Abstract

By using an operator description we derive the spectrum of a symmetric two-mirror resonator in the presence of spherical aberration.

© 2005 Optical Society of America

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References

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  5. V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, 2001).
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  6. H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
    [CrossRef]
  7. T. Klaassen, A. Hoogeboom, M. P. van Exter, J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A 21, 1689–1693 (2004).
    [CrossRef]
  8. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
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  9. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
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  10. H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
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  17. R. Borghi, M. Santarsiero, R. Simon, “Shape invariance and a universal form for the Gouy phase,” J. Opt. Soc. Am. A 21, 572–579 (2004).
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2005 (1)

2004 (2)

1999 (1)

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

1998 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1985 (1)

D. R. Truax, “Baker–Campbell–Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators,” Phys. Rev. D 31, 1988–1991 (1985).
[CrossRef]

1981 (2)

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

D. Stoler, “Operator methods in physical optics,” J. Opt. Soc. Am. 71, 334–341 (1981).
[CrossRef]

1980 (1)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1969 (1)

1963 (1)

J. Wei, E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[CrossRef]

Arnaud, J. A.

Bacry, H.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Borghi, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
[CrossRef]

Cadilhac, M.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Friberg, A. T.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

Hoogeboom, A.

Klaassen, T.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Laabs, H.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, 2001).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, 1972).

Mukunda, N.

Nazarathy, M.

Nienhuis, G.

Norman, E.

J. Wei, E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[CrossRef]

Santarsiero, M.

Shamir, J.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, 1986).

Simon, R.

Stoler, D.

Truax, D. R.

D. R. Truax, “Baker–Campbell–Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators,” Phys. Rev. D 31, 1988–1991 (1985).
[CrossRef]

van Exter, M. P.

Visser, J.

Wei, J.

J. Wei, E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[CrossRef]

Woerdman, J. P.

T. Klaassen, A. Hoogeboom, M. P. van Exter, J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A 21, 1689–1693 (2004).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
[CrossRef]

Zelders, N. J.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

J. Math. Phys. (1)

J. Wei, E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Phys. Rev. A (2)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Phys. Rev. D (1)

D. R. Truax, “Baker–Campbell–Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators,” Phys. Rev. D 31, 1988–1991 (1985).
[CrossRef]

Other (4)

A. E. Siegman, Lasers (Oxford U. Press, 1986).

A. Messiah, Quantum Mechanics (North-Holland, 1972).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, 2001).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Symmetric two-mirror resonator for which the distance between the mirrors is L and the radius of curvature of the spherical mirrors is R. (a) The resonator configuration, (b) the corresponding lens guide, which is obtained by unfolding the resonator. The input plane of the lens guide is the plane in the middle of the resonator.

Fig. 2
Fig. 2

In the case that the mirror with radius of curvature R has aberrations, it is represented in the lens guide by two lenses with focal distance R between which there is a phase plate that introduces the aberrations ϵ V ( x , y ) of the mirror. Both lenses and the phase plate lie in the plane of the mirror.

Equations (56)

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E ( x , y , z , t ) = E 0 u ( x , y , z ) exp ( i k z i ω t ) ,
( 2 x 2 + 2 y 2 + 2 i k z ) u ( x , y , z ) = 0 .
u ( z ) = U ̂ f ( z ) u ( 0 ) , U ̂ f ( z ) = exp ( i z 2 k p ̂ x 2 ) .
u ( z + ) = U ̂ l ( f ) u ( z ) , U ̂ l ( f ) = exp ( i k 2 f x ̂ 2 ) ,
T ̂ 1 = 1 4 ( x ̂ p ̂ x + p ̂ x x ̂ ) , T ̂ 2 = 1 4 ( 1 γ 2 x ̂ 2 γ 2 p ̂ x 2 ) ,
T ̂ 3 = 1 4 ( 1 γ 2 x ̂ 2 + γ 2 p ̂ x 2 ) ,
[ T ̂ 1 , T ̂ 2 ] = i T ̂ 3 , [ T ̂ 2 , T ̂ 3 ] = i T ̂ 1 , [ T ̂ 3 , T ̂ 1 ] = i T ̂ 2 .
U ̂ = exp [ i ( β 1 T ̂ 1 + β 2 T ̂ 2 + β 3 T ̂ 3 ) ] ,
U ̂ f ( z ) ( x ̂ p ̂ x k ) U ̂ f ( z ) = M f ( z ) ( x ̂ p ̂ x k ) , M f ( z ) = [ 1 z 0 1 ] ,
U ̂ l ( f ) ( x ̂ p ̂ x k ) U ̂ l ( f ) = M l ( f ) ( x ̂ p ̂ x k ) , M l ( f ) = [ 1 0 1 f 1 ] ,
U ̂ ( x ̂ p ̂ x k ) U ̂ = M ( x ̂ p ̂ x k ) .
[ T ̂ m , ( x ̂ p ̂ x k ) ] = J m ( x ̂ p ̂ x k ) , m = 1 , 2 , 3 ,
J 1 = i 2 [ 1 0 0 1 ] , J 2 = i 2 [ 0 b 1 b 0 ] ,
J 3 = i 2 [ 0 b 1 b 0 ] ,
M = exp [ i ( β 1 J 1 + β 2 J 2 + β 3 J 3 ) ] .
U ̂ = exp [ i α ( T ̂ 2 + T ̂ 3 ) ] exp ( i ξ T ̂ 1 ) exp ( i θ T ̂ 3 ) .
M [ A B C D ] = exp [ i α ( J 2 + J 3 ) ] exp ( i ξ J 1 ) exp ( i θ J 3 ) .
cos ( θ 2 ) = b A b 2 A 2 + B 2 ,
exp ( ξ ) = A 2 + B 2 b 2 ,
α = b b 2 A C + B D b 2 A 2 + B 2 .
tan [ θ ( z ) 2 ] = z b , exp [ ξ ( z ) ] = 1 + z 2 b 2 , α ( z ) = b z b 2 + z 2 .
exp ( i ξ T ̂ 1 ) x ̂ exp ( i ξ T ̂ 1 ) = exp ( ξ 2 ) x ̂ .
d ( x , y ) = 2 k ( R 2 x 2 y 2 R ) ,
U ̂ = U ̂ f ( L 2 ) U ̂ l ( R 2 ) U ̂ f ( L 2 ) .
U ̂ = exp ( i H ̂ ) ,
b 2 = B D A C ,
cos ( θ 2 ) = sign ( A ) A D , exp ( ξ ) = A D ,
M = M f ( L 2 ) M l ( R 2 ) M f ( L 2 ) = [ g L 2 ( 1 + g ) 2 L ( 1 g ) g ] ,
b = k γ 2 = L 2 1 + g 1 g ,
H ̂ = 1 2 arccos ( g ) [ γ 2 ( p ̂ x 2 + p ̂ y 2 ) + 1 γ 2 ( x ̂ 2 + y ̂ 2 ) ] ,
a ̂ x = 1 2 ( x ̂ γ + i γ p ̂ x ) , a ̂ x = 1 2 ( x ̂ γ i γ p ̂ x ) ,
a ̂ y = 1 2 ( y ̂ γ + i γ p ̂ y ) , a ̂ y = 1 2 ( y ̂ γ i γ p ̂ y ) ,
H ̂ = arccos ( g ) ( n ̂ x + n ̂ y + 1 ) ,
H ̂ u n m = arccos ( g ) ( n + m + 1 ) u n m .
k L = arccos ( g ) ( n + m + 1 ) + π q ,
l ̂ z = x ̂ p ̂ y y ̂ p ̂ x ,
a ̂ ± = 1 2 ( a ̂ x i a ̂ y ) , a ̂ ± = 1 2 ( a ̂ x ± i a ̂ y ) .
H ̂ = arccos ( g ) ( n ̂ + + n ̂ + 1 ) , l ̂ z = n ̂ + n ̂ ,
d ( x , y ) = k ( x 2 + y 2 ) R k ( x 2 + y 2 ) 2 4 R 3 + .
d ( x , y ) = 1 2 1 g 2 ( x 2 + y 2 γ 2 ) ϵ 16 ( 1 g 2 ) ( x 2 + y 2 γ 2 ) 2 + O ( ϵ 2 ) ,
U ̂ l ( R ) exp [ i ϵ V ( x ̂ , y ̂ ) ] U ̂ l ( R ) ,
V ( x , y ) = 1 16 ( 1 g 2 ) ( x 2 + y 2 γ 2 ) 2 .
U ̂ = U ̂ 2 exp [ i ϵ V ( x ̂ , y ̂ ) ] U ̂ 1 ,
U ̂ 1 = U ̂ l ( R ) U ̂ f ( L 2 ) , U ̂ 2 = U ̂ f ( L 2 ) U ̂ l ( R ) .
U ̂ 1 = exp [ i ξ ( L 2 ) T ̂ 1 ] exp [ i θ ( L 2 ) T ̂ 3 ] .
U ̂ 2 = exp [ i θ ( L 2 ) T ̂ 3 ] exp [ i ξ ( L 2 ) T ̂ 1 ] .
exp [ i ξ ( L 2 ) T ̂ 1 ] x ̂ exp [ i ξ ( L 2 ) T ̂ 1 ] = 2 1 + g x ̂ .
U ̂ = exp ( i 2 H ̂ ) exp ( i ϵ V ̂ ) exp ( i 2 H ̂ ) ,
V ̂ = V ( 2 1 + g x ̂ , 2 1 g y ̂ ) = 1 g 4 ( 1 + g ) ( x ̂ 2 + y ̂ 2 γ 2 ) 2 .
exp ( A ̂ ) exp ( B ̂ ) = exp ( A ̂ + B ̂ + 1 2 [ A ̂ , B ̂ ] + 1 12 [ A ̂ , [ A ̂ , B ̂ ] ] + 1 12 [ B ̂ , [ B ̂ , A ̂ ] ] + ) .
U ̂ = exp [ i H ̂ i ϵ Δ H ̂ + O ( ϵ 2 ) ] ,
Δ H ̂ = V ̂ 1 6 [ H ̂ , [ H ̂ , V ̂ ] ] + ,
x ̂ 2 + y ̂ 2 γ 2 = n ̂ + + n ̂ + 1 + a ̂ + a ̂ + a ̂ + a ̂ .
Δ H ( n + , n ) = u n + n V ̂ u n + n = 1 g 4 ( 1 + g ) [ ( n + + n + 2 ) ( n + + n + 1 ) + 2 n + n ] .
k L = arccos ( g ) ( n + + n + 1 ) + ϵ Δ H ( n + , n ) + π q ,
1 g 1 + g ( n + + n + 1 ) k R .

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