Abstract

Thin flat mirrors are often used in designing various optical measurement systems. Such mirrors are generally deformed by environmental conditions during measurements. A detailed theory of deformation of a thin flat mirror that oscillates harmonically in the direction of the normal to its surface is introduced in our work. The mirror is treated as a vibrating membrane, and the time-dependent effect of the mirror deformation on the properties of reflected light is studied. A relation is derived for a dynamic wave aberration. On the basis of this relation, calculation of the Strehl definition of the deformed mirror is performed both by exact integration and by approximation. The results obtained can be used for analysis of the influence of mechanical vibrations on the accuracy of optical measurement systems in various practical applications where thin flat mirrors are used.

© 2004 Optical Society of America

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References

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  1. L. D. Landau, E. M. Lifshitz, Theory of Elasticity. Course in Theoretical Physics, Vol. 7 (Pergamon, Oxford, UK, 1970).
  2. K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge U. Press, Cambridge, UK, 2002).
  3. A. P. S. Selvadurai, Partial Differential Equations in Mechanics, Vol. 1 (Springer, Berlin, 2000).
  4. A. P. S. Selvadurai, Partial Differential Equations in Mechanics, Vol. 2 (Springer, Berlin, 2000).
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).
  6. A. Miks, “Dependence of the wave-front aberration on the radius of the reference sphere,” J. Opt. Soc. Am. A 19, 1187–1190 (2002).
    [CrossRef]
  7. A. Miks, Applied Optics (Czech Technical University Press, Prague, 2000).
  8. C. Lanczos, Applied Analysis (Prentice Hall, New York, 1956).
  9. G. Meinardus, Approximation von Funktionen und ihre numerische Behandlung (Springer, Berlin, 1964).
  10. Melles Griot Catalogue 2003, Chap. 36.2.

2002 (1)

Bence, S. J.

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge U. Press, Cambridge, UK, 2002).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).

Hobson, M. P.

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge U. Press, Cambridge, UK, 2002).

Lanczos, C.

C. Lanczos, Applied Analysis (Prentice Hall, New York, 1956).

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Theory of Elasticity. Course in Theoretical Physics, Vol. 7 (Pergamon, Oxford, UK, 1970).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Theory of Elasticity. Course in Theoretical Physics, Vol. 7 (Pergamon, Oxford, UK, 1970).

Meinardus, G.

G. Meinardus, Approximation von Funktionen und ihre numerische Behandlung (Springer, Berlin, 1964).

Miks, A.

Riley, K. F.

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge U. Press, Cambridge, UK, 2002).

Selvadurai, A. P. S.

A. P. S. Selvadurai, Partial Differential Equations in Mechanics, Vol. 1 (Springer, Berlin, 2000).

A. P. S. Selvadurai, Partial Differential Equations in Mechanics, Vol. 2 (Springer, Berlin, 2000).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).

J. Opt. Soc. Am. A (1)

Other (9)

A. Miks, Applied Optics (Czech Technical University Press, Prague, 2000).

C. Lanczos, Applied Analysis (Prentice Hall, New York, 1956).

G. Meinardus, Approximation von Funktionen und ihre numerische Behandlung (Springer, Berlin, 1964).

Melles Griot Catalogue 2003, Chap. 36.2.

L. D. Landau, E. M. Lifshitz, Theory of Elasticity. Course in Theoretical Physics, Vol. 7 (Pergamon, Oxford, UK, 1970).

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge U. Press, Cambridge, UK, 2002).

A. P. S. Selvadurai, Partial Differential Equations in Mechanics, Vol. 1 (Springer, Berlin, 2000).

A. P. S. Selvadurai, Partial Differential Equations in Mechanics, Vol. 2 (Springer, Berlin, 2000).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1968).

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Figures (6)

Fig. 1
Fig. 1

Scheme of the oscillating mirror.

Fig. 2
Fig. 2

Deformation of the mirror.

Fig. 3
Fig. 3

Aberration of the thin vibrating mirror.

Fig. 4
Fig. 4

Approximated wave aberration.

Fig. 5
Fig. 5

Time dependence of the Strehl definition.

Fig. 6
Fig. 6

Dependence of the Strehl definition on the amplitude of the oscillations α and the parameter β=ΩR/c.

Tables (1)

Tables Icon

Table 1 Calculation of Strehl Definition

Equations (58)

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2u(x, y, t)t2=c22u(x, y, t),
2u(r, θ, t)t2=c22u(r, θ, t),
2=2r2+1rr+1r22θ2
u(R, θ, t)=α sin Ωt,
u(r, θ, 0)=0,u(r, θ, 0)/t=0;
u=f+g,
f(r, t)=α J0Ωc rJ0Ωc Rsin Ωt,
2g(r, θ, t)t2=c22g(r, θ, t).
u(R, θ, t)=f(R, t)+g(R, θ, t)=F(R)sin Ωt+g(R, θ, t)=α sin Ωt.
u(r, θ, 0)=f(r, 0)+g(r, θ, 0)=0+g(r, θ, 0)=0,
g(r, θ, 0)=0.
u(r, θ, 0)/t=f(r, 0)/t+g(r, θ, 0)/t=0,
g(r, θ, 0)/t=-f(r, 0)/t.
g(r, θ, 0)=0,g(r, θ, 0)/t=-f(r, 0)/t.
g=n=0m=1gn,m,
gn,m=An,msin c μn,mR t+Bn,mcos c μn,mR t×(Cnsin nθ+Dncos nθ)Jnμn,mrR
u(r, t)=α J0β rRJ0(β)sin Ωt+m=1AmJ0μmrRsin Ωmt,
β=Ωc R,Am=-2αβJ1(μm)(μm2-β2),
Ωm=Ωμmβ.
W(r, t)=2δ(r, t)=P(r)sin Ωt+m=1Qm(r)sin Ωmt,
P(r)=2αJ0rR βJ0(β)-1,
Qm(r)=2AmJ0rR μm.
Jn(x)=s=0(-1)ss!(n+s)!x2n+2s,
Wm(r, t)=P(r)sin Ωt+Qm(r)sin Ωmt=W00+s=1W2s0rR2s,
W2s0=(-1)s22s(s!)2 (Uβ2s+Umμm2s),
U=2αJ0(β)sin Ωt,Um=2Amsin Ωmt,
W00=-2α sin Ωt+U+Um.
W20=-Uβ2+Umμm24,W40=Uβ4+Umμm464,
W60=-Uβ6+Umμm62304,W80=Uβ8+Umμm8147456,W100=-Uβ10+Umμm1014745600.
I(t)=1SSexp[ikW(r, t)]dS2,
I=I(t)=1T0TI(t)dt,
I(t)=1-k2(W2(t)¯-W(t)¯2),
W2(t)¯=1SSW2(r, t)dS,
W(t)¯=1SSW(r, t)dS.
Wm(r, t)=P(r)sin Ωt+Qm(r)sin Ωmt,
P¯=1SSP(r)dS,
P2¯=1SSP2(r)dS,
Qm¯=1SSQm(r)dS,
Qm2¯=1SSQm2(r)dS,
P.Qm¯=1SSP(r)Qm(r)dS,
Wm2¯-Wm¯2=(P2¯-P¯2)sin2 Ωt+(Qm2¯-Qm¯2)sin2 Ωmt+2(P.Qm¯-P.Qm¯)sin Ωt sin Ωmt.
P¯=-2α1-2βJ1(β)J0(β),
P2¯=8α21-2βJ1(β)J0(β)+12J12(β)J02(β),
Qm¯=-8αβμm(μm2-β2),
Qm2¯=16αβ(μm2-β2)2,
P.Qm¯=-16α2βμm(μm2-β2)μm2μm2-β2-1.
E0(t)=Wm2¯-Wm¯2=E1sin2 Ωt+E2sin2 Ωmt+2E12sin Ωt sin Ωmt,
E1=P2¯-P¯2,E2=Qm2¯-Qm¯2,E12=P.Qm¯-P.Qm¯,
Im(t)=1-k2E0(t).
I=Im(t)=1-k2E0(t),
=1T0T()dt.
sin2 Ωt=12-12sin 2ΩT2ΩT,
sin2 Ωmt=12-12sin 2ΩmT2ΩmT,
sin Ωt sin Ωmt=12sin(Ω-Ωm)T(Ω-Ωm)T-sin(Ω+Ωm)T(Ω+Ωm)T.
I=1-k2E0(t)=1-k22 (E1+E2).
W=W20ξ2+W40ξ4+W60ξ6+W80ξ8+,
ξ=r/R.
W2¯-W¯2=112 W202+445 W402+9112 W602+16225 W802+16 W20W40+320 W20W60+215 W20W80+16 W40W60+16105 W40W80+320 W60W80.

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