Abstract

This paper describes the behavior of a cat’s eye retroreflector, which is incorporated in a novel way in a double-pass homodyne polarization interferometer. The amount of mirror tilt immunity a cat’s eye provides is calculated within the paraxial approximation using 4×4 ABCD matrices. It is found that there is a position of the target mirror in which the tilt immunity is at a maximum. A real cat’s eye, which is affected by aberrations, is optimized and examined using Zemax software for optical design. The maximum amount of mirror tilt immunity is numerically calculated and written in terms of defocus and spherical aberration. Finally, for the purposes of comparison, the amplitude of the Lissajous pattern as the target mirror tilts is calculated for both an interferometer with an integrated cat’s eye and an interferometer with a cube corner.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. P. Hariharan and D. Sen, “The separation of symmetrical and asymmetrical wave-front aberrations in the Twyman interferometer,” Proc. Phys. Soc. London 77, 328–334 (1961).
    [CrossRef]
  3. S. J. Bennett, “A double-passed Michelson interferometer,” Opt. Commun. 4, 428–430 (1972).
    [CrossRef]
  4. M. J. Downs and K W. Raine, “An unmodulated bi-directional fringe counting interferometer system for measuring displacements,” Precis. Eng. 1, 85–88 (1979).
    [CrossRef]
  5. M. J. Downs and J. W. Nunn, “Verification of the sub-nanometric capability of an NPL differential plane mirror interferometer with a capacitance probe,” Meas. Sci. Technol. 9, 1437–1440 (1998).
    [CrossRef]
  6. P. de Groot, “Jones matrix analysis of high-precision displacement measuring interferometers,” in ODIMAP II (Pavia, 1999), pp. 9–14.
  7. H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17, 746–752 (2006).
    [CrossRef]
  8. J. J. Snyder, “Paraxial ray analysis of a cat’s-eye retroreflector,” Appl. Opt. 14, 1825–1828 (1975).
    [CrossRef] [PubMed]
  9. M. L. Biermann, W. S. Rabinovich, R. Mahon, and G. C. Gilbreath, “Design and analysis of a diffraction-limited cat’s-eye retroreflector,” Opt. Eng. 41, 1655–1660 (2002).
    [CrossRef]
  10. S. E. Segre and V. Zanza, “Mueller calculus of polarization change in the cube-corner retroreflector,” J. Opt. Soc. Am. A 20, 1804–1811 (2003).
    [CrossRef]
  11. J. Dyson, Interferometry as a Measuring Tool (The Machinery Publishing Company, 1970).
  12. F. Petrů and O. Čip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
    [CrossRef]
  13. G. D. Hammond, A. Pulido Patón, C. C. Speake, and C. Trenkel, “Novel torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 75, 955–961(2004).
    [CrossRef]
  14. G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
    [CrossRef] [PubMed]
  15. F. E. Peña-Arellano, C. C. Speake, H. Panjwani, and L. Carbone “An interferometer for measuring angular motion,” in preparation.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  18. K. W. Raine and M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” J. Mod. Opt. 25, 549–558 (1978).
    [CrossRef]
  19. V. Greco, C. Iemmi, S. Ledesma, G. Molesini, and F. Quercioli, “Multiphase homodyne interferometry: analysis of some error sources,” Appl. Opt. 34, 2207–2213 (1995).
    [CrossRef] [PubMed]
  20. M. A. Zumberge, J. Berger, M. A. Dzieciuch, and R. L. Parker, “Resolving quadrature fringes in real time,” Appl. Opt. 43, 771–775 (2004).
    [CrossRef] [PubMed]
  21. F. E. Peña-Arellano, “Characterization of polarization homodyne interferometers,” Ph.D. dissertation (University of Birmingham, 2008).
  22. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
    [CrossRef]
  23. F. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, 1996).
  24. S. Wang and L. Ronchi, “Principles and design of optical arrays,” Prog. Opt. 25, 279–310 (1988).
    [CrossRef]
  25. C. C. Speake and S. M. Aston, “An interferometric sensor for satellite drag-free control,” Class. Quant. Grav. 22, S269–S277 (2005).
    [CrossRef]
  26. F. E. Peña-Arellano, C. C. Speake, and S. M. Aston are preparing a manuscript to be called “Experimental realization of an interferometer with mirror tilt immunity.”
  27. C. C. Speake, S. M. Aston, F. E. Peña-Arellano, and T. P. E. Copland, “Improved interferometer,” International patent, pub. no. WO 2009/010750 A1 (22 January 2009).

2008 (1)

G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
[CrossRef] [PubMed]

2006 (1)

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17, 746–752 (2006).
[CrossRef]

2005 (1)

C. C. Speake and S. M. Aston, “An interferometric sensor for satellite drag-free control,” Class. Quant. Grav. 22, S269–S277 (2005).
[CrossRef]

2004 (2)

G. D. Hammond, A. Pulido Patón, C. C. Speake, and C. Trenkel, “Novel torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 75, 955–961(2004).
[CrossRef]

M. A. Zumberge, J. Berger, M. A. Dzieciuch, and R. L. Parker, “Resolving quadrature fringes in real time,” Appl. Opt. 43, 771–775 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

M. L. Biermann, W. S. Rabinovich, R. Mahon, and G. C. Gilbreath, “Design and analysis of a diffraction-limited cat’s-eye retroreflector,” Opt. Eng. 41, 1655–1660 (2002).
[CrossRef]

1999 (1)

F. Petrů and O. Čip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

1998 (1)

M. J. Downs and J. W. Nunn, “Verification of the sub-nanometric capability of an NPL differential plane mirror interferometer with a capacitance probe,” Meas. Sci. Technol. 9, 1437–1440 (1998).
[CrossRef]

1995 (1)

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

1988 (1)

S. Wang and L. Ronchi, “Principles and design of optical arrays,” Prog. Opt. 25, 279–310 (1988).
[CrossRef]

1981 (1)

1979 (1)

M. J. Downs and K W. Raine, “An unmodulated bi-directional fringe counting interferometer system for measuring displacements,” Precis. Eng. 1, 85–88 (1979).
[CrossRef]

1978 (1)

K. W. Raine and M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” J. Mod. Opt. 25, 549–558 (1978).
[CrossRef]

1975 (1)

1972 (1)

S. J. Bennett, “A double-passed Michelson interferometer,” Opt. Commun. 4, 428–430 (1972).
[CrossRef]

1961 (1)

P. Hariharan and D. Sen, “The separation of symmetrical and asymmetrical wave-front aberrations in the Twyman interferometer,” Proc. Phys. Soc. London 77, 328–334 (1961).
[CrossRef]

1953 (1)

1948 (1)

Aston, S. M.

C. C. Speake and S. M. Aston, “An interferometric sensor for satellite drag-free control,” Class. Quant. Grav. 22, S269–S277 (2005).
[CrossRef]

F. E. Peña-Arellano, C. C. Speake, and S. M. Aston are preparing a manuscript to be called “Experimental realization of an interferometer with mirror tilt immunity.”

C. C. Speake, S. M. Aston, F. E. Peña-Arellano, and T. P. E. Copland, “Improved interferometer,” International patent, pub. no. WO 2009/010750 A1 (22 January 2009).

Bennett, S. J.

S. J. Bennett, “A double-passed Michelson interferometer,” Opt. Commun. 4, 428–430 (1972).
[CrossRef]

Berger, J.

Biermann, M. L.

M. L. Biermann, W. S. Rabinovich, R. Mahon, and G. C. Gilbreath, “Design and analysis of a diffraction-limited cat’s-eye retroreflector,” Opt. Eng. 41, 1655–1660 (2002).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Büchner, H. J.

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17, 746–752 (2006).
[CrossRef]

Carbone, L.

F. E. Peña-Arellano, C. C. Speake, H. Panjwani, and L. Carbone “An interferometer for measuring angular motion,” in preparation.

Cip, O.

F. Petrů and O. Čip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Copland, T. P. E.

C. C. Speake, S. M. Aston, F. E. Peña-Arellano, and T. P. E. Copland, “Improved interferometer,” International patent, pub. no. WO 2009/010750 A1 (22 January 2009).

de Groot, P.

P. de Groot, “Jones matrix analysis of high-precision displacement measuring interferometers,” in ODIMAP II (Pavia, 1999), pp. 9–14.

Downs, M. J.

M. J. Downs and J. W. Nunn, “Verification of the sub-nanometric capability of an NPL differential plane mirror interferometer with a capacitance probe,” Meas. Sci. Technol. 9, 1437–1440 (1998).
[CrossRef]

M. J. Downs and K W. Raine, “An unmodulated bi-directional fringe counting interferometer system for measuring displacements,” Precis. Eng. 1, 85–88 (1979).
[CrossRef]

K. W. Raine and M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” J. Mod. Opt. 25, 549–558 (1978).
[CrossRef]

Dyson, J.

J. Dyson, Interferometry as a Measuring Tool (The Machinery Publishing Company, 1970).

Dzieciuch, M. A.

Gilbreath, G. C.

M. L. Biermann, W. S. Rabinovich, R. Mahon, and G. C. Gilbreath, “Design and analysis of a diffraction-limited cat’s-eye retroreflector,” Opt. Eng. 41, 1655–1660 (2002).
[CrossRef]

Greco, V.

Hammond, G. D.

G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
[CrossRef] [PubMed]

G. D. Hammond, A. Pulido Patón, C. C. Speake, and C. Trenkel, “Novel torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 75, 955–961(2004).
[CrossRef]

Hariharan, P.

P. Hariharan and D. Sen, “The separation of symmetrical and asymmetrical wave-front aberrations in the Twyman interferometer,” Proc. Phys. Soc. London 77, 328–334 (1961).
[CrossRef]

Heydemann, P. L. M.

Iemmi, C.

Jäger, G.

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17, 746–752 (2006).
[CrossRef]

Ledesma, S.

Mahon, R.

M. L. Biermann, W. S. Rabinovich, R. Mahon, and G. C. Gilbreath, “Design and analysis of a diffraction-limited cat’s-eye retroreflector,” Opt. Eng. 41, 1655–1660 (2002).
[CrossRef]

Matthews, A. J.

G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
[CrossRef] [PubMed]

Molesini, G.

Nunn, J. W.

M. J. Downs and J. W. Nunn, “Verification of the sub-nanometric capability of an NPL differential plane mirror interferometer with a capacitance probe,” Meas. Sci. Technol. 9, 1437–1440 (1998).
[CrossRef]

Obetz, S. W.

Panjwani, H.

F. E. Peña-Arellano, C. C. Speake, H. Panjwani, and L. Carbone “An interferometer for measuring angular motion,” in preparation.

Parker, R. L.

Patón, A. Pulido

G. D. Hammond, A. Pulido Patón, C. C. Speake, and C. Trenkel, “Novel torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 75, 955–961(2004).
[CrossRef]

Peck, E. R.

Pedrotti, F.

F. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, 1996).

Pedrotti, L.

F. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, 1996).

Peña-Arellano, F. E.

G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
[CrossRef] [PubMed]

F. E. Peña-Arellano, C. C. Speake, H. Panjwani, and L. Carbone “An interferometer for measuring angular motion,” in preparation.

F. E. Peña-Arellano, “Characterization of polarization homodyne interferometers,” Ph.D. dissertation (University of Birmingham, 2008).

F. E. Peña-Arellano, C. C. Speake, and S. M. Aston are preparing a manuscript to be called “Experimental realization of an interferometer with mirror tilt immunity.”

Petru, F.

F. Petrů and O. Čip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Quercioli, F.

Rabinovich, W. S.

M. L. Biermann, W. S. Rabinovich, R. Mahon, and G. C. Gilbreath, “Design and analysis of a diffraction-limited cat’s-eye retroreflector,” Opt. Eng. 41, 1655–1660 (2002).
[CrossRef]

Raine, K W.

M. J. Downs and K W. Raine, “An unmodulated bi-directional fringe counting interferometer system for measuring displacements,” Precis. Eng. 1, 85–88 (1979).
[CrossRef]

Raine, K. W.

K. W. Raine and M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” J. Mod. Opt. 25, 549–558 (1978).
[CrossRef]

Rocco, E.

G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
[CrossRef] [PubMed]

Ronchi, L.

S. Wang and L. Ronchi, “Principles and design of optical arrays,” Prog. Opt. 25, 279–310 (1988).
[CrossRef]

Segre, S. E.

Sen, D.

P. Hariharan and D. Sen, “The separation of symmetrical and asymmetrical wave-front aberrations in the Twyman interferometer,” Proc. Phys. Soc. London 77, 328–334 (1961).
[CrossRef]

Snyder, J. J.

Speake, C. C.

G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
[CrossRef] [PubMed]

C. C. Speake and S. M. Aston, “An interferometric sensor for satellite drag-free control,” Class. Quant. Grav. 22, S269–S277 (2005).
[CrossRef]

G. D. Hammond, A. Pulido Patón, C. C. Speake, and C. Trenkel, “Novel torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 75, 955–961(2004).
[CrossRef]

F. E. Peña-Arellano, C. C. Speake, H. Panjwani, and L. Carbone “An interferometer for measuring angular motion,” in preparation.

F. E. Peña-Arellano, C. C. Speake, and S. M. Aston are preparing a manuscript to be called “Experimental realization of an interferometer with mirror tilt immunity.”

C. C. Speake, S. M. Aston, F. E. Peña-Arellano, and T. P. E. Copland, “Improved interferometer,” International patent, pub. no. WO 2009/010750 A1 (22 January 2009).

Trenkel, C.

G. D. Hammond, A. Pulido Patón, C. C. Speake, and C. Trenkel, “Novel torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 75, 955–961(2004).
[CrossRef]

Wang, S.

S. Wang and L. Ronchi, “Principles and design of optical arrays,” Prog. Opt. 25, 279–310 (1988).
[CrossRef]

Zanza, V.

Zumberge, M. A.

Appl. Opt. (4)

Class. Quant. Grav. (1)

C. C. Speake and S. M. Aston, “An interferometric sensor for satellite drag-free control,” Class. Quant. Grav. 22, S269–S277 (2005).
[CrossRef]

J. Mod. Opt. (1)

K. W. Raine and M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” J. Mod. Opt. 25, 549–558 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Meas. Sci. Technol. (3)

M. J. Downs and J. W. Nunn, “Verification of the sub-nanometric capability of an NPL differential plane mirror interferometer with a capacitance probe,” Meas. Sci. Technol. 9, 1437–1440 (1998).
[CrossRef]

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17, 746–752 (2006).
[CrossRef]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Opt. Commun. (1)

S. J. Bennett, “A double-passed Michelson interferometer,” Opt. Commun. 4, 428–430 (1972).
[CrossRef]

Opt. Eng. (1)

M. L. Biermann, W. S. Rabinovich, R. Mahon, and G. C. Gilbreath, “Design and analysis of a diffraction-limited cat’s-eye retroreflector,” Opt. Eng. 41, 1655–1660 (2002).
[CrossRef]

Precis. Eng. (2)

M. J. Downs and K W. Raine, “An unmodulated bi-directional fringe counting interferometer system for measuring displacements,” Precis. Eng. 1, 85–88 (1979).
[CrossRef]

F. Petrů and O. Čip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Proc. Phys. Soc. London (1)

P. Hariharan and D. Sen, “The separation of symmetrical and asymmetrical wave-front aberrations in the Twyman interferometer,” Proc. Phys. Soc. London 77, 328–334 (1961).
[CrossRef]

Prog. Opt. (1)

S. Wang and L. Ronchi, “Principles and design of optical arrays,” Prog. Opt. 25, 279–310 (1988).
[CrossRef]

Rev. Sci. Instrum. (2)

G. D. Hammond, A. Pulido Patón, C. C. Speake, and C. Trenkel, “Novel torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 75, 955–961(2004).
[CrossRef]

G. D. Hammond, C. C. Speake, A. J. Matthews, E. Rocco, and F. E. Peña-Arellano, “Development of a second generation torsion balance based on a spherical superconducting suspension,” Rev. Sci. Instrum. 79, 025103 (2008).
[CrossRef] [PubMed]

Other (7)

F. E. Peña-Arellano, C. C. Speake, H. Panjwani, and L. Carbone “An interferometer for measuring angular motion,” in preparation.

P. de Groot, “Jones matrix analysis of high-precision displacement measuring interferometers,” in ODIMAP II (Pavia, 1999), pp. 9–14.

F. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, 1996).

F. E. Peña-Arellano, “Characterization of polarization homodyne interferometers,” Ph.D. dissertation (University of Birmingham, 2008).

J. Dyson, Interferometry as a Measuring Tool (The Machinery Publishing Company, 1970).

F. E. Peña-Arellano, C. C. Speake, and S. M. Aston are preparing a manuscript to be called “Experimental realization of an interferometer with mirror tilt immunity.”

C. C. Speake, S. M. Aston, F. E. Peña-Arellano, and T. P. E. Copland, “Improved interferometer,” International patent, pub. no. WO 2009/010750 A1 (22 January 2009).

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Figures (11)

Fig. 1
Fig. 1

Four different configurations of the polarization Michelson interferometer.

Fig. 2
Fig. 2

Misalignment cancellation of the retroreflected beam in the configurations shown in Figs. 1c, 1d.

Fig. 3
Fig. 3

Unfolded propagation diagram of a cat’s eye retroreflector with a flat mirror. It is a symmetric system.

Fig. 4
Fig. 4

Cube corners produce virtual images of objects. For simplicity, refraction at the entrance face of the cube corner was not considered in the diagram.

Fig. 5
Fig. 5

(a) Afocal and (b) focal configurations of the cat’s eye retroreflector with a flat mirror.

Fig. 6
Fig. 6

Optimized cat’s eye retroreflector. The optics are 5 mm in size.

Fig. 7
Fig. 7

Root-mean-square wavefront error in waves with respect to the principal ray for both configurations.

Fig. 8
Fig. 8

Cat’s eye retroreflector within a homodyne polarization interferometer. A, laser; B, E, and K, polarizing beam splitters; C, beam splitter; D, half-wave-plate; F, F , quarter-wave-plate; G, G , mirrors; H, aspheric lens; I, mirror; J, quarter-wave plate; P 1 , P 2 , and P 3 , photodiodes.

Fig. 9
Fig. 9

Optical path difference measured by the interferometer when the mirror rotates around the sweet spot. The abscissa is the normalized pupil coordinate of the chief ray of the Gaussian beam.

Fig. 10
Fig. 10

Parameter C follows a linear dependence with s as predicted by the paraxial approximation [see Eq. (22)]. Compared with the defocus, the spherical aberration remains constant.

Fig. 11
Fig. 11

From the standpoint of the amplitude of the Lissajous pattern, a cat’s eye provides a better mirror tilt immunity than a cube corner. Parameter s is the distance from the mirror to the sweet plane. It is negative when the mirror is closer to the interferometer than the sweet plane and positive when it is further away.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

r out = ( D d ) ( RR ) ( D d ) ( r in ) ,
r in = [ x i y i ] , r out = [ x o y o ] , D d = [ 1 d 0 1 ] , RR = [ A B C D ] ,
r out = ( D d ) ( RR ) ( D d ) ( r ) = [ A ( A + 1 A ) d + B 0 1 A ] [ x i y i ] .
d = A B A 2 + 1 .
r out = [ A ( A + 1 A ) ϵ 0 1 A ] [ x i y i ] .
r out = [ x o y o ] = [ 1 2 ϵ 0 1 ] [ x i y i ] ,
RR = [ 1 B 0 1 ] ,
L f = [ 1 0 1 f 1 ] ,
D f = [ 1 f 0 1 ] ,
M S = [ 1 0 2 S 1 ] .
RR = ( L f ) ( D f ) ( M S ) ( D f ) ( L f ) ,
B = { 2 f flat mirror , 2 ( f + f 2 | S | ) convex mirror , 2 ( f f 2 | S | ) concave mirror .
CC = [ 1 2 h n 0 1 ] ,
x o = x i + ( y i 2 θ ) ( B rr 2 d ) ,
y o = y i ,
L = L 0 4 s θ 2 + s y i 2 ,
W f ( h , θ ) = W a ( h , θ ) + Δ C ( h , θ ) ,
d = 1 4 ( λ 2 π ) arctan ( x 1 y 2 x 2 y 1 x 1 x 2 + y 1 y 2 ) ,
ρ = tan 2 θ tan 2.5 ° ,
W ( ρ ) = C ρ 2 + A ρ 4 ,
W ( ρ ) = C ρ 2 + A ρ 4 4 C ( tan 2.5 ° ) 2 θ 2 .
C = ( tan 2.5 ° ) 2 ( s + d f λ ) ,

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