text: are

text: not

text: available,

text: and

text: because

text: the

text: issues

text: are

text: not

text: widely

text: recognised.

text: We

text: solve

text: this

text: problem

text: for

text: single-factor

text: experiments

text: in

text: which

text: treatments,

text: taking

text: categorical

text: values

text: only,

text: are

text: arranged

text: in

text: blocks

text: and

text: responses

text: drawn

text: from

text: a

text: Poisson

text: distribution.

text: We

text: derive

text: expressions

text: for

text: two

text: objective

text: functions,

text: based

text: on

text: D

text: and

text: A

text: C-optimality,

text: with

text: eﬃcient

text: estimation

text: of

text: linear

text: contrasts

text: of

text: the

text: ﬁxed

text: eﬀects

text: parameters

text: in

text: a

text: Poisson

text: generalised

text: linear

text: mixed

text: model

text: (GLMM)

text: being

text: the

text: objective.

text: These

text: objective

text: functions

text: are

text: shown

text: to

text: be

text: computational

text: eﬃcient,

text: requiring

text: no

text: matrix

text: inversion.

text: Using

text: simulated

text: annealing

text: to

text: generate

text: Poisson

text: GLMM-based

text: locally

text: optimal

text: designs,

text: we

text: show

text: that

text: the

text: replication

text: numbers

text: of

text: treatments

text: in

text: these

text: designs

text: are

text: inversely

text: proportional

text: to

text: the

text: relative

text: magnitudes

text: of

text: the

text: treatments’

text: expected

text: counts.

text: Importantly,

text: for

text: non-negligible

text: treatment

text: eﬀect

text: sizes,

text: Poisson

text: GLMM-based

text: optimal

text: designs

text: may

text: be

text: substantially

text: more

text: eﬃcient

text: than

text: their

text: classically

text: optimal

text: counterparts.

text: Introduction

text: The

text: introduction

text: of

text: gene

text: expression

text: microarrays

text: (Schena

text: et

text: al.,

text: towards

text: the

text: end

text: of

text: the

text: twentieth

text: century

text: initiated

text: the

text: start

text: of

text: a

text: biotechnology

text: revolution

text: of

text: rapidly

text: evolving

text: instruments

text: capable

text: of

text: proﬁling

text: a

text: wide

text: range

text: of

text: diﬀerent

text: molecular

text: species

text: at

text: the

text: cellular

text: level.

text: Today,

text: high

text: resolution

text: instruments,

text: such

text: as

text: next-generation

text: sequencing

text: (NGS)

text: technologies

text: (Craig

text: et

text: al.,

text: are

text: capable

text: of

text: generating

text: counts

text: of

text: individual

text: copies

text: of,

text: for

text: example,

text: diﬀerent

text: gene

text: transcripts.

text: Experiments

text: using

text: these

text: technologies

text: are

text: relatively

text: expensive,

text: resulting

text: in

text: studies

text: with

text: low

text: numbers

text: of

text: biological

text: replicates

text: of

text: treatments,

text: making

text: the

text: eﬃcient

text: statistical

text: design

text: of

text: such

text: experiment

text: critical.

text: Yet,

text: with

text: few

text: exceptions,

text: classically

text: optimal

text: designs

text: based

text: on

text: the

text: assumption

text: of

text: unit-treatment

text: additivity

text: (i.e.

text: functional

text: independence

text: of

text: variances

text: and

text: covariances

text: on

text: their

text: means)

text: continue

text: to

text: be

text: used.

text: Optimal

text: statistical

text: designs

text: are

text: central

text: to

text: conducting

text: eﬃcient

text: comparative

text: experiments,

text: enabling

text: contrasts

text: of

text: treatment

text: parameters

text: to

text: be

text: estimated

text: without

text: bias

text: and

text: minimum

text: variance

text: while

text: requiring

text: minimum

text: eﬀort,

text: subjects,

text: or

text: other

text: resources.

text: A

text: rich

text: body

text: of

text: literature

text: on

text: optimal

text: designs

text: (John

text: and

text: Williams,

text: Atkinson

text: et

text: al.,

text: has

text: grown

text: since

text: the

text: creation

text: of

text: this

text: ﬁeld

text: of

text: statistics

text: (Smith,

text: Until

text: very

text: recently

text: however,

text: the

text: criteria

text: used

text: to

text: deﬁne

text: optimal

text: designs

text: has

text: depended

text: on

text: the

text: assumption

text: of

text: unit-treatment

text: additivity.

text: Under

text: additivity,

text: optimal

text: designs

text: achieve

text: their

text: eﬃciency

text: by

text: dividing

text: the

text: experimental

text: material

text: into

text: relatively

text: homogeneous

text: blocks

text: and

text: allocating

text: treatments

text: equally

text: to

text: blocks.

text: Full

text: eﬃciency,

text: whereby

text: a

text: model’s

text: treatment

text: parameters

text: can

text: be

text: estimated

text: independently

text: of

text: its

text: block

text: parameters,

text: is

text: attained

text: when

text: treatments

text: can

text: be

text: arranged

text: in

text: a

text: complete

text: block

text: design

text: (Fisher,

text: i.e.

text: each

text: treatment

text: occurs

text: equally

text: frequently,

text: usually

text: once,

text: in

text: each

text: block.

text: However,

text: as

text: already

text: noted,

text: responses

text: in

text: many

text: modern

text: experiments

text: are

text: drawn

text: from

text: distributions

text: such

text: as

text: the

text: one

text: and

text: two

text: parameter

text: exponential

text: families,

text: e.g.,

text: RNA

text: sequence

text: counts

text: from

text: a

text: negative

text: binomial

text: distribution

text: (Auer

text: and

text: Doerge,

text: These

text: violate

text: additivity.

text: Our

text: current

text: work

text: on


text: Relative D−efficiency of various designs for different genes 1.00 l l l l l l l l 0.75 Gene l Gene A 0.50 Gene B Gene C Gene D 0.25 0.00 D1 D2 D3 D4 D5 D6 D7 D8 Design Figure 1: Per-gene relative eﬃciencies of eight designs, D – D , deﬁned in Table 4 based on point priors of 1 8 four genes, A – D, given in Table 3. 15 ycneiciffE evitaleR ##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##

text: Relative

font: default

text: D−efficiency

text: of

text: various

text: designs

text: for

text: different

text: genes

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: Gene

text: l

text: Gene

text: A

text: Gene

text: B

text: Gene

text: C

text: Gene

text: D

text: Design

text: Figure

text: Per-gene

text: relative

text: eﬃciencies

text: of

text: eight

text: designs,

text: D

text: D

text: deﬁned

text: in

text: Table

text: based

text: on

text: point

text: priors

text: of

text: four

text: genes,

text: A

text: D,

text: given

text: in

text: Table

text: ycneiciffE

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text: Table 4: Eight block designs for t = 2 strains arranged in b = 3 blocks of size k = 7. Treatments within a block are denoted by 1r12r2, where rh denotes the number of replicates of strain h, h = 1, 2. The blocks in designs D and D each have two diﬀerent combinations of replicates of strains 1 and 2, indicated by the 4 ×5 multiplier c in c 1r12r2, c = 1, 2. Design D D D D D D D D 1 2 3 4 5 6 7 8 × × 1621 1522 1423 2 1423 2 1324 1324 1225 1126 × × 1 1324 1 1423 19 ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Eight

text: block

text: designs

text: for

text: t

text: strains

text: arranged

text: in

text: b

text: blocks

text: of

text: size

text: k

text: Treatments

text: within

text: a

text: block

text: are

text: denoted

text: by

text: where

text: rh

text: denotes

text: the

text: number

text: of

text: replicates

text: of

text: strain

text: h,

text: h

text: The

text: blocks

text: in

text: designs

text: D

text: and

text: D

text: each

text: have

text: two

text: diﬀerent

text: combinations

text: of

text: replicates

text: of

text: strains

text: and

text: indicated

text: by

text: the

text: ×5

text: multiplier

text: c

text: in

text: c

text: c

text: Design

label: table

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: D

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