Abstract

A generalized method to accurately calculate astigmatism of the unit-magnification multipass system (UMS) is proposed. A practical coaxial optical transmission model is developed for the UMS. Astigmatism analysis is then made convenient by a 4 by 4 general transfer matrix. Astigmatism correction is significantly promoted, and hence further improvement in imaging quality can be expected. Good agreement between numerical simulations and Zemax ray tracing results verifies the effectiveness of this method. The resulted RMS spot size of this method is only 25% to 64% of other previous methods based on the golden section search for minimum astigmatism in real design cases. This method is helpful for the optical design of the UMS.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. R. P. Blickensderfer, G. E. Ewing, and R. Leonard, “A long path, low temperature cell,” Appl. Opt. 7, 2214-2217 (1968).
    [CrossRef]
  4. J. U. White, “Very long optical paths in air,” J. Opt. Soc. Am. 66, 411-416 (1976).
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  5. T. H. Edwards, “Multiple-traverse absorption cell design,” J. Opt. Soc. Am. 51, 98-102 (1961).
    [CrossRef]
  6. A. L. Vitushkin and L. F. Vitushkin, “Design of a multipass optical cell based on the use of shifted corner cubes and right-angle prisms,” Appl. Opt. 37, 162-165 (1998).
    [CrossRef]
  7. D. Horn and G. C. Pimentel, “2.5 km Low-temperature multiple-reflection cell,” Appl. Opt. 10, 1892-1898 (1971).
    [CrossRef]
  8. J.-F. Doussin, R. Dominique, and C. Patrick, “Multiple-pass cell for very-long-path infrared spectrometry,” Appl. Opt. 38, 4145-4150 (1999).
    [CrossRef]
  9. J. T. K. McCubbin and R. P. Grosso, “White-type multiple-pass absorption cell of simple construction,” Appl. Opt. 2, 764-765(1963).
    [CrossRef]
  10. H. M. Pickett, G. M. Bradley, and H. L. Strauss, “White type multiple pass absorption cell,” Appl. Opt. 9, 2397-2398 (1970).
    [CrossRef]
  11. S. M. Chernin, “Promising version of the three-objective multipass matrix system,” Opt. Express 10, 104-107 (2002).
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    [CrossRef]
  13. S. M. Chernin, S. B. Mikhailov, and E. G. Barskaya, “Aberrations of a multipass matrix system,” Appl. Opt. 31, 765-769(1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. D. C. Tobin, L. L. Strow, W. J. Lafferty, and W. B. Olson, “Experimental investigation of the self- and N2-broadened continuum within the N2 band of water vapor,” Appl. Opt. 35, 4724-4734 (1996).
    [CrossRef]
  18. H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys. 16, 30-39 (1948).
    [CrossRef]
  19. C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE 7156, 71560G (2008),
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. B. Macukow and H. H. Arsenault, “Extension of the matrix theory for nonsymmetrical optical system,” J. Opt. (Paris) 15, 145-151 (1984).
    [CrossRef]
  24. G. V. Reklaitis, A. Ravindran, and K. M. Ragsdell, Engineering Optimization: Methods and Applications (Wiley, 1983), pp. 43-47.

2008

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE 7156, 71560G (2008),

X. Liu and K.-H. Brenner, “Minimal optical decomposition of ray transfer matrices,” Appl. Opt. 47, E88-E98 (2008).
[CrossRef]

2007

2005

2002

1999

1998

1996

1992

1991

1984

B. Macukow and H. H. Arsenault, “Extension of the matrix theory for nonsymmetrical optical system,” J. Opt. (Paris) 15, 145-151 (1984).
[CrossRef]

1983

1976

1971

Y. G. Barskaya, “Aberrations of a multipass cell,” Opt. Technol. 38, 278-280 (1971).

D. Horn and G. C. Pimentel, “2.5 km Low-temperature multiple-reflection cell,” Appl. Opt. 10, 1892-1898 (1971).
[CrossRef]

1970

1968

1963

1961

1951

1948

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys. 16, 30-39 (1948).
[CrossRef]

Arsenault, H. H.

Barskaya, E. G.

Barskaya, Y. G.

Y. G. Barskaya, “Aberrations of a multipass cell,” Opt. Technol. 38, 278-280 (1971).

Bernstein, H. J.

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys. 16, 30-39 (1948).
[CrossRef]

Blickensderfer, R. P.

Bradley, G. M.

Brenner, K.-H.

Chernin, S. M.

Dominique, R.

Doussin, J.-F.

Edwards, T. H.

Ewing, G. E.

Grosso, R. P.

Guofan, J.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE 7156, 71560G (2008),

Herzberg, G.

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys. 16, 30-39 (1948).
[CrossRef]

Horn, D.

Huaidong, Y.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE 7156, 71560G (2008),

Kexin, C.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE 7156, 71560G (2008),

Kohn, W. H.

Lafferty, W. J.

Leonard, R.

Liqun, S.

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE 7156, 71560G (2008),

Liu, X.

Macukow, B.

McCubbin, J. T. K.

Mikhailov, S. B.

Olson, W. B.

Patrick, C.

Pickett, H. M.

Pimentel, G. C.

Ragsdell, K. M.

G. V. Reklaitis, A. Ravindran, and K. M. Ragsdell, Engineering Optimization: Methods and Applications (Wiley, 1983), pp. 43-47.

Ravindran, A.

G. V. Reklaitis, A. Ravindran, and K. M. Ragsdell, Engineering Optimization: Methods and Applications (Wiley, 1983), pp. 43-47.

Reesor, T. R.

Reklaitis, G. V.

G. V. Reklaitis, A. Ravindran, and K. M. Ragsdell, Engineering Optimization: Methods and Applications (Wiley, 1983), pp. 43-47.

Robert, C.

Silver, J.

Strauss, H. L.

Strow, L. L.

Tobin, D. C.

Vitushkin, A. L.

Vitushkin, L. F.

White, J. U.

Appl. Opt.

R. P. Blickensderfer, G. E. Ewing, and R. Leonard, “A long path, low temperature cell,” Appl. Opt. 7, 2214-2217 (1968).
[CrossRef]

S. M. Chernin and E. G. Barskaya, “Optical multipass matrix systems,” Appl. Opt. 30, 51-58 (1991).
[CrossRef]

S. M. Chernin, S. B. Mikhailov, and E. G. Barskaya, “Aberrations of a multipass matrix system,” Appl. Opt. 31, 765-769(1992).
[CrossRef]

W. H. Kohn, “Astigmatism and White cells: theoretical considerations on the construction of an anastigmatic White cell,” Appl. Opt. 31, 6757-6764 (1992).
[CrossRef]

J.-F. Doussin, R. Dominique, and C. Patrick, “Multiple-pass cell for very-long-path infrared spectrometry,” Appl. Opt. 38, 4145-4150 (1999).
[CrossRef]

D. C. Tobin, L. L. Strow, W. J. Lafferty, and W. B. Olson, “Experimental investigation of the self- and N2-broadened continuum within the N2 band of water vapor,” Appl. Opt. 35, 4724-4734 (1996).
[CrossRef]

A. L. Vitushkin and L. F. Vitushkin, “Design of a multipass optical cell based on the use of shifted corner cubes and right-angle prisms,” Appl. Opt. 37, 162-165 (1998).
[CrossRef]

D. Horn and G. C. Pimentel, “2.5 km Low-temperature multiple-reflection cell,” Appl. Opt. 10, 1892-1898 (1971).
[CrossRef]

J. Silver, “Simple dense-pattern optical multipass cells,” Appl. Opt. 44, 6545-6556 (2005).
[CrossRef]

C. Robert, “Simple, stable, and compact multiple-reflection optical cell for very long optical paths,” Appl. Opt. 46, 5408-5418 (2007).
[CrossRef]

J. T. K. McCubbin and R. P. Grosso, “White-type multiple-pass absorption cell of simple construction,” Appl. Opt. 2, 764-765(1963).
[CrossRef]

H. M. Pickett, G. M. Bradley, and H. L. Strauss, “White type multiple pass absorption cell,” Appl. Opt. 9, 2397-2398 (1970).
[CrossRef]

X. Liu and K.-H. Brenner, “Minimal optical decomposition of ray transfer matrices,” Appl. Opt. 47, E88-E98 (2008).
[CrossRef]

J. Chem. Phys.

H. J. Bernstein and G. Herzberg, “Rotation-vibration spectra of diatomic and simple polyatomic molecules with long absorbing paths,” J. Chem. Phys. 16, 30-39 (1948).
[CrossRef]

J. Opt. (Paris)

B. Macukow and H. H. Arsenault, “Extension of the matrix theory for nonsymmetrical optical system,” J. Opt. (Paris) 15, 145-151 (1984).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Opt. Technol.

Y. G. Barskaya, “Aberrations of a multipass cell,” Opt. Technol. 38, 278-280 (1971).

Proc. SPIE

C. Kexin, Y. Huaidong, S. Liqun, and J. Guofan, “Astigmatism analysis by matrix methods in White cells,” Proc. SPIE 7156, 71560G (2008),

Other

G. V. Reklaitis, A. Ravindran, and K. M. Ragsdell, Engineering Optimization: Methods and Applications (Wiley, 1983), pp. 43-47.

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Figures (11)

Fig. 1
Fig. 1

Diagram of UMS: 1.1, system layout of the UMS; 1.2, frontal views of the objective mirrors in the UMS; 1.2(a), two objectives; 1.2(b), three objectives; 1.3, frontal views of the field mirrors in the UMS; 1.3(a), the BHWC; 1.3(b), the PBWC; 1.3(c), modified White cell; 1.3(d), MMS. C x , the curvature center of the objective mirror M x where subscripts x = 1 , 2, 3 represent the index of objective mirrors: In, the entrance aperture; Out, the exit aperture; filled circles, images of the entrance aperture; FT, spherical field mirror with T-shape aperture; FR, spherical field mirror with rectangular aperture; FP, prisms or corner mirrors.

Fig. 2
Fig. 2

Rotation angle of the kth generalized astigmatic thin lens: (a) reflections in the UMS and (b) corresponding partial model for astigmatism of the UMS.

Fig. 3
Fig. 3

Overall model for astigmatism of the UMS. The kth reflection is equivalent to the kth GATL; z k , the distance between ( k 1 ) th and kth GATLs; z 1 , the distance between the position of the entrance aperture and the first GATL; z m + 1 , the distance between the mth GATL and the position of the exit aperture.

Fig. 4
Fig. 4

System layout of the 16-pass BHWC. F, the field mirror with the center of curvature C F ; M 1 and M 2 , two spherical objective mirrors with centers of curvature C 1 and C 2 , respectively; In, the entrance aperture; Out, the exit aperture; Tags 1 to 7, images of the entrance aperture in sequence.

Fig. 5
Fig. 5

Astigmatism of a 40-pass BHWC varying with h: solid curve, simulations by our method; dot–dash curve, simulations by the Kohn method; discrete cubic, ray tracing results by Zemax; subplot in figure, enlarged part in rectangle.

Fig. 6
Fig. 6

MTF of a 40-pass BHWC optimized by our method (solid curves), the Kohn method (dashed curves), and MCE (dash–dot curves): (a)  MTF in the meridional plane and (b)  MTF in the sagittal plane.

Fig. 7
Fig. 7

Spot diagram in Zemax for output focusing image of the BHWC optimized by the three methods: (a)  h = 42.38 mm optimized by our method, (b)  h = 43.55 mm optimized by the Kohn method; (c)  h = 46.3 mm optimized by MCE. Scale bars are 1000 μm .

Fig. 8
Fig. 8

System layout of the 18-pass PBWC: F, field mirror with the curvature center C F ; M 1 and M 2 , two spherical objective mirrors with curvature centers C 1 and C 2 , respectively; In, the entrance aperture; Out, the exit aperture; Tags 1 to 8, images of the entrance aperture in sequence.

Fig. 9
Fig. 9

Astigmatism of the PBWC for four cases: (a) astigmatism varying with h in a 30-pass PBWC with R = 625 mm , p = 40 mm , Δ = 50 mm ; (b) astigmatism varying with Δ in a 30-pass PBWC with R = 625 mm , p = 40 mm , h = 20 mm ; (c) astigmatism varying with p in a 30-pass PBWC with R = 625 mm , h = 20 mm , Δ = 50 mm ; (d) astigmatism in the varying n-pass PBWC with R = 625 mm , p = 40 mm , h = 30 mm and Δ = 50 mm . Solid curves and pluses, astigmatism calculated by our method; cubic and multiplication sign, ray tracing results by Zemax.

Fig. 10
Fig. 10

MTF of a 30-pass PBWC optimized by Tobin et al. and us. Solid (dashed) curve, meridional (sagittal) response optimized by us; dot (dash–dot) curve, meridional (sagittal) response optimized by Tobin et al.

Fig. 11
Fig. 11

Spot diagram in Zemax to focus the image of the PBWC optimized by Tobin et al. and us: (a)  h = 46.27 mm optimized by our method and (b)  h = 49.37 mm optimized by Tobin et al. Scale bars are 1000 μm .

Tables (2)

Tables Icon

Table 1 Spot Size of Output Focusing Image for the BHWC by Zemax Ray Tracing Optimized by Three Methods

Tables Icon

Table 2 Spot Size of Output Focusing Image for the PBWC by Zemax Ray Tracing Optimized by Tobin et al. and Our Method

Equations (11)

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α k = α 1 + j = 1 k 1 ε j ,
P ( z ) = [ I z I 0 I ] ,
Λ g ( θ , α ) = R ( α ) Λ ( θ ) R ( α ) = [ S ( α ) 0 0 S ( α ) ] [ I 0 L ( θ ) I ] [ S ( α ) 0 0 S ( α ) ] , S ( α ) = [ cos α sin α sin α cos α ] , L ( θ ) = [ f s 1 0 0 f m 1 ] = [ 2 cos θ R 0 0 2 R cos θ ] .
P ( f ) [ I 0 f 1 I I ] ,
M = P ( z m + 1 ) { k = 1 m Λ g ( θ k , α k ) P ( z k ) } P ( R / 2 ) Λ g ( 0 , 0 ) .
M = [ A B C D ] = [ I E 0 I ] [ 0 F F * 0 ] [ I G 0 I ] ,
l s m = ( λ s ) ( λ m ) = λ m λ s .
E = [ e 11 e 12 e 12 e 22 ] = S ( β ) [ λ s 0 0 λ m ] S ( β ) = [ cos 2 β λ s + sin 2 β λ m ( λ s λ m ) sin β cos β ( λ s λ m ) sin β cos β sin 2 β λ s + cos 2 β λ m ] ,
( λ s λ m ) cos 2 β = e 11 e 22 , ( λ s λ m ) sin 2 β = 2 e 12 .
l s m = λ m λ s = ( e 11 e 22 ) cos 2 β 2 e 12 sin 2 β , 2 β = arctan ( 2 e 12 e 11 e 22 ) .
l s m = ( e 11 e 22 ) 2 + 4 e 12 2 , β = { 1 2 arctan ( 2 e 12 e 11 e 22 ) + π 2 , for     e 11 e 22 0 1 2 arctan ( 2 e 12 e 11 e 22 ) , for     e 11 e 22 < 0 .

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