Abstract

We derive the behavior of the average exit time (i.e., the number of reflections before escape) of a ray path traveling between two perfect mirrors subject to dynamic random-tilt aberrations. Our calculation is performed in the paraxial approximation. When small random tilts are taken into account, we may consider an asymptotic regime that generically reduces the problem to the study of the exit time from an interval for a harmonic, frictionless oscillator driven by Gaussian white noise. Despite its apparent simplicity, the exact solution of this problem remains an open mathematical challenge, and we propose here a simple approximation scheme. For flat mirrors, the natural frequency of the oscillator vanishes, and, in this case, the average exit time is known exactly. It exhibits a 2/3 scaling-law behavior in terms of the variance of the random tilts. This behavior also follows from our approximation scheme, which establishes the consistency of the scaling law. Our mathematical results are confirmed with simulation experiments.

© 2003 Optical Society of America

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References

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  1. V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics, Vol. PM45 of SPIE Monographs (SPIE Press, Bellingham, Wash., 1998), Chap. 7.
  2. A. J. F. Siegert, “On the first passage time probability problem,” Phys. Rev. 81, 617–623 (1951).
    [CrossRef]
  3. H. P. McKean, “A winding problem for a resonator driven by a white noise,” J. Math. Kyoto Univ. 2, 227–235 (1963).
  4. S. H. Crandall, “First-crossing probabilities of the linear oscillator,” J. Sound Vib. 12, 282–299 (1970).
    [CrossRef]
  5. J. B. Roberts, “First passage time for the envelope of a randomly excited linear oscillator,” J. Sound Vib. 46, 1–14 (1976).
    [CrossRef]
  6. J. B. Roberts, “First passage time for randomly excited non-linear oscillators,” J. Sound Vib. 109, 31–50 (1986).
    [CrossRef]
  7. J. N. Franklin, E. R. Rodemich, “Numerical analysis of an elliptic–parabolic partial differential equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 680–716 (1968).
    [CrossRef]
  8. J. Masoliver, J. P. Porra, “Exact solution to the exit-time problem for free particle driven by Gaussian white noise,” Phys. Rev. E 53, 2243–2256 (1996).
    [CrossRef]
  9. A. Lachal, “Sur le premier instant de passage de l’intégrale du mouvement Brownien,” Ann. Inst. Henri Poincaré Sect. B 27, 385–405 (1991).
  10. A. Lachal, “Temps de sortie d’un intervalle borné pour l’intégrale du mouvement Brownien,” C. R. Acad. Sci. Ser. I: Math. 324, 559–564 (1997).
  11. A. Lachal, “First exit time from a bounded interval for a certain class of additive functional of Brownian motion,” J. Theor. Prob. 13, 733–775 (2000).
  12. S. Spuler, M. Linne, “Numerical analysis of beam propagation in pulsed cavity ring-down spectroscopy,” Appl. Opt. 41, 2858–2868 (2002).
    [CrossRef] [PubMed]
  13. A. E. Siegmann, Lasers (University Science, Mill Valley, Calif., 1986).
  14. J. M. Burch, A. Gerrard, Introduction to Matrix Methods in Optics (Dover, New York, 1975).
  15. S. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer-Verlag, Berlin, 1983).

2002 (1)

2000 (1)

A. Lachal, “First exit time from a bounded interval for a certain class of additive functional of Brownian motion,” J. Theor. Prob. 13, 733–775 (2000).

1997 (1)

A. Lachal, “Temps de sortie d’un intervalle borné pour l’intégrale du mouvement Brownien,” C. R. Acad. Sci. Ser. I: Math. 324, 559–564 (1997).

1996 (1)

J. Masoliver, J. P. Porra, “Exact solution to the exit-time problem for free particle driven by Gaussian white noise,” Phys. Rev. E 53, 2243–2256 (1996).
[CrossRef]

1991 (1)

A. Lachal, “Sur le premier instant de passage de l’intégrale du mouvement Brownien,” Ann. Inst. Henri Poincaré Sect. B 27, 385–405 (1991).

1986 (1)

J. B. Roberts, “First passage time for randomly excited non-linear oscillators,” J. Sound Vib. 109, 31–50 (1986).
[CrossRef]

1976 (1)

J. B. Roberts, “First passage time for the envelope of a randomly excited linear oscillator,” J. Sound Vib. 46, 1–14 (1976).
[CrossRef]

1970 (1)

S. H. Crandall, “First-crossing probabilities of the linear oscillator,” J. Sound Vib. 12, 282–299 (1970).
[CrossRef]

1968 (1)

J. N. Franklin, E. R. Rodemich, “Numerical analysis of an elliptic–parabolic partial differential equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 680–716 (1968).
[CrossRef]

1963 (1)

H. P. McKean, “A winding problem for a resonator driven by a white noise,” J. Math. Kyoto Univ. 2, 227–235 (1963).

1951 (1)

A. J. F. Siegert, “On the first passage time probability problem,” Phys. Rev. 81, 617–623 (1951).
[CrossRef]

Burch, J. M.

J. M. Burch, A. Gerrard, Introduction to Matrix Methods in Optics (Dover, New York, 1975).

Crandall, S. H.

S. H. Crandall, “First-crossing probabilities of the linear oscillator,” J. Sound Vib. 12, 282–299 (1970).
[CrossRef]

Franklin, J. N.

J. N. Franklin, E. R. Rodemich, “Numerical analysis of an elliptic–parabolic partial differential equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 680–716 (1968).
[CrossRef]

Gardiner, S. W.

S. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer-Verlag, Berlin, 1983).

Gerrard, A.

J. M. Burch, A. Gerrard, Introduction to Matrix Methods in Optics (Dover, New York, 1975).

Lachal, A.

A. Lachal, “First exit time from a bounded interval for a certain class of additive functional of Brownian motion,” J. Theor. Prob. 13, 733–775 (2000).

A. Lachal, “Temps de sortie d’un intervalle borné pour l’intégrale du mouvement Brownien,” C. R. Acad. Sci. Ser. I: Math. 324, 559–564 (1997).

A. Lachal, “Sur le premier instant de passage de l’intégrale du mouvement Brownien,” Ann. Inst. Henri Poincaré Sect. B 27, 385–405 (1991).

Linne, M.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics, Vol. PM45 of SPIE Monographs (SPIE Press, Bellingham, Wash., 1998), Chap. 7.

Masoliver, J.

J. Masoliver, J. P. Porra, “Exact solution to the exit-time problem for free particle driven by Gaussian white noise,” Phys. Rev. E 53, 2243–2256 (1996).
[CrossRef]

McKean, H. P.

H. P. McKean, “A winding problem for a resonator driven by a white noise,” J. Math. Kyoto Univ. 2, 227–235 (1963).

Porra, J. P.

J. Masoliver, J. P. Porra, “Exact solution to the exit-time problem for free particle driven by Gaussian white noise,” Phys. Rev. E 53, 2243–2256 (1996).
[CrossRef]

Roberts, J. B.

J. B. Roberts, “First passage time for randomly excited non-linear oscillators,” J. Sound Vib. 109, 31–50 (1986).
[CrossRef]

J. B. Roberts, “First passage time for the envelope of a randomly excited linear oscillator,” J. Sound Vib. 46, 1–14 (1976).
[CrossRef]

Rodemich, E. R.

J. N. Franklin, E. R. Rodemich, “Numerical analysis of an elliptic–parabolic partial differential equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 680–716 (1968).
[CrossRef]

Siegert, A. J. F.

A. J. F. Siegert, “On the first passage time probability problem,” Phys. Rev. 81, 617–623 (1951).
[CrossRef]

Siegmann, A. E.

A. E. Siegmann, Lasers (University Science, Mill Valley, Calif., 1986).

Spuler, S.

Ann. Inst. Henri Poincaré Sect. B (1)

A. Lachal, “Sur le premier instant de passage de l’intégrale du mouvement Brownien,” Ann. Inst. Henri Poincaré Sect. B 27, 385–405 (1991).

Appl. Opt. (1)

C. R. Acad. Sci. Ser. I: Math. (1)

A. Lachal, “Temps de sortie d’un intervalle borné pour l’intégrale du mouvement Brownien,” C. R. Acad. Sci. Ser. I: Math. 324, 559–564 (1997).

J. Math. Kyoto Univ. (1)

H. P. McKean, “A winding problem for a resonator driven by a white noise,” J. Math. Kyoto Univ. 2, 227–235 (1963).

J. Sound Vib. (3)

S. H. Crandall, “First-crossing probabilities of the linear oscillator,” J. Sound Vib. 12, 282–299 (1970).
[CrossRef]

J. B. Roberts, “First passage time for the envelope of a randomly excited linear oscillator,” J. Sound Vib. 46, 1–14 (1976).
[CrossRef]

J. B. Roberts, “First passage time for randomly excited non-linear oscillators,” J. Sound Vib. 109, 31–50 (1986).
[CrossRef]

J. Theor. Prob. (1)

A. Lachal, “First exit time from a bounded interval for a certain class of additive functional of Brownian motion,” J. Theor. Prob. 13, 733–775 (2000).

Phys. Rev. (1)

A. J. F. Siegert, “On the first passage time probability problem,” Phys. Rev. 81, 617–623 (1951).
[CrossRef]

Phys. Rev. E (1)

J. Masoliver, J. P. Porra, “Exact solution to the exit-time problem for free particle driven by Gaussian white noise,” Phys. Rev. E 53, 2243–2256 (1996).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

J. N. Franklin, E. R. Rodemich, “Numerical analysis of an elliptic–parabolic partial differential equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 680–716 (1968).
[CrossRef]

Other (4)

V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics, Vol. PM45 of SPIE Monographs (SPIE Press, Bellingham, Wash., 1998), Chap. 7.

A. E. Siegmann, Lasers (University Science, Mill Valley, Calif., 1986).

J. M. Burch, A. Gerrard, Introduction to Matrix Methods in Optics (Dover, New York, 1975).

S. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer-Verlag, Berlin, 1983).

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Figures (7)

Fig. 1
Fig. 1

Perfectly aligned resonator.

Fig. 2
Fig. 2

Resonator with small misalignment angles β1 and β2.

Fig. 3
Fig. 3

Log–log plot and linear regression of mean exit time versus standard deviation of the stochastic tilt. The resonator parameters are g1=g2=1.0, L=1 m, and A=0.05 m, the simulation is made with 100,000 samples, and the regression line slope is m=-0.664±0.009.

Fig. 4
Fig. 4

Log–log plot and linear regression of mean exit time versus standard deviation of the stochastic tilt. The resonator parameters are (●) g1=g2=0.7, L=1 m, and A=0.004 m and (+) g1=g2=0.9, L=1 m, and A=0.006 m; the simulation is made with 100,000 samples; and the regression line slopes are m0.7=-1.887±0.023 and m0.9=-1.861±0.020.

Fig. 5
Fig. 5

Average sojourn time in the resonator. Simulated results are shown for three values of σ: 2.0×10-4, 2.1×10-4, 2.2×10-4 (from top to bottom).

Fig. 6
Fig. 6

Average sojourn time in the resonator. Approximate results are obtained by solving Eq. (26).

Fig. 7
Fig. 7

Reflection on a spherical mirror slightly tilted from its optical axis A.

Equations (39)

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g=1-LR,0g1.
ynαn=(Tn)(Rn)(T2)(R2)(T1)(R1)y0α0,
ynαn=(Tn)(Rn)(T1)(R1)y0α0+Δy1Δα1+ΔynΔαn,
αm=αm-1+2(g-1)k=1m-1αk-2βm-1,
αm=αm-1-2βm-1=α1-2k=1m-1βk.
Δ(g, A; {σ})=:min{n:|τ(n)|=A;τ(0)=0},
τ(n)=m=0nL tan(αm),
Prob[0Δ(g, A, {β})M]=:Φ(M),MN+,
paraxialconfigurationLA.
τ(n)Lm=1n(αm).
nξRk=0m-1αk0ξα(s)ds,
2βm-1=2k=0k=m-1βk-k=0k=m-2βkσ dds Ws,
τ(ξ)=L0α(s)ds,
d2dξ2 τ(ξ)+2(1-g)τ(ξ)=LσddξWξ=LσNξ,
Nξ; Nξ=δ(ξ-ξ),
Δ(g, A; {σ})=inf[ξ0:|τ(ξ)|=A].
Φ(x)=:Prob[0Δ(g, A; {σ})x],xR+.
Δ(1, A; σ)=0xd[Φ(x)]=CALσ2/3,
H(ξ)=:1ωA2 [ωτ2(ξ)+V2(ξ)],θ(ξ)=:arctanV(ξ)ωτ(ξ),
τ(ξ)=AH(ξ)cos[θ(ξ)],V(ξ)=AωH(ξ)sin[θ(ξ)],
dH(ξ)dξ=2LσH(ξ)ωA sin[θ(ξ)]Nξ,dθ(ξ)dξ=-ω+σLAωH(ξ) cos[θ(ξ)]Nξ.
dH(ξ)dξ=2LσH(ξ)ωA sin(ωξ)Nξ,H(ξ)0.
ξ P(H, ξ|0, 0)=2σ2L2ωA2 sin2(ωξ)×H H H [HP(H, ξ|0, 0)].
P(H, ξ|0, 0)=12πη(ξ)H exp-H2η(ξ),H0,
η(ξ)=L2σ22ωA2 ξ-12ω sin(2ωξ)=1Γ ξ-12ω sin(2ωξ).
H(ξ)n=0HnP(H, ξ|0, 0)dH=(2n-1)!![η(ξ)]n.
H(ξ)=η(ξ)H(Δ(g, A; {σ}))=1.
η(ξ)η(Δ(g, A; {σ}))=1Γ-12ωΔ(g, A; {σ})Γ+12ω,
Δ(g2, A; {σ})=ΓΔ(g2, A; {σ})A2ωL2σ2=A2RLσ2.
L2σ23A2 Δ(1, A; {σ})3=1Δ(1, A; {σ})A2L2σ21/3,
Proby¯+sn z0.005μy¯-sn z0.005=0.99,
α0+γ+[π-(β0+ε)]=π.
γ+(β0+ε)+[π-(-α1)]=π.
dist(B, F)=dist(B, E)cos(δ).
dist(B, F)sin(ε)=dist(O, F)sin[(π/2-δ)+β0].
γ=β0+ε-α0[from(A1)]-α1=γ+β0+ε[from(A2)]dist(B, F)=y0cos(δ)=R sin(ε)cos(δ-β0) [from(A3), (A4)]γ=β0+ε-α0-α1=-α0+2β0+2εsin(ε)=y0R cos(δ-β0)cos(δ).
α1=α0-2 y0R-2β0.
y1α1=(T)(R)y0α0y1α1=1L0110-2R1×y0α0-2β0.
ym=ym-1+αmLαm=αm-1-2R ym-1-2βm-1ym=y0+Lk=1mαkαm=αm-1-2R y0+2(g-1)k=1m-1αk-2βm-1.

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