Abstract

Manufacturing and misalignment errors are present in every optical system. Usually these errors lead to intolerable wavefront deviations and system inaccuracies if they are not characterized and taken into consideration. In the interferometric measurement of surfaces, the characterization of the interferometer aberrations plays a central role, since unknown phase contributions lead to an erroneous assessment of the test surface and therefore an incorrect estimation of the performance of an optical system. In this work, we present a method for the interferometric characterization of surfaces based on the principles of Hamilton’s characteristic functions and perturbation theory. The application of the proposed method to an interferometer for the measurement of aspherical surfaces is shown.

© 2009 Optical Society of America

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  1. H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253-282 (1989).
    [CrossRef]
  2. B. Braunecker, R. Hentschel, and H. J. Tiziani, “Advanced Optics Using Aspherical Elements,” SPIE Press Monograph, Vol. PM173 (2008).
  3. E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2002).
  4. J. L. Synge, Geometrical Optics: An Introduction to Hamilton's Method (Cambridge Univ. Press, 1937).
  5. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964), Chap. 2.
  6. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993), Chap. 2 and 3.
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), Chap. 3.
  8. H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
    [CrossRef]
  9. M. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533-537 (1970).
    [CrossRef] [PubMed]
  10. B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A 14, 2837-2849 (1997).
    [CrossRef]
  11. R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39, 2198-2209 (2000).
    [CrossRef]
  12. W. Osten, “Messung von Verschiebungsfeldern und Objektkoordinaten mit optimierten holografischen Interferometern,” Ph.D. dissertation (Martin-Luther-Universitaet, Halle, Germany, 1983).
  13. W. Osten, “Some considerations on the statistical error analysis in holographic interferometry,” J. Mod. Opt. 32, 827-838 (1985).
    [CrossRef]
  14. J. Liesener, E. Garbusi, C. Pruss, and W. Osten, “Verfahren und Messvorrichtung zur Vermessung einer optisch glatten Oberflaeche,” Deutsches Patent und Markenamt: 10 2006 057 606.3.
  15. E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973-2975 (2008).
    [CrossRef] [PubMed]
  16. C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology , 42, 577-580 (1993).
    [CrossRef]
  17. G. Schulz, “Aspheric surfaces,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (North-Holland, 1988), pp. 349-415.
    [CrossRef]
  18. J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43, 5143-5151 (2004).
    [CrossRef] [PubMed]
  19. M. Andrews, “Concatenation of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. 72, 1493-1497 (1982).
    [CrossRef]
  20. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005), Chap. 4.
    [CrossRef]
  21. A. W. Lohmann and D. P. Paris, “Space-variant image formation,” J. Opt. Soc. Am. 55, 1007-1013 (1965).
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  24. C. Progler and A. Wong, “Zernike coefficients: are they really enough?” Proc. SPIE 4000, 40-52 (2000).
    [CrossRef]
  25. H. A. Buchdahl, “Perturbations of the point characteristic,” J. Opt. Soc. Am. A 7, 2260-2263 (1990).
    [CrossRef]
  26. H. A. Buchdahl, “Perturbed characteristic functions II,” Int. J. Theor. Phys. 24, 457-465 (1985).
    [CrossRef]
  27. H. A. Buchdahl, “Perturbed characteristic functions III,” Int. J. Theor. Phys. 29, 209-213 (1990).
    [CrossRef]
  28. W. S. S. Blaschke, “A procedure for the differential correction of optical systems allowing large parameter changes,” J. Mod. Opt. 3, 10-23 (1956).
    [CrossRef]
  29. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

2008 (1)

2004 (1)

2000 (2)

1997 (1)

1993 (1)

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology , 42, 577-580 (1993).
[CrossRef]

1990 (2)

H. A. Buchdahl, “Perturbed characteristic functions III,” Int. J. Theor. Phys. 29, 209-213 (1990).
[CrossRef]

H. A. Buchdahl, “Perturbations of the point characteristic,” J. Opt. Soc. Am. A 7, 2260-2263 (1990).
[CrossRef]

1989 (1)

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253-282 (1989).
[CrossRef]

1985 (2)

W. Osten, “Some considerations on the statistical error analysis in holographic interferometry,” J. Mod. Opt. 32, 827-838 (1985).
[CrossRef]

H. A. Buchdahl, “Perturbed characteristic functions II,” Int. J. Theor. Phys. 24, 457-465 (1985).
[CrossRef]

1984 (1)

1982 (1)

1970 (1)

1966 (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

1965 (1)

1956 (1)

W. S. S. Blaschke, “A procedure for the differential correction of optical systems allowing large parameter changes,” J. Mod. Opt. 3, 10-23 (1956).
[CrossRef]

Andrews, M.

Blaschke, W. S. S.

W. S. S. Blaschke, “A procedure for the differential correction of optical systems allowing large parameter changes,” J. Mod. Opt. 3, 10-23 (1956).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), Chap. 3.

Braunecker, B.

B. Braunecker, R. Hentschel, and H. J. Tiziani, “Advanced Optics Using Aspherical Elements,” SPIE Press Monograph, Vol. PM173 (2008).

Bryan, J. B.

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology , 42, 577-580 (1993).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, “Perturbations of the point characteristic,” J. Opt. Soc. Am. A 7, 2260-2263 (1990).
[CrossRef]

H. A. Buchdahl, “Perturbed characteristic functions III,” Int. J. Theor. Phys. 29, 209-213 (1990).
[CrossRef]

H. A. Buchdahl, “Perturbed characteristic functions II,” Int. J. Theor. Phys. 24, 457-465 (1985).
[CrossRef]

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993), Chap. 2 and 3.

Chow, W. W.

Evans, C. J.

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology , 42, 577-580 (1993).
[CrossRef]

Gappinger, R. O.

Garbusi, E.

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973-2975 (2008).
[CrossRef] [PubMed]

J. Liesener, E. Garbusi, C. Pruss, and W. Osten, “Verfahren und Messvorrichtung zur Vermessung einer optisch glatten Oberflaeche,” Deutsches Patent und Markenamt: 10 2006 057 606.3.

Greivenkamp, J. E.

Hentschel, R.

B. Braunecker, R. Hentschel, and H. J. Tiziani, “Advanced Optics Using Aspherical Elements,” SPIE Press Monograph, Vol. PM173 (2008).

Hinch, E. J.

E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2002).

Hopkins, H. H.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

Lawrence, G. N.

Liesener, J.

J. Liesener, E. Garbusi, C. Pruss, and W. Osten, “Verfahren und Messvorrichtung zur Vermessung einer optisch glatten Oberflaeche,” Deutsches Patent und Markenamt: 10 2006 057 606.3.

Lohmann, A. W.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964), Chap. 2.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005), Chap. 4.
[CrossRef]

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005), Chap. 4.
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

Osten, W.

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973-2975 (2008).
[CrossRef] [PubMed]

W. Osten, “Some considerations on the statistical error analysis in holographic interferometry,” J. Mod. Opt. 32, 827-838 (1985).
[CrossRef]

J. Liesener, E. Garbusi, C. Pruss, and W. Osten, “Verfahren und Messvorrichtung zur Vermessung einer optisch glatten Oberflaeche,” Deutsches Patent und Markenamt: 10 2006 057 606.3.

W. Osten, “Messung von Verschiebungsfeldern und Objektkoordinaten mit optimierten holografischen Interferometern,” Ph.D. dissertation (Martin-Luther-Universitaet, Halle, Germany, 1983).

Paris, D. P.

Progler, C.

C. Progler and A. Wong, “Zernike coefficients: are they really enough?” Proc. SPIE 4000, 40-52 (2000).
[CrossRef]

Pruss, C.

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973-2975 (2008).
[CrossRef] [PubMed]

J. Liesener, E. Garbusi, C. Pruss, and W. Osten, “Verfahren und Messvorrichtung zur Vermessung einer optisch glatten Oberflaeche,” Deutsches Patent und Markenamt: 10 2006 057 606.3.

Rimmer, M.

Schulz, G.

G. Schulz, “Aspheric surfaces,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (North-Holland, 1988), pp. 349-415.
[CrossRef]

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005), Chap. 4.
[CrossRef]

Stone, B. D.

Synge, J. L.

J. L. Synge, Geometrical Optics: An Introduction to Hamilton's Method (Cambridge Univ. Press, 1937).

Tiziani, H. J.

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253-282 (1989).
[CrossRef]

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

B. Braunecker, R. Hentschel, and H. J. Tiziani, “Advanced Optics Using Aspherical Elements,” SPIE Press Monograph, Vol. PM173 (2008).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), Chap. 3.

Wong, A.

C. Progler and A. Wong, “Zernike coefficients: are they really enough?” Proc. SPIE 4000, 40-52 (2000).
[CrossRef]

Youngworth, R. N.

Appl. Opt. (3)

Br. J. Appl. Phys. (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

CIRP Annals Manufacturing Technology (1)

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe densities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology , 42, 577-580 (1993).
[CrossRef]

Int. J. Theor. Phys. (2)

H. A. Buchdahl, “Perturbed characteristic functions II,” Int. J. Theor. Phys. 24, 457-465 (1985).
[CrossRef]

H. A. Buchdahl, “Perturbed characteristic functions III,” Int. J. Theor. Phys. 29, 209-213 (1990).
[CrossRef]

J. Mod. Opt. (2)

W. S. S. Blaschke, “A procedure for the differential correction of optical systems allowing large parameter changes,” J. Mod. Opt. 3, 10-23 (1956).
[CrossRef]

W. Osten, “Some considerations on the statistical error analysis in holographic interferometry,” J. Mod. Opt. 32, 827-838 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253-282 (1989).
[CrossRef]

Proc. SPIE (1)

C. Progler and A. Wong, “Zernike coefficients: are they really enough?” Proc. SPIE 4000, 40-52 (2000).
[CrossRef]

Other (12)

B. Braunecker, R. Hentschel, and H. J. Tiziani, “Advanced Optics Using Aspherical Elements,” SPIE Press Monograph, Vol. PM173 (2008).

E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2002).

J. L. Synge, Geometrical Optics: An Introduction to Hamilton's Method (Cambridge Univ. Press, 1937).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964), Chap. 2.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993), Chap. 2 and 3.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), Chap. 3.

J. Liesener, E. Garbusi, C. Pruss, and W. Osten, “Verfahren und Messvorrichtung zur Vermessung einer optisch glatten Oberflaeche,” Deutsches Patent und Markenamt: 10 2006 057 606.3.

W. Osten, “Messung von Verschiebungsfeldern und Objektkoordinaten mit optimierten holografischen Interferometern,” Ph.D. dissertation (Martin-Luther-Universitaet, Halle, Germany, 1983).

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

G. Schulz, “Aspheric surfaces,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (North-Holland, 1988), pp. 349-415.
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005), Chap. 4.
[CrossRef]

ZEMAX Development Corporation, www.zemax.com.

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Figures (9)

Fig. 1
Fig. 1

Interferometer setup. L, HeNe laser; SF, spatial filter and beam expander; M 1 and M 2 , fold mirrors; L 1 , focusing lens; B S 1 and B S 2 , beam splitters; PSA, point source array; L 2 , collimation lens; TS, transmission sphere; T, test surface; B, interferometer aperture; L 3 , imaging lens; C, camera.

Fig. 2
Fig. 2

Fermat’s principle. Perturbation of the intermediate points ( A and B ) in the ray trajectories have, to first order, no influence on the optical path of the ray connecting A and B.

Fig. 3
Fig. 3

Typical misalignments of a spherical optical surface. (a) Tilt, (b) decentering, (c) displacement along the axis, (d) change of the curvature radius. Solid lines indicate nominal surfaces and rays, dotted ones their perturbed states.

Fig. 4
Fig. 4

Schematic representation of the test path of the interferometer. L, light source; T, test surface; W T , test wavefront; W R , wavefront after transmission (reflection) through the test object (surface) T; W D , wavefront reaching the detector plane; E Q , reference plane for the Q characteristic; E P , reference plane for the P characteristic.

Fig. 5
Fig. 5

Perturbation of the test surface in reflection configuration.

Fig. 6
Fig. 6

Positioning of the reference planes. Incoming wavefronts to the test space are characterized at the reference plane E Q by the point characteristic Q. Analogously, the characteristic P describes the outgoing wavefronts with respect to the reference plane E P .

Fig. 7
Fig. 7

Aspherical test surface. (a) Convex aspheric with 860 μ m deviation (PV) from the best-fit sphere over a 36 mm aperture. (b) Zernike polynomial fit of the surface (the offset has been removed for visualization purposes).

Fig. 8
Fig. 8

Retrieved surface deviation. (a) Original figure error due to the coefficient perturbation shown in (b). (b) Simulated and calculated Zernike perturbations. (c) Surface figure error after four iterations (difference between calculated and simulated perturbations). (d) Evolution of the residual surface figure error (PV).

Fig. 9
Fig. 9

Interferometer calibration. The real point characteristics Q and P are determined by means of corrections of their nominal states Q 0 and P 0 . A known surface (reference sphere) is used for the calibration measurements.

Equations (70)

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V A B = OPL [ A , A ] + OPL [ A , B ] + OPL [ B , B ] ,
V A B + δ V = OPL [ A + δ A , A + δ A ] + OPL [ A + δ A , B + δ B ] + OPL [ B + δ B , B + δ B ] .
V A B + δ V = OPL [ A + δ A , A ] + OPL [ A , B ] + OPL [ B , B + δ B ] ,
V A B A = 0 and V A B B = 0 .
δ W Tilt = W Pert W Nom = OPL [ E F ] OPL [ E G ] ,
δ W Tilt = E F ¯ cos α ( n cos α n cos α ) + E F ¯ sin α ( n sin α n sin α ) ,
δ W Tilt = E F ¯ cos α ( n cos α n cos α ) .
ϵ E = E F ¯ cos α .
δ W Tilt = ϵ E ( n cos α n cos α ) .
δ x = z δ β , δ z = x δ β ,
δ W Tilt = ( n cos α n cos α ) ( K E z M E x ) δ β ,
ϵ E = K E δ x ,
δ W Decx = ( n cos α n cos α ) K E δ x .
ϵ E = M E δ z ,
δ W Dispz = ( n cos α n cos α ) M E δ z .
F ( x , y , z , R ) = x 2 + y 2 + z 2 R 2 ,
{ x x + δ x = x + S k , y y + δ y = y + S l , z z + δ z = z + S m , } S = E F ¯ ,
S = 1 F x k + F y l + F z m F R δ R .
δ W Radius = 2 R ( n cos α n cos α ) δ R .
δ W = ( δ r g ) Γ ,
Γ = n i g n i g ,
W Pert = W Nom + ϵ W 1 + ϵ 2 W 2 + ϵ 3 W 3 + ,
g 0 ( x , y , z ) = z f 0 ( x , y ) ,
g 0 ( x , y , z ) = 0 ,
V ( x S , x D ) = Q { x S ; X ; g ( h , z ) , h } + P { g ( h , z ) , h ; x ; x D } ,
V ( x S , x D ) = V 0 + ϵ V 1 + ϵ 2 V 2 + = t ϵ t V t ( x S , x D ) ,
g ( h , z ) = g 0 ( h , z ) + ϵ g 1 ( h , z ) + ϵ 2 g 2 ( h , z ) + = t ϵ t g ( h , z ) ,
h ( x S , x D ) = h 0 ( x S , x D ) + ϵ h 1 ( x S , x D ) + ϵ 2 h 2 ( x S , x D ) + = t ϵ t h t ( x S , x D ) .
V ( x S , x D ) = Q 0 { x S ; X ; r ϵ r g r [ t ϵ t h t ( x S , x D ) , z ] , r ϵ r h r ( x S , x D ) } + + P 0 { r ϵ r g r [ t ϵ t h t ( x S , x D ) , z ] , r ϵ r h r ( x S , x D ) ; x ; x D } .
| [ ( Q 0 + P 0 ) h + ( Q 0 + P 0 ) r g 0 h ] | h = h 0 ( x S , x D ) = 0 .
V FO ( x S ; h ; x D ) = V 0 ( x S , x D ) + ϵ ( Q 0 + P 0 ) ϵ = V 0 ( x S , x D ) + ϵ { h 1 [ ( Q 0 + P 0 ) h + ( Q 0 + P 0 ) r g 0 h ] + ( Q 0 + P 0 ) r g 1 } + O ( ϵ 2 ) ,
V FO ( x S ; h 0 ; x D ) = V 0 ( x S , x D ) + ϵ | [ g 1 ( Q 0 + P 0 ) r ] | h = h 0 ( x S , x D ) ,
V O ( x S ; x D ) = Q 0 { x S ; X ; g 0 ( h 0 , z 0 ) , h 0 } + P 0 { g 0 ( h 0 , z 0 ) , h 0 ; x ; x D } .
V FO ( x S ; h 0 ; x D ) V 0 ( x S , x D ) = ϵ A ( x S , x D ) ,
A ( x S , x D ) = | [ g 1 ( Q 0 + P 0 ) r ] | h = h 0 ( x S , x D ) .
| Q 0 r | h = h 0 = n | ( k , l , m ) | h 0 ( x S , x D ) ,
| P 0 r | h = h 0 = ± n | ( k , l , m ) | h 0 ( x S , x D ) ,
N E = ( f x , f y , 1 ) 1 + ( f x ) 2 + ( f y ) 2 ,
A ( x S , x D ) = | g 1 ( h 0 , z 0 ) [ ± n N E ( k , l , m ) n N E ( k , l , m ) ] | h 0 ( x S , x D ) .
V FO ( x S ; h 0 ; x D ) V 0 ( x S , x D ) = ϵ | g 1 ( h 0 , z 0 ) ( ± n cos α n cos α ) | h 0 ( x S , x D ) ,
δ W Refl = Δ OPL ( h 0 , z 0 ) = n E F ¯ n E G ¯ ,
δ W Refl = E F ¯ cos α ( n cos α + n cos α ) + E F ¯ sin α ( n sin α + n sin α ) .
δ W Refl = n E F ¯ cos α ( cos α + cos α ) ,
z ( r ) = c r 2 1 + 1 ( 1 + k ) c 2 r 2 + i A 2 i r 2 i ,
W T ( ρ , θ ) = j N z a j Z j ( ρ , θ ) ,
B Z = { Z 1 , Z 2 , , Z F } , Z i , Z j = 0 , i j ;
T ( B Z ) B T ( z ) = { T ( Z 1 ) , T ( Z 2 ) , , T ( Z F ) } , T ( Z i ) , T ( Z j ) = 0 , i j ,
A = { a i j } , where a i j = δ OPL ( Ray i , ϵ j ) , ϵ = { ϵ j } .
A ϵ = V FO V 0 ,
Q { x S ; X ; g ( h , z ) , h } = Q 0 { x S ; X ; g ( h , z ) , h } + ϵ Q 1 { x S ; X ; g ( h , z ) , h } + + ϵ 2 Q 2 { x S ; X ; g ( h , z ) , h } + = t ϵ t Q t { x S ; X ; g ( h , z ) , h } ,
P { g ( h , z ) , h ; x ; x D } = P 0 { g ( h , z ) , h ; x ; x D } + ϵ P 1 { g ( h , z ) , h ; x ; x D } + + ϵ 2 P 2 { g ( h , z ) , h ; x ; x D } + = t ϵ t P t { g ( h , z ) , h ; x ; x D } .
W n m ( r , θ ) = R n m ( r ) { cos sin } ( m θ ) ,
W n m , W n m = 1 2 π 0 1 0 2 π W n m ( r , θ ) W n m ( r , θ ) r d r d θ .
r ( 1 r 2 ) d 2 R n m d r 2 + ( 1 3 r 2 ) d R n m d r + [ n ( n + 2 ) r m 2 r ] R n m = 0 ,
R n m d d r { r ( 1 r 2 ) d R n m d r } + [ n ( n + 2 ) r m 2 r ] R n m R n m R n m d d r { r ( 1 r 2 ) d R n m d r } + + [ n ( n + 2 ) r m 2 r ] R n m R n m = d d r { r ( 1 r 2 ) ( R n m d R n m d r R n m d R n m d r ) } .
0 1 R n m ( r ) R n m ( r ) r d r = 1 n ( n + 2 ) n ( n + 2 ) 0 1 d d r { r ( 1 r 2 ) ( R n m d R n m d r R n m d R n m d r ) } d r = 1 n ( n + 2 ) n ( n + 2 ) | r ( 1 r 2 ) ( R n m d R n m d r R n m d R n m d r ) | 0 1 ,
E ( x ) = E ( x ) A ( x x , x ) d x ,
E 1 ( x ) = E 1 ( x ) A ( x x , x ) d x ,
E 2 ( x ) = E 2 ( x ) A ( x x , x ) d x ,
E 1 ( x ) E 2 * ( x ) d x = E 1 ( x ) E 2 * ( x ) C ( x x , x , x ) d x d x ,
C ( x x , x , x ) = A ( x x , x ) A * ( x x , x ) d x .
exp [ i Φ j ( x ) ] exp [ i Φ j ( x ) ] A ( x x ; x ) d x ,
Φ j ( x ) Φ j ( x ) A ( x x ; x ) d x .
Φ 1 ( x ) Φ 2 * ( x ) d x Φ 1 ( x ) Φ 2 * ( x ) A ( x x ; x ) A * ( x x ; x ) d x d x d x Φ 1 ( x ) Φ 2 * ( x ) C ( x x ; x ; x ) d x d x .
x = ξ + η 2 ,
x = ξ η 2 ,
Φ 1 ( x ) Φ 2 * ( x ) d x Φ 1 ( ξ + η 2 ) Φ 2 * ( ξ η 2 ) C ( η ; ξ + η 2 , ξ η 2 ) d ξ d η .
Φ 1 ( x ) Φ 2 * ( x ) d x Δ η Φ 1 ( ξ ) Φ 2 * ( ξ ) C ( 0 ; ξ ) d ξ .
A ( x x ; x ) A * ( x x ; x ) d x = 1 , x R
Φ 1 ( x ) Φ 2 * ( x ) d x K Φ 1 ( ξ ) Φ 2 * ( ξ ) d ξ ,

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