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Frequency fluctuations of nonlinear origin in self-sustained optical oscillators
29 January 2001
Physics Letters A 279 (2001) 154–162 www.elsevier.nl/locate/pla
Frequency fluctuations of nonlinear origin in self-sustained optical oscillators V.B. Braginsky ∗ , S.P. Vyatchanin Physics Faculty, Moscow State University, Moscow 119899, Russia Received 16 November 2000; accepted 27 November 2000 Communicated by V.M. Agranovich
Abstract We present the results obtained from the analysis of the laser frequency fluctuations caused by ponderomotive nonlinearity (the standard quantum limit (SQL) of frequency) and several kinds of thermorefractive and thermoexpansion effects. These effects are shown not to be small enough and have to be taken into account in high precision experiments (for example, in the laser gravitational antennae). 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The impressive achievements in quantum optical and spectroscopic experiments during the last two decades in many cases have its origin in the invention and implementation of self-sustained optical oscillators (lasers) with very small frequency fluctuations. The record high accuracy obtained in the comparison of the values of eigenfrequencies of two Fabri–Perot optical resonators performed in the gravitational wave antennae prototype [1,2], is reasonable to be mentioned as a good example of such an achievement. The possibility to detect the relative difference between the two eigenfrequencies at the level of 1ω0 /ω0 ' 3 × 10−19 with averaging time τ ' 10−2 s was demonstrated in this experiment. There is another “source” of achievements in this area of experimental physics: the steady rise of the optical mirrors finesse F . The obtained few years ago value of F ' 2 × 106 [3] per* Corresponding author.
E-mail address: [email protected] (V.B. Braginsky).
mits to realize in the table-top experiment an optical Fabri–Perot resonator with the eigenmode quality factor Q ' 1013 . Using √ this value of Q the Townes limit 1ω0 /ω0 ' 1/(Q N ) “permits” to measure the level of relative frequency fluctuation s ' 10−21 if only N = 1016 optical photons will be “spend”. It is worth noting that nobody has formulated any fundamental limit for the value of F yet. Recently, Gorodetsky together with the authors of this Letter [4,5] and Liu with Thorne [6] have analyzed the random deviations of the eigenfrequency ωr of Fabri–Perot resonator produced by thermodynamic fluctuations of temperature T and by the temperature fluctuations appearing due to the random absorption of optical photons in the resonator mirrors. These effects are in essence of nonlinear origin: namely, due to thermal expansion, characterized by coefficient α = (1/ l)(dl/dT ), and thermorefractivity (dependence of the refraction index n(T ) on temperature T ) which can be characterized by coefficient β = (1/n)(dn/dT ). The results presented in these articles show that the thermal fluctuations in the mirrors may be a serious obstacle when experimentalist wants to reach
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 8 2 4 - 0
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
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or even to overcome standard quantum limit (SQL) of frequency. The goal of this Letter is to present the results of the analysis of random fluctuations of the frequency in the output radiation of a selfsustained optical oscillator (laser) which are of similar nonlinear origin as in passive Fabri–Perot resonator. All estimates presented in text are given for parameters listed in Appendix A. 2. Analyzed model of laser For the analysis we have chosen the laser scheme in which the solid body acts as the amplifying part. The inversion of population in certain optical transition in this solid body is provided by the radiation of the nearby set of photodiodes. One of this laser types (based on Nd:YAG) has a very narrow line width, very high efficiency of photodiode-to-output radiation conversion and high mean output power W (several tens or even hundreds of Watts). The output radiation wavelength mean value in this laser type is λ = 1.06 µm [7,8]. This laser is being used now in the laser interferometer gravitational wave observatory (the LIGO project) [1,2,9] in the first stage (LIGO-I) and is planned to be used in the second stage (LIGO-II) when the value of W will be at the level of >100 W ' 1021 photon/s. The phase difference between oscillations in two Fabri–Perot resonators pumped by such a laser is expected to be measured with the error of '10−10 rad. The scheme of laser which we analyze is presented in Fig. 1: the resonator AB, containing solid state active media (shown by dashed line on the Fig. 1), is coupled with reference cavity CD. We assume that: 1. All the mirrors have no losses and the mirror A is ideally reflecting one. Thus the power W is irradiated only through the mirror D. 2. The mirrors A, B and the solid state active media are rigidly assembled and the length of solid state active media are equal to the distance between mirrors A and B. 3. The finesses F of the mirrors B, C and D are assumed to be equal for simplicity, but the length lAB of AB is much smaller than the length lCD of CD. Thus the resonator AB cavity decay rate (bandwidth) γAB is much larger than the resonator CD cavity decay rate γCD (γAB /γCD = lCD /2lAB 1).
Fig. 1. The laser scheme. Resonator formed by the mirrors A and B contains the active media shown by the dashed line. The reference cavity is formed by the mirrors C and D. The mirror A is fully reflecting thus the power is irradiated only through the mirror D.
4. The resonators AB and CD are optimally coupled and their mean frequencies ω0 coincide. Therefore the output frequency mean value is very close to the CD resonance frequency and the frequency random deviations have to be substantially reduced due to the high quality Q = ω/(2γCD ) of the CD. 5. The solid state active media have negligible losses near the operating wavelength λ, and we also assume the random distortions of this solid body due to Brownian motion in itself are possible not to be taken into account. Under these assumptions we can write down the general expression for the output radiation frequency fluctuations spectral density Sω (Ω) for the laser working far above the threshold: 1 Sω (Ω) S0ω (Ω) S1ω (Ω) = + , ω02 ω02 ω02 " # γCD 2 1 Ω 2 S0ω (Ω) h¯ ω0 ' + , W ω0 2 ω0 ω02 ( ) 1 lAB 2 S1ω (Ω) = 2 S1ωAB + S1ωCD . 2lCD ω02 ω0
(1) (2)
(3)
1 We use “one-sided” spectral density, defined only for positive frequencies, which may be calculated from correlation function hX(t)X(t + τ )i using formula
Z∞ SX (Ω) = 2 −∞
dτ hX(t)X(t + τ )i cos(Ωτ ).
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(See details of calculations in Appendix B.) Here Ω is the observation frequency. The first term S0ω (Ω) in (1) describes the fluctuations caused by vacuum fluctuations (penetrating through mirror D) and spontaneous emission in active media: in case Ω γCD it is practically the Townes formula in which the cavity decay rate γCD of reference cavity plays the decisive role and in opposite case when Ω γCD this term corresponds to the phase fluctuations 1ϕ 2 ' 1/(4N) of the wave irradiated through the mirror D where N = W τ/(h¯ ω0 ) is the mean number of photons irradiated during the time τ . The second term S1ω describes the resonators AB and CD frequency fluctuations: S1ωAB and S1ωCD are spectral densities of eigenfrequencies fluctuations of these resonators (the resonator AB eigenfrequency fluctuations contribution is suppressed by the factor lAB /2lCD ). The first term (S0ω (Ω)) gives the major limit which for parameters listed in Appendix A is of the order √ S0ω (Ω) 1 ' 1.1 × 10−21 √ . (4) ω0 Hz
in thermal equilibrium. The second ones are of quantum origin: the optical photons from the pumping photodiodes being absorbed in the laser solid body produces random local jumps of temperature — photothermal shot noise (SN). We assume that radius r0 of light beam in solid state active media is much smaller than cross dimensions of active media and for calculations one can expand the cross dimensions of media to infinity. Therefore we have the problem to calculate the fluctuations of temperature in cylindrical volume with radius r0 and length lAB in the infinite layer with the width lAB (the cylinder axis is perpendicular to the layer). The results of calculations for photo-thermal shot noise (noted by superscript SN) and the temperature thermodynamic fluctuations (noted by superscript TD) may be presented in the following form:
In the following sections we analyze several intrinsic effects which define the value of second term (S1ω (Ω)).
SN 1 lAB S1ωAB (Ω) ' 2.2 × 10−21 √ , 2lCD ω0 Hz q s TD (Ω) S1ω 4κT 2 λ∗ AB = , ω0 (Cρ)2 πr04 lAB Ω 2 q TD 1 βlAB S1ωAB (Ω) ' 1.8 × 10−20 √ . lCD ω0 Hz
3. Thermorefractive fluctuations in laser resonator Consider that the only sources of frequency fluctuations are the temperature fluctuations which together with nonzero β produce random changes of optical length in AB active media and thus produce the AB resonance frequency random changes. (In this section we assume that the positions of all the mirrors A, B, C, and D do not fluctuate.) These frequency fluctuations will be substantially reduced in accordance with formula (3) due to the assumed small ratio lAB / lCD value. In this section we do not consider the contribution of the temperature fluctuations through thermoexpansion (coefficient α) because these effects are relatively small due to the small values of lAB / lCD and α. There are two types of temperature fluctuations in the discussed scheme. The first ones are thermodynamic (TD) fluctuations (see details in [4,5] and references therein). These ones are of pure classical origin
q
SN (Ω) S1ω AB
ω0
s =β
h¯ ω0 W , 2 (Cρ) πr02 lAB V Ω 2
q
(5)
(6)
Here λ∗ is thermal conductivity, ρ is density and C is specific heat capacity of active media, κ is Boltzmann constant, T is absolute temperature, V is effective volume, W is absorbed power (all over below we assume power absorbed in active media is equal to optical power irradiated by laser for simplicity). Here we also use the adiabatic approximation Ω λ∗ /(ρCr02 ). For numerical estimates we used material parameters of nondoped Nd:YAG. Details of calculations are presented in Appendix C. We see that thermorefractive effect caused by the temperature TD fluctuations increases considerably the frequency fluctuations — its contribution is about 16 times larger than Townes limit (compare (6) and (4)).
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
4. Fluctuations of reference cavity 4.1. SQL of frequency The standard quantum limit (SQL) existence for self-sustained oscillator was predicted more that 20 years ago [12]. This limit origin is very simple: the output optical radiation “brings out” the information about the coordinate (in our scheme the frequency is linearly connected with the distance between the mirrors C and D). During the continuous coordinate measurement process the momentum of the masses (in our case they are the mirrors C and D) have to be inevitably perturbed. In other words, the Heisenberg uncertainty principle has to be fulfilled. The momentum perturbation origin is well known: these are the random “kicks” of optical photons on the mirrors. This effect in essence is nonlinear because it is created by the fluctuations of the ponderomotive pressure of the optical field inside the Fabri–Perot resonator. If the mirrors C and D can be regarded as free equal masses mC = mD (the eigenfrequency of the mechanical suspension is much smaller than the observation frequency Ω) then the frequency fluctuations SQL is equal to 2 q s SQL S1ω (Ω0 ) 8h¯ CD = 2 Ω2 ω0 mC lCD 0 1 ' 1.4 × 10−21 √ , (7) Hz which is achieved at the optimal output power Wopt : Wopt =
π 2 mC c02 Ω02 ' 12 W, 16ω0 F 2
(8)
where c0 is the velocity of light. These formulae are obtained in approximation γCD Ω, however for opposite case γCD Ω it differs by factor about unity only. It is important that frequency SQL (7) decreases when the distance lCD increases, whereas optimal 2 The similar AB ponderomotive pressure fluctuations are substantially smaller and were not taken into account in previous section because the mirrors A, B and active media had been assumed to be rigidly assembled, and the lowest mechanical eigenfrequency of such a set is much higher than Ω. The frequency fluctuations SQL of such a set (without reference cavity) was analyzed in [12].
157
power Wopt does not depend on the length lCD and depends on the mirror finesse F . It is worth to note that limit (7) is valid only for specific frequency Ω0 defining the optimal power (8). In general case for arbitrary frequency Ω and power W the frequency fluctuations are bigger: q PM (Ω) S1ω CD ω0 v u 4 ! u W W 4 h Ω0 opt ¯ + =t . 2 Ω2 W W mC lCD opt Ω 0 4.2. Mirrors C and D surface fluctuations Unfortunately, apart from mentioned sources of noise there are additional ones which change the effective distance lCD : 1. Thermodynamic temperature fluctuations cause the mirror surface fluctuations due to nonzero thermal expansion coefficient α. These fluctuations are also called as thermoelastic noise. 2. Thermal shot noise temperature fluctuations cause the mirror surface fluctuations due to nonzero thermal expansion coefficient α. 3. The noise associated with the mirrors’ material is inevitably related to the mechanical losses which can be originated by different physical mechanisms. We use the model of structural damping and denote it as Brownian motion of the surface. In this model the loss angle φ does not depend on frequency [10]. These effects are analyzed in details in [4] and we give only final formulae denoted by superscripts TD and B for thermodynamic fluctuations and Brownian fluctuations correspondingly (the thermal shot noise effect is substantially smaller and we do not present formula for it): TD (Ω) S1ω CD
ω02
16 α 2 κT 2 (1 + σ )2 λ∗ =√ , 2 2π (ρC)2 r03 Ω 2 lCD
fused silica: v u TD u S1ω (Ω) CD t ω02
1 ' 6.6 × 10−23 √ , Hz
(9)
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sapphire: v u TD u S1ω (Ω) CD t ω02
B (Ω) S1ω CD
ω02
1 ' 1.6 × 10−20 √ , Hz
8 κT (1 − σ 2 ) '√ φ, 2 2π r0 ΩElCD
(10)
Ta2 O5 –SiO2 we obtain the following estimates: q TR (Ω) S1ω 1 CD (TiO2 –SiO2 ), ' 1.4 × 10−23 √ ω0 Hz q TR (Ω) S1ω 1 CD ' 2.5 × 10−23 √ (Ta2 O5 –SiO2 ). ω0 Hz
fused silica: v u B u S1ω (Ω) 1 CD t ' 5.4 × 10−21 √ , 2 ω0 Hz
This effect seems to be small enough for presented parameters, however it has weak dependence q TR (Ω) ∼ √1 S1ω 4 CD Ω
sapphire: v u B u S1ω (Ω) CD t
on observation frequency Ω and can be significant for the frequencies above 1 kHz.
ω02
1 ' 5.4 × 10−22 √ . Hz
Here σ is the Poison ratio and E is the Young modulus. For thermodynamic fluctuations we also use adiabatic approximation Ω λ∗ /(ρCr02 ). The estimations are given for two kinds of mirror material: fused silica and sapphire. We see that these fluctuations are not small and may exceed the SQL fluctuations for chosen numerical parameters. 4.3. Thermorefractive noise in mirrors Thermodynamic temperature fluctuations together with photothermal shot noise also originate the thermorefractive fluctuations: the mirror coating optical layers effective refractive indexes fluctuations lead to these layers optical thickness fluctuations and hence to the phase noise in the reflected wave. The detailed calculations of these effects are given in [5] and below we write down only the final formulae for thermorefractive effect caused by thermodynamic temperature fluctuations (we do not present formula for thermorefractive shot noise because of its smaller numerical value): √ 2 2 2 TR (Ω) S1ω λ κT 2 2 βeff CD = 22 √ , (11) 2 ω0 πr0 lCD ΩρCλ∗ βeff =
n22 β1 + n21 β2 4(n21 − n22 )
,
λ=
ω0 . 2πc0
(12)
Here β1 and β2 are thermorefractive coefficients for layers with refractive indexes n1 and n2 correspondingly. For often used pairs of layers TiO2 –SiO2 and
5. Conclusion We see that for used parameters thermorefractive fluctuations in laser resonator makes the largest contribution into frequency instability. The frequency instability caused by the effects considered in this Letter is inversely proportional to distance lCD and therefore their negative influence can be suppressed by increasing the length of the reference cavity. The effects analyzed in this Letter can be analytically calculated. However, there are a lot of other processes responsible for additional frequency instability, which may provide substantial contribution especially within the band of observation frequency near 100 Hz. They are usually called as “1/f ” noise or excess noise. For example, as “the candidate” responsible for such process, we may point to the random jumps of vacancies or the birth of dislocations in the solid objects. Unfortunately, there are no reliable theoretical model for such processes which would allow to obtain analytical formulae and numerical estimates. Thus the presented in this Letter analysis may offer to experimentalists only the lower limits of the frequency instability. For real experiments this value has to be larger. In the same time none of the noise sources analyzed in this Letter may be called as the fundamental one. This statement is also correct for the frequency SQL: the ponderomotive nonlinearity (which is equivalent to positive cubic nonlinearity of optical material) was
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
emphasized in [13] and in principle can be compensated by a nonlinearity with opposite sign in solid. In this case the output radiation will “bring out” information not only about the coordinate, and the frequency deviation may be smaller. However, this potential possibility has not been seriously investigated yet. We also think that the analysis presented above may be useful for the final choice of the optimal LIGO topology. In the ideal case when both arms of the LIGO interferometer are identical the laser frequency fluctuations are completely subtracted in output signal. For real case the symmetry level of two interferometer arms may play an important role. The requirements for symmetry level became more tough when the power recycling is used as it was planned in the LIGO-II and the LIGO-III.
Acknowledgement This material is based upon work supported by the Russian Foundation for Basic Research under Grant #96-15-96780, the Russian Ministry of Science under State Grants #106-13/00 and #108-23/00, the National Science Foundation of USA under Grant No. PHY9800097.
159
Fused silica: C = 6.7 × 107 erg/(g K),
ρ = 2.2 g/cm3 ,
λ∗ = 1.4 × 105 erg cm/(s K), σ = 0.17,
E = 7.2 × 1011 erg/cm3 , φ = 5 × 10
−8
α = 5.5 × 10−7 K−1 ;
,
Sapphire: ρ = 4.0 g/cm3 ,
C = 7.9 × 106 erg/(g K), λ∗ = 4.0 × 106 erg cm/(s K),
σ = 0.29,
E = 4 × 1012 erg/cm3 , α = 5.0 × 10
−6
K
−1
φ = 3 × 10−9 .
,
The material parameters (C, ρ, λ∗ , α, β) are given for fused silica, sapphire [4], and nondoped YAG [11]. For estimates of the thermorefractive effect we used the following optical parameters for the mirror coatings layers: n1 = 2.2, n1 = 2.2, n2 = 1.45,
β1 = 4 × 10−5 K−1 β1 = 6 × 10
−5
K
β2 = 1.5 × 10
−1
−5
K
(TiO2 ), (Ta2 O5 ), −1
(SiO2 ).
Appendix B. Fluctuations of frequency Appendix A. Parameters For estimates we used the following parameters: ω0 = 2 × 1015 s−1 ,
mc = 10 kg,
T = 300 K, −1
Ω = 2π × 100 s lAB = 30 cm,
W = 10 W,
,
r0 = 0.5 cm,
lCD = 103 cm,
V = 2 × πr02 lAB ' 50 cm3 , πc γCD = ' 1.5 × 104 s−1 ; F = 3 × 103 , 2F lCD Nd:YAG: C = 1.4 × 107 erg/(g K), λ∗ = 1.4 × 106 erg cm/(s K), β = 0.7 × 10−5 K−1 ;
ρ = 4.55 g/cm3 ,
In this appendix we present the calculations of general formula (1) for frequency fluctuations. Below we use the complex amplitudes (A, B, C, D, shown in Fig. 1) related to the electric field E and the mean power W by the following equations: s h¯ ω0 −iω0 t e + h.c., E = A(t) Sc0 W=
Sc0 hE 2 i h¯ ω0 A2 = , 4π 2π
where S is the cross-section of the light beam, c0 is the velocity of light. Complex amplitude (for example, A(t)) is written below as a sum of the constant mean amplitude (denoted by capital letter with zero superscript) and the fluctuation part (denoted by small letter): A(t) = e−iω0 t A0 + a(t) + h.c.,
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Z∞ s a(t) = −∞
Ω 1+ a(Ω)e−iΩt dΩ, ω0
a(Ω), a(Ω 0) = 0, a(Ω), a +(Ω 0 ) = δ(Ω − Ω 0 ), a(t), a + (t 0 ) = 2πδ(t − t 0 ).
√ 1 − T ' 1: (B.1) ∂t a + (δ − γ ) a + a
r =
(B.2)
c0 γCD (d2 + b1 ) + iC 0 1ωmc , 2lCD √ √ b2 = c T − b1 , b1 = −b2 − a T , √ d1 = −d2 + c T . c˙ + γCD c =
Fourier transform of the output radiation phase comph ponent D1 (Ω) may be obtained after long but simple calculations: ph
(B.3)
Here δ and k are the constants describing negative nonlinear losses, esp is the additional noise caused by spontaneous emission in the active media. The last term describes the fluctuations of the resonator frequency. For the complex amplitude C (in the resonator CD) we have the following equation: r c0 γCD (D2 + B1 ) + iB3 1ωCD . (B.4) C˙ + γCD C = 2lCD These equations have to be supplemented by the boundary conditions on the mirrors B, C and D: √ √ B1 = −B2 1 − T + A T , √ √ B2 = −C T − B1 1 − T , √ √ D1 = −D2 1 − T + C T . Here T is the transparency coefficient (remind that the mirrors B, C and D have the same transparency). Now we consider every amplitude as a sum of the mean constant value and the small fluctuation part. For the mean amplitudes we have r δ−γ 0 0 , A =C = k √ D10 = B10 = A0 T ,
c0 γAB (b2 + esp ) lAB
+ iA0 1ωAB ,
r
We assume below that Ω ω0 and drop the term √ 1 + Ω/ω0 under the integral in (B.1). We write down the equation for the complex amplitude A describing the field within the resonator AB: A˙ + γ − δ + k|A|2 A r c0 γAB (B2 + esp ) + iA1ωAB. = lAB
+
D1 (Ω) =
2 (2γ Ω 2 (γCD + 2γAB ) + γCD AB + iΩ) (γCD − iΩ)(γCD + 2γAB) ph
D2 (Ω) 2γABγCD ph + Esp (Ω) (−iΩ) −iΩ(γCD + 2γAB) D10 1ωAB γCD + (−iΩ) (γCD + 2γAB) 21ωCD γAB , + (γCD + 2γAB) ×
d1 + d1+ , 2i + esp + esp ph . Esp = 2i ph
D1 =
ph
D2 =
d2 + d2+ , 2i
(B.5) ph
Assuming that γAB γCD the expression for D1 (Ω) can be simplified resulting in ph
D1 (Ω) '
i(γCD − iΩ) ph γCD ph D2 (Ω) + Esp (Ω) Ω −iΩ D10 γCD 1ωAB + 1ωCD . + (−iΩ) 2γAB ph
ph
Assume the spectral density SD2 (Ω) of D2 fluctuations corresponds to the vacuum fluctuations and is ph equal to the spectral density SEsp (Ω) of spontaneous ph
noise Esp :
B20 = 0.
2π
ph ph δ(t − t 0 ), D2 (t)D2 (t 0 ) = 4 Z∞
ph ph ph dt e−iΩt C3 (0)C3 (t) = π, SD2 (Ω) = 2
For the fluctuation components√we use linear approximation keeping the terms ∼ T and assuming
−∞ ph ph SEsp (Ω) = SD2 (Ω) = π.
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
Then the output radiation phase fluctuations spectral density can be obtained as follows: ph SD2 (Ω) |D 0 |2
Sϕ (Ω) =
(γCD /2γAB)2 S1ωAB + S1ωCD . Ω2 Using (B.6) formula (1) can be easy obtained.
0 wn ω, kE⊥ wn0 ω0 , kE⊥ = h¯ ω0 W0 2πδ(ω − ω0 )
2 − δ0n δnn0 lAB
0 × (2π)2 δ kE⊥ − kE⊥ .
hω0 γCD 2 = +1 2 2W Ω +
161
We are interested in the fluctuations of temperature u¯ averaged over the beam volume: ZlAB
(B.6) u(t) ¯ =
dz lAB
0
Z∞ Appendix C. Thermorefractive fluctuations in infinite layer In this appendix we present calculations for the laser resonator AB thermorefractive fluctuations formulae (5), (6).
u(ω) ¯ = −∞
Z∞ −∞
dxdy −(x 2 +y 2 )/r 2 0 u t, r e E, 2 πr0
dkx dky −(ky2 +kz2 )r 2 /4 w0 (ω, kE⊥ ) 0 , e 2 + iω) (2π)2 Cρ(a 2 k⊥
h¯ ω0 W0 u(ω) ¯ u(ω ¯ 0 ) = 2πδ(ω − ω0 ) (Cρ)2 lAB 2 2 Z∞ dkx dky e−k⊥ r0 /2 × . 4 + ω2 (2π)2 a 4 k⊥ −∞
C.1. Thermal shot noise For calculation of temperature u fluctuations in infinite layer with thermoisolated boundary that caused by shot noise fluctuations, we have the equation of thermal conductivity (for this case z-axis is perpendicular to the layer): Cρ∂t u − λ∗ 1u = w rE, t ,
∂z u|z=0 = 0, ∂z u|z=h = 0, w rE, t w rE0 , t 0 = h¯ ω0 W0 δ(t − t 0 )δ rE − rE0 .
(C.1) (C.2)
Here W0 is averaged power absorbed in a unit volume. We assume that W0 = W/V , where W is the total averaged power absorbed in whole volume V of the laser active media. We write the solution as a Fourierseries expansion:
Z∞
u t, rE⊥ , z = −∞
∞
dkx dky dω X cos(bn z) (2π)3 n=0
× eiωt −ikx x−iky y bn =
πn , lAB
a2 =
rE⊥ = eEx x + eEy y,
wn (ω, kE⊥ ) , Cρ(a 2 k 2 + iω)
λ∗ , Cρ kE⊥ = eEx kx + eEy ky ,
Using adiabatic approximation ω a 2 /r02 the term ∼a 4 k 4 in the last fraction denominator can be dropped. Then the expression for spectral density SuTS ¯ (ω) of averaged temperature u¯ will be the following: SuTS ¯ (ω) '
2h¯ ω0 W0 (CρlAB )2 ω2
Z∞ −∞
dkx dky −k 2 r 2 /2 e ⊥0 (2π)2
h¯ ω0 W = . 2 (Cρ) πr02 lAB V ω2
(C.3)
Formula (5) can be easily obtained from (C.3).
C.2. TD temperature fluctuations For the TD temperature u fluctuations calculation in infinite layer we use the Langevin approach, i.e., we add fluctuation forces into the thermal conductivity equation right-hand side (see details in [4]): Cρ∂t u − λ∗ 1u = F rE, t , ∂z u|z=0 = 0, ∂z u|z=lAB = 0, (C.4)
F rE, t F rE0 , t 0 = −2κT 2 λ∗ δ(t − t 0 )1δ rE − rE0 . (C.5)
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V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
We write the solution as a Fourier-series expansion: Z∞
u t, rE⊥ , z = −∞
∞
dkx dky dω X cos(bn z) (2π)3
bn =
πn , lAB
a2 =
rE⊥ = eEx x + eEy y,
SuTD ¯ (ω) '
n=0
× eiωt −ikx x−iky y
Fn (ω, kE⊥ ) , Cρ(a 2 k 2 + iω)
λ∗ , Cρ kE⊥ = eEx kx + eEy ky ,
(C.6)
0
(C.7)
We are interested in fluctuations of temperature u¯ averaged over the beam volume: u(t) ¯ = −lAB /2 Z∞
u(ω) ¯ = −∞
dz lAB
Z∞ −∞
Z∞
−∞ 2 4κT λ∗
2 k⊥ dk⊥ −k 2 r 2 /2 k⊥ e ⊥0 2π ω2
(Cρ)2 πr04 lAB ω2
.
(C.8)
References
2 − δ0n δnn0 = 2κT λ 2πδ(ω − ω ) lAB 0 2 × (2π)2 δ kE⊥ − kE⊥ bn2 + k⊥ .
lZ AB /2
=
4κT 2 λ∗ (Cρ)2 lAB
Formula (6) can be easily obtained from (C.8).
0 Fn ω, kE⊥ Fn0 ω0 , kE⊥ 2 ∗
Then the expression for spectral density SuTD ¯ (ω) of averaged temperature u¯ will be the following:
dxdy −(x 2+y 2 )/r 2 0 u t, r e E, 2 πr0
dkx dky −(ky2 +kz2 )r 2 /4 F0 (ω, kE⊥ ) 0 , e 2 + iω) (2π)2 Cρ(a 2 k⊥
2κT 2 λ∗ u(ω) ¯ u(ω ¯ 0 ) = 2πδ(ω − ω0 ) (Cρ)2 lAB 2 r 2 /2 Z∞ 2 e −k⊥ 0 dkx dky k⊥ × . 2 4 4 (2π) a k⊥ + ω2 −∞
Using adiabatic approximation ω a 2 /r02 the term ∼a 4 k 4 in the last fraction denominator can be dropped.
[1] A. Abramovici et al., Science 256 (1992) 325. [2] A. Abramovici et al., Phys. Lett. A 218 (1996) 157. [3] G. Rempe, R. Thompson, Y.J. Kimble, Opt. Lett. 17 (1992) 362. [4] V.B. Braginsky, M.L. Gorodetsky, S.P. Vyatchanin, Phys. Lett. A 264 (1999) 1, cond-mat/9912139. [5] V.B. Braginsky, M.L. Gorodetsky, S.P. Vyatchanin, Phys. Lett. A 271 (2000) 303. [6] Yu.T. Liu, K.S. Thorne, Phys. Rev. D, submitted. [7] B. Zhou, T.J. Kane, G.J. Dixon, R.L. Byer, Opt. Lett. 10 (1985) 62. [8] T.J. Kane, R.L. Byer, Opt. Lett. 10 (1985) 65. [9] W. Wiechnann, T.J. Kane, D. Haserot, F. Adams, G. Tuong, J.D. Kmetec, in: Proceedings of CLEO ’98, 1998, p. 432. [10] P.R. Saulson, Phys. Rev. D 42 (1990) 2437; G.I. Gonzalez, P.R. Saulson, J. Acoust. Soc. Am. 96 (1994) 207. [11] F. Bondu, Report on Aspen Winter Conference on Gravitational Waves and Their Detection, February 20–26, 2000. [12] V.B. Braginsky, S.P. Vyatchanin, Zh. Eksp. Teor. Fiz. 74 (1978) 828, English translation: Sov. Phys. JETP 47 (1978) 433. [13] V.B. Braginsky, V.I. Panov, S.P. Vyatchanin, Dokl. Akad. Nauk SSSR 247 (1979) 583, English translation: Phys. Dokl. 24 (7) (1979).
Physics Letters A 279 (2001) 154–162 www.elsevier.nl/locate/pla
Frequency fluctuations of nonlinear origin in self-sustained optical oscillators V.B. Braginsky ∗ , S.P. Vyatchanin Physics Faculty, Moscow State University, Moscow 119899, Russia Received 16 November 2000; accepted 27 November 2000 Communicated by V.M. Agranovich
Abstract We present the results obtained from the analysis of the laser frequency fluctuations caused by ponderomotive nonlinearity (the standard quantum limit (SQL) of frequency) and several kinds of thermorefractive and thermoexpansion effects. These effects are shown not to be small enough and have to be taken into account in high precision experiments (for example, in the laser gravitational antennae). 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The impressive achievements in quantum optical and spectroscopic experiments during the last two decades in many cases have its origin in the invention and implementation of self-sustained optical oscillators (lasers) with very small frequency fluctuations. The record high accuracy obtained in the comparison of the values of eigenfrequencies of two Fabri–Perot optical resonators performed in the gravitational wave antennae prototype [1,2], is reasonable to be mentioned as a good example of such an achievement. The possibility to detect the relative difference between the two eigenfrequencies at the level of 1ω0 /ω0 ' 3 × 10−19 with averaging time τ ' 10−2 s was demonstrated in this experiment. There is another “source” of achievements in this area of experimental physics: the steady rise of the optical mirrors finesse F . The obtained few years ago value of F ' 2 × 106 [3] per* Corresponding author.
E-mail address: [email protected] (V.B. Braginsky).
mits to realize in the table-top experiment an optical Fabri–Perot resonator with the eigenmode quality factor Q ' 1013 . Using √ this value of Q the Townes limit 1ω0 /ω0 ' 1/(Q N ) “permits” to measure the level of relative frequency fluctuation s ' 10−21 if only N = 1016 optical photons will be “spend”. It is worth noting that nobody has formulated any fundamental limit for the value of F yet. Recently, Gorodetsky together with the authors of this Letter [4,5] and Liu with Thorne [6] have analyzed the random deviations of the eigenfrequency ωr of Fabri–Perot resonator produced by thermodynamic fluctuations of temperature T and by the temperature fluctuations appearing due to the random absorption of optical photons in the resonator mirrors. These effects are in essence of nonlinear origin: namely, due to thermal expansion, characterized by coefficient α = (1/ l)(dl/dT ), and thermorefractivity (dependence of the refraction index n(T ) on temperature T ) which can be characterized by coefficient β = (1/n)(dn/dT ). The results presented in these articles show that the thermal fluctuations in the mirrors may be a serious obstacle when experimentalist wants to reach
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 8 2 4 - 0
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
155
or even to overcome standard quantum limit (SQL) of frequency. The goal of this Letter is to present the results of the analysis of random fluctuations of the frequency in the output radiation of a selfsustained optical oscillator (laser) which are of similar nonlinear origin as in passive Fabri–Perot resonator. All estimates presented in text are given for parameters listed in Appendix A. 2. Analyzed model of laser For the analysis we have chosen the laser scheme in which the solid body acts as the amplifying part. The inversion of population in certain optical transition in this solid body is provided by the radiation of the nearby set of photodiodes. One of this laser types (based on Nd:YAG) has a very narrow line width, very high efficiency of photodiode-to-output radiation conversion and high mean output power W (several tens or even hundreds of Watts). The output radiation wavelength mean value in this laser type is λ = 1.06 µm [7,8]. This laser is being used now in the laser interferometer gravitational wave observatory (the LIGO project) [1,2,9] in the first stage (LIGO-I) and is planned to be used in the second stage (LIGO-II) when the value of W will be at the level of >100 W ' 1021 photon/s. The phase difference between oscillations in two Fabri–Perot resonators pumped by such a laser is expected to be measured with the error of '10−10 rad. The scheme of laser which we analyze is presented in Fig. 1: the resonator AB, containing solid state active media (shown by dashed line on the Fig. 1), is coupled with reference cavity CD. We assume that: 1. All the mirrors have no losses and the mirror A is ideally reflecting one. Thus the power W is irradiated only through the mirror D. 2. The mirrors A, B and the solid state active media are rigidly assembled and the length of solid state active media are equal to the distance between mirrors A and B. 3. The finesses F of the mirrors B, C and D are assumed to be equal for simplicity, but the length lAB of AB is much smaller than the length lCD of CD. Thus the resonator AB cavity decay rate (bandwidth) γAB is much larger than the resonator CD cavity decay rate γCD (γAB /γCD = lCD /2lAB 1).
Fig. 1. The laser scheme. Resonator formed by the mirrors A and B contains the active media shown by the dashed line. The reference cavity is formed by the mirrors C and D. The mirror A is fully reflecting thus the power is irradiated only through the mirror D.
4. The resonators AB and CD are optimally coupled and their mean frequencies ω0 coincide. Therefore the output frequency mean value is very close to the CD resonance frequency and the frequency random deviations have to be substantially reduced due to the high quality Q = ω/(2γCD ) of the CD. 5. The solid state active media have negligible losses near the operating wavelength λ, and we also assume the random distortions of this solid body due to Brownian motion in itself are possible not to be taken into account. Under these assumptions we can write down the general expression for the output radiation frequency fluctuations spectral density Sω (Ω) for the laser working far above the threshold: 1 Sω (Ω) S0ω (Ω) S1ω (Ω) = + , ω02 ω02 ω02 " # γCD 2 1 Ω 2 S0ω (Ω) h¯ ω0 ' + , W ω0 2 ω0 ω02 ( ) 1 lAB 2 S1ω (Ω) = 2 S1ωAB + S1ωCD . 2lCD ω02 ω0
(1) (2)
(3)
1 We use “one-sided” spectral density, defined only for positive frequencies, which may be calculated from correlation function hX(t)X(t + τ )i using formula
Z∞ SX (Ω) = 2 −∞
dτ hX(t)X(t + τ )i cos(Ωτ ).
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(See details of calculations in Appendix B.) Here Ω is the observation frequency. The first term S0ω (Ω) in (1) describes the fluctuations caused by vacuum fluctuations (penetrating through mirror D) and spontaneous emission in active media: in case Ω γCD it is practically the Townes formula in which the cavity decay rate γCD of reference cavity plays the decisive role and in opposite case when Ω γCD this term corresponds to the phase fluctuations 1ϕ 2 ' 1/(4N) of the wave irradiated through the mirror D where N = W τ/(h¯ ω0 ) is the mean number of photons irradiated during the time τ . The second term S1ω describes the resonators AB and CD frequency fluctuations: S1ωAB and S1ωCD are spectral densities of eigenfrequencies fluctuations of these resonators (the resonator AB eigenfrequency fluctuations contribution is suppressed by the factor lAB /2lCD ). The first term (S0ω (Ω)) gives the major limit which for parameters listed in Appendix A is of the order √ S0ω (Ω) 1 ' 1.1 × 10−21 √ . (4) ω0 Hz
in thermal equilibrium. The second ones are of quantum origin: the optical photons from the pumping photodiodes being absorbed in the laser solid body produces random local jumps of temperature — photothermal shot noise (SN). We assume that radius r0 of light beam in solid state active media is much smaller than cross dimensions of active media and for calculations one can expand the cross dimensions of media to infinity. Therefore we have the problem to calculate the fluctuations of temperature in cylindrical volume with radius r0 and length lAB in the infinite layer with the width lAB (the cylinder axis is perpendicular to the layer). The results of calculations for photo-thermal shot noise (noted by superscript SN) and the temperature thermodynamic fluctuations (noted by superscript TD) may be presented in the following form:
In the following sections we analyze several intrinsic effects which define the value of second term (S1ω (Ω)).
SN 1 lAB S1ωAB (Ω) ' 2.2 × 10−21 √ , 2lCD ω0 Hz q s TD (Ω) S1ω 4κT 2 λ∗ AB = , ω0 (Cρ)2 πr04 lAB Ω 2 q TD 1 βlAB S1ωAB (Ω) ' 1.8 × 10−20 √ . lCD ω0 Hz
3. Thermorefractive fluctuations in laser resonator Consider that the only sources of frequency fluctuations are the temperature fluctuations which together with nonzero β produce random changes of optical length in AB active media and thus produce the AB resonance frequency random changes. (In this section we assume that the positions of all the mirrors A, B, C, and D do not fluctuate.) These frequency fluctuations will be substantially reduced in accordance with formula (3) due to the assumed small ratio lAB / lCD value. In this section we do not consider the contribution of the temperature fluctuations through thermoexpansion (coefficient α) because these effects are relatively small due to the small values of lAB / lCD and α. There are two types of temperature fluctuations in the discussed scheme. The first ones are thermodynamic (TD) fluctuations (see details in [4,5] and references therein). These ones are of pure classical origin
q
SN (Ω) S1ω AB
ω0
s =β
h¯ ω0 W , 2 (Cρ) πr02 lAB V Ω 2
q
(5)
(6)
Here λ∗ is thermal conductivity, ρ is density and C is specific heat capacity of active media, κ is Boltzmann constant, T is absolute temperature, V is effective volume, W is absorbed power (all over below we assume power absorbed in active media is equal to optical power irradiated by laser for simplicity). Here we also use the adiabatic approximation Ω λ∗ /(ρCr02 ). For numerical estimates we used material parameters of nondoped Nd:YAG. Details of calculations are presented in Appendix C. We see that thermorefractive effect caused by the temperature TD fluctuations increases considerably the frequency fluctuations — its contribution is about 16 times larger than Townes limit (compare (6) and (4)).
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
4. Fluctuations of reference cavity 4.1. SQL of frequency The standard quantum limit (SQL) existence for self-sustained oscillator was predicted more that 20 years ago [12]. This limit origin is very simple: the output optical radiation “brings out” the information about the coordinate (in our scheme the frequency is linearly connected with the distance between the mirrors C and D). During the continuous coordinate measurement process the momentum of the masses (in our case they are the mirrors C and D) have to be inevitably perturbed. In other words, the Heisenberg uncertainty principle has to be fulfilled. The momentum perturbation origin is well known: these are the random “kicks” of optical photons on the mirrors. This effect in essence is nonlinear because it is created by the fluctuations of the ponderomotive pressure of the optical field inside the Fabri–Perot resonator. If the mirrors C and D can be regarded as free equal masses mC = mD (the eigenfrequency of the mechanical suspension is much smaller than the observation frequency Ω) then the frequency fluctuations SQL is equal to 2 q s SQL S1ω (Ω0 ) 8h¯ CD = 2 Ω2 ω0 mC lCD 0 1 ' 1.4 × 10−21 √ , (7) Hz which is achieved at the optimal output power Wopt : Wopt =
π 2 mC c02 Ω02 ' 12 W, 16ω0 F 2
(8)
where c0 is the velocity of light. These formulae are obtained in approximation γCD Ω, however for opposite case γCD Ω it differs by factor about unity only. It is important that frequency SQL (7) decreases when the distance lCD increases, whereas optimal 2 The similar AB ponderomotive pressure fluctuations are substantially smaller and were not taken into account in previous section because the mirrors A, B and active media had been assumed to be rigidly assembled, and the lowest mechanical eigenfrequency of such a set is much higher than Ω. The frequency fluctuations SQL of such a set (without reference cavity) was analyzed in [12].
157
power Wopt does not depend on the length lCD and depends on the mirror finesse F . It is worth to note that limit (7) is valid only for specific frequency Ω0 defining the optimal power (8). In general case for arbitrary frequency Ω and power W the frequency fluctuations are bigger: q PM (Ω) S1ω CD ω0 v u 4 ! u W W 4 h Ω0 opt ¯ + =t . 2 Ω2 W W mC lCD opt Ω 0 4.2. Mirrors C and D surface fluctuations Unfortunately, apart from mentioned sources of noise there are additional ones which change the effective distance lCD : 1. Thermodynamic temperature fluctuations cause the mirror surface fluctuations due to nonzero thermal expansion coefficient α. These fluctuations are also called as thermoelastic noise. 2. Thermal shot noise temperature fluctuations cause the mirror surface fluctuations due to nonzero thermal expansion coefficient α. 3. The noise associated with the mirrors’ material is inevitably related to the mechanical losses which can be originated by different physical mechanisms. We use the model of structural damping and denote it as Brownian motion of the surface. In this model the loss angle φ does not depend on frequency [10]. These effects are analyzed in details in [4] and we give only final formulae denoted by superscripts TD and B for thermodynamic fluctuations and Brownian fluctuations correspondingly (the thermal shot noise effect is substantially smaller and we do not present formula for it): TD (Ω) S1ω CD
ω02
16 α 2 κT 2 (1 + σ )2 λ∗ =√ , 2 2π (ρC)2 r03 Ω 2 lCD
fused silica: v u TD u S1ω (Ω) CD t ω02
1 ' 6.6 × 10−23 √ , Hz
(9)
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sapphire: v u TD u S1ω (Ω) CD t ω02
B (Ω) S1ω CD
ω02
1 ' 1.6 × 10−20 √ , Hz
8 κT (1 − σ 2 ) '√ φ, 2 2π r0 ΩElCD
(10)
Ta2 O5 –SiO2 we obtain the following estimates: q TR (Ω) S1ω 1 CD (TiO2 –SiO2 ), ' 1.4 × 10−23 √ ω0 Hz q TR (Ω) S1ω 1 CD ' 2.5 × 10−23 √ (Ta2 O5 –SiO2 ). ω0 Hz
fused silica: v u B u S1ω (Ω) 1 CD t ' 5.4 × 10−21 √ , 2 ω0 Hz
This effect seems to be small enough for presented parameters, however it has weak dependence q TR (Ω) ∼ √1 S1ω 4 CD Ω
sapphire: v u B u S1ω (Ω) CD t
on observation frequency Ω and can be significant for the frequencies above 1 kHz.
ω02
1 ' 5.4 × 10−22 √ . Hz
Here σ is the Poison ratio and E is the Young modulus. For thermodynamic fluctuations we also use adiabatic approximation Ω λ∗ /(ρCr02 ). The estimations are given for two kinds of mirror material: fused silica and sapphire. We see that these fluctuations are not small and may exceed the SQL fluctuations for chosen numerical parameters. 4.3. Thermorefractive noise in mirrors Thermodynamic temperature fluctuations together with photothermal shot noise also originate the thermorefractive fluctuations: the mirror coating optical layers effective refractive indexes fluctuations lead to these layers optical thickness fluctuations and hence to the phase noise in the reflected wave. The detailed calculations of these effects are given in [5] and below we write down only the final formulae for thermorefractive effect caused by thermodynamic temperature fluctuations (we do not present formula for thermorefractive shot noise because of its smaller numerical value): √ 2 2 2 TR (Ω) S1ω λ κT 2 2 βeff CD = 22 √ , (11) 2 ω0 πr0 lCD ΩρCλ∗ βeff =
n22 β1 + n21 β2 4(n21 − n22 )
,
λ=
ω0 . 2πc0
(12)
Here β1 and β2 are thermorefractive coefficients for layers with refractive indexes n1 and n2 correspondingly. For often used pairs of layers TiO2 –SiO2 and
5. Conclusion We see that for used parameters thermorefractive fluctuations in laser resonator makes the largest contribution into frequency instability. The frequency instability caused by the effects considered in this Letter is inversely proportional to distance lCD and therefore their negative influence can be suppressed by increasing the length of the reference cavity. The effects analyzed in this Letter can be analytically calculated. However, there are a lot of other processes responsible for additional frequency instability, which may provide substantial contribution especially within the band of observation frequency near 100 Hz. They are usually called as “1/f ” noise or excess noise. For example, as “the candidate” responsible for such process, we may point to the random jumps of vacancies or the birth of dislocations in the solid objects. Unfortunately, there are no reliable theoretical model for such processes which would allow to obtain analytical formulae and numerical estimates. Thus the presented in this Letter analysis may offer to experimentalists only the lower limits of the frequency instability. For real experiments this value has to be larger. In the same time none of the noise sources analyzed in this Letter may be called as the fundamental one. This statement is also correct for the frequency SQL: the ponderomotive nonlinearity (which is equivalent to positive cubic nonlinearity of optical material) was
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
emphasized in [13] and in principle can be compensated by a nonlinearity with opposite sign in solid. In this case the output radiation will “bring out” information not only about the coordinate, and the frequency deviation may be smaller. However, this potential possibility has not been seriously investigated yet. We also think that the analysis presented above may be useful for the final choice of the optimal LIGO topology. In the ideal case when both arms of the LIGO interferometer are identical the laser frequency fluctuations are completely subtracted in output signal. For real case the symmetry level of two interferometer arms may play an important role. The requirements for symmetry level became more tough when the power recycling is used as it was planned in the LIGO-II and the LIGO-III.
Acknowledgement This material is based upon work supported by the Russian Foundation for Basic Research under Grant #96-15-96780, the Russian Ministry of Science under State Grants #106-13/00 and #108-23/00, the National Science Foundation of USA under Grant No. PHY9800097.
159
Fused silica: C = 6.7 × 107 erg/(g K),
ρ = 2.2 g/cm3 ,
λ∗ = 1.4 × 105 erg cm/(s K), σ = 0.17,
E = 7.2 × 1011 erg/cm3 , φ = 5 × 10
−8
α = 5.5 × 10−7 K−1 ;
,
Sapphire: ρ = 4.0 g/cm3 ,
C = 7.9 × 106 erg/(g K), λ∗ = 4.0 × 106 erg cm/(s K),
σ = 0.29,
E = 4 × 1012 erg/cm3 , α = 5.0 × 10
−6
K
−1
φ = 3 × 10−9 .
,
The material parameters (C, ρ, λ∗ , α, β) are given for fused silica, sapphire [4], and nondoped YAG [11]. For estimates of the thermorefractive effect we used the following optical parameters for the mirror coatings layers: n1 = 2.2, n1 = 2.2, n2 = 1.45,
β1 = 4 × 10−5 K−1 β1 = 6 × 10
−5
K
β2 = 1.5 × 10
−1
−5
K
(TiO2 ), (Ta2 O5 ), −1
(SiO2 ).
Appendix B. Fluctuations of frequency Appendix A. Parameters For estimates we used the following parameters: ω0 = 2 × 1015 s−1 ,
mc = 10 kg,
T = 300 K, −1
Ω = 2π × 100 s lAB = 30 cm,
W = 10 W,
,
r0 = 0.5 cm,
lCD = 103 cm,
V = 2 × πr02 lAB ' 50 cm3 , πc γCD = ' 1.5 × 104 s−1 ; F = 3 × 103 , 2F lCD Nd:YAG: C = 1.4 × 107 erg/(g K), λ∗ = 1.4 × 106 erg cm/(s K), β = 0.7 × 10−5 K−1 ;
ρ = 4.55 g/cm3 ,
In this appendix we present the calculations of general formula (1) for frequency fluctuations. Below we use the complex amplitudes (A, B, C, D, shown in Fig. 1) related to the electric field E and the mean power W by the following equations: s h¯ ω0 −iω0 t e + h.c., E = A(t) Sc0 W=
Sc0 hE 2 i h¯ ω0 A2 = , 4π 2π
where S is the cross-section of the light beam, c0 is the velocity of light. Complex amplitude (for example, A(t)) is written below as a sum of the constant mean amplitude (denoted by capital letter with zero superscript) and the fluctuation part (denoted by small letter): A(t) = e−iω0 t A0 + a(t) + h.c.,
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Z∞ s a(t) = −∞
Ω 1+ a(Ω)e−iΩt dΩ, ω0
a(Ω), a(Ω 0) = 0, a(Ω), a +(Ω 0 ) = δ(Ω − Ω 0 ), a(t), a + (t 0 ) = 2πδ(t − t 0 ).
√ 1 − T ' 1: (B.1) ∂t a + (δ − γ ) a + a
r =
(B.2)
c0 γCD (d2 + b1 ) + iC 0 1ωmc , 2lCD √ √ b2 = c T − b1 , b1 = −b2 − a T , √ d1 = −d2 + c T . c˙ + γCD c =
Fourier transform of the output radiation phase comph ponent D1 (Ω) may be obtained after long but simple calculations: ph
(B.3)
Here δ and k are the constants describing negative nonlinear losses, esp is the additional noise caused by spontaneous emission in the active media. The last term describes the fluctuations of the resonator frequency. For the complex amplitude C (in the resonator CD) we have the following equation: r c0 γCD (D2 + B1 ) + iB3 1ωCD . (B.4) C˙ + γCD C = 2lCD These equations have to be supplemented by the boundary conditions on the mirrors B, C and D: √ √ B1 = −B2 1 − T + A T , √ √ B2 = −C T − B1 1 − T , √ √ D1 = −D2 1 − T + C T . Here T is the transparency coefficient (remind that the mirrors B, C and D have the same transparency). Now we consider every amplitude as a sum of the mean constant value and the small fluctuation part. For the mean amplitudes we have r δ−γ 0 0 , A =C = k √ D10 = B10 = A0 T ,
c0 γAB (b2 + esp ) lAB
+ iA0 1ωAB ,
r
We assume below that Ω ω0 and drop the term √ 1 + Ω/ω0 under the integral in (B.1). We write down the equation for the complex amplitude A describing the field within the resonator AB: A˙ + γ − δ + k|A|2 A r c0 γAB (B2 + esp ) + iA1ωAB. = lAB
+
D1 (Ω) =
2 (2γ Ω 2 (γCD + 2γAB ) + γCD AB + iΩ) (γCD − iΩ)(γCD + 2γAB) ph
D2 (Ω) 2γABγCD ph + Esp (Ω) (−iΩ) −iΩ(γCD + 2γAB) D10 1ωAB γCD + (−iΩ) (γCD + 2γAB) 21ωCD γAB , + (γCD + 2γAB) ×
d1 + d1+ , 2i + esp + esp ph . Esp = 2i ph
D1 =
ph
D2 =
d2 + d2+ , 2i
(B.5) ph
Assuming that γAB γCD the expression for D1 (Ω) can be simplified resulting in ph
D1 (Ω) '
i(γCD − iΩ) ph γCD ph D2 (Ω) + Esp (Ω) Ω −iΩ D10 γCD 1ωAB + 1ωCD . + (−iΩ) 2γAB ph
ph
Assume the spectral density SD2 (Ω) of D2 fluctuations corresponds to the vacuum fluctuations and is ph equal to the spectral density SEsp (Ω) of spontaneous ph
noise Esp :
B20 = 0.
2π
ph ph δ(t − t 0 ), D2 (t)D2 (t 0 ) = 4 Z∞
ph ph ph dt e−iΩt C3 (0)C3 (t) = π, SD2 (Ω) = 2
For the fluctuation components√we use linear approximation keeping the terms ∼ T and assuming
−∞ ph ph SEsp (Ω) = SD2 (Ω) = π.
V.B. Braginsky, S.P. Vyatchanin / Physics Letters A 279 (2001) 154–162
Then the output radiation phase fluctuations spectral density can be obtained as follows: ph SD2 (Ω) |D 0 |2
Sϕ (Ω) =
(γCD /2γAB)2 S1ωAB + S1ωCD . Ω2 Using (B.6) formula (1) can be easy obtained.
0 wn ω, kE⊥ wn0 ω0 , kE⊥ = h¯ ω0 W0 2πδ(ω − ω0 )
2 − δ0n δnn0 lAB
0 × (2π)2 δ kE⊥ − kE⊥ .
hω0 γCD 2 = +1 2 2W Ω +
161
We are interested in the fluctuations of temperature u¯ averaged over the beam volume: ZlAB
(B.6) u(t) ¯ =
dz lAB
0
Z∞ Appendix C. Thermorefractive fluctuations in infinite layer In this appendix we present calculations for the laser resonator AB thermorefractive fluctuations formulae (5), (6).
u(ω) ¯ = −∞
Z∞ −∞
dxdy −(x 2 +y 2 )/r 2 0 u t, r e E, 2 πr0
dkx dky −(ky2 +kz2 )r 2 /4 w0 (ω, kE⊥ ) 0 , e 2 + iω) (2π)2 Cρ(a 2 k⊥
h¯ ω0 W0 u(ω) ¯ u(ω ¯ 0 ) = 2πδ(ω − ω0 ) (Cρ)2 lAB 2 2 Z∞ dkx dky e−k⊥ r0 /2 × . 4 + ω2 (2π)2 a 4 k⊥ −∞
C.1. Thermal shot noise For calculation of temperature u fluctuations in infinite layer with thermoisolated boundary that caused by shot noise fluctuations, we have the equation of thermal conductivity (for this case z-axis is perpendicular to the layer): Cρ∂t u − λ∗ 1u = w rE, t ,
∂z u|z=0 = 0, ∂z u|z=h = 0, w rE, t w rE0 , t 0 = h¯ ω0 W0 δ(t − t 0 )δ rE − rE0 .
(C.1) (C.2)
Here W0 is averaged power absorbed in a unit volume. We assume that W0 = W/V , where W is the total averaged power absorbed in whole volume V of the laser active media. We write the solution as a Fourierseries expansion:
Z∞
u t, rE⊥ , z = −∞
∞
dkx dky dω X cos(bn z) (2π)3 n=0
× eiωt −ikx x−iky y bn =
πn , lAB
a2 =
rE⊥ = eEx x + eEy y,
wn (ω, kE⊥ ) , Cρ(a 2 k 2 + iω)
λ∗ , Cρ kE⊥ = eEx kx + eEy ky ,
Using adiabatic approximation ω a 2 /r02 the term ∼a 4 k 4 in the last fraction denominator can be dropped. Then the expression for spectral density SuTS ¯ (ω) of averaged temperature u¯ will be the following: SuTS ¯ (ω) '
2h¯ ω0 W0 (CρlAB )2 ω2
Z∞ −∞
dkx dky −k 2 r 2 /2 e ⊥0 (2π)2
h¯ ω0 W = . 2 (Cρ) πr02 lAB V ω2
(C.3)
Formula (5) can be easily obtained from (C.3).
C.2. TD temperature fluctuations For the TD temperature u fluctuations calculation in infinite layer we use the Langevin approach, i.e., we add fluctuation forces into the thermal conductivity equation right-hand side (see details in [4]): Cρ∂t u − λ∗ 1u = F rE, t , ∂z u|z=0 = 0, ∂z u|z=lAB = 0, (C.4)
F rE, t F rE0 , t 0 = −2κT 2 λ∗ δ(t − t 0 )1δ rE − rE0 . (C.5)
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We write the solution as a Fourier-series expansion: Z∞
u t, rE⊥ , z = −∞
∞
dkx dky dω X cos(bn z) (2π)3
bn =
πn , lAB
a2 =
rE⊥ = eEx x + eEy y,
SuTD ¯ (ω) '
n=0
× eiωt −ikx x−iky y
Fn (ω, kE⊥ ) , Cρ(a 2 k 2 + iω)
λ∗ , Cρ kE⊥ = eEx kx + eEy ky ,
(C.6)
0
(C.7)
We are interested in fluctuations of temperature u¯ averaged over the beam volume: u(t) ¯ = −lAB /2 Z∞
u(ω) ¯ = −∞
dz lAB
Z∞ −∞
Z∞
−∞ 2 4κT λ∗
2 k⊥ dk⊥ −k 2 r 2 /2 k⊥ e ⊥0 2π ω2
(Cρ)2 πr04 lAB ω2
.
(C.8)
References
2 − δ0n δnn0 = 2κT λ 2πδ(ω − ω ) lAB 0 2 × (2π)2 δ kE⊥ − kE⊥ bn2 + k⊥ .
lZ AB /2
=
4κT 2 λ∗ (Cρ)2 lAB
Formula (6) can be easily obtained from (C.8).
0 Fn ω, kE⊥ Fn0 ω0 , kE⊥ 2 ∗
Then the expression for spectral density SuTD ¯ (ω) of averaged temperature u¯ will be the following:
dxdy −(x 2+y 2 )/r 2 0 u t, r e E, 2 πr0
dkx dky −(ky2 +kz2 )r 2 /4 F0 (ω, kE⊥ ) 0 , e 2 + iω) (2π)2 Cρ(a 2 k⊥
2κT 2 λ∗ u(ω) ¯ u(ω ¯ 0 ) = 2πδ(ω − ω0 ) (Cρ)2 lAB 2 r 2 /2 Z∞ 2 e −k⊥ 0 dkx dky k⊥ × . 2 4 4 (2π) a k⊥ + ω2 −∞
Using adiabatic approximation ω a 2 /r02 the term ∼a 4 k 4 in the last fraction denominator can be dropped.
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