Abstract

A new approach to the analysis of grating systems, based on an investigation of the changes of the Fourier spectrum during field propagation through the gratings, is applied to analyze the Fresnel field of an arbitrary system. Detailed considerations of Talbot and grating-shearing interferometers used in various metrological problems are presented. A qualitative analysis of the output fringe patterns is given. The application of the phase-stepping method with the phase shift introduced by lateral and longitudinal translations of gratings is discussed. These theoretical considerations are checked by experiment.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Lohmann, O. Bryngdahl, “A lateral wavefront shearing interferometer with variable shear,” Appl. Opt. 6, 1934–1937 (1967).
    [CrossRef] [PubMed]
  2. J. Schwider, “Single sideband Ronchi test,” Appl. Opt. 20, 2635–2642 (1981).
    [CrossRef] [PubMed]
  3. V. A. Komissaruk, “Investigation of wave front aberration of optical systems using three-beam interference,” Opt. Spek-trosk. 16, 571–575 (1964).
  4. N. M. Spornik, V. I. Yanichkin, “Grating interferometer with variable shearing wavefronts and arbitrary fringe detection,” Sov. J. Opt. Technol. 38, 487–489 (1971).
  5. P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–322 (1974).
    [CrossRef]
  6. K. Patorski, “Grating shearing interferometer with variable shear and fringe orientation,” Appl. Opt. 25, 4192–4198 (1986), and references therein.
    [CrossRef] [PubMed]
  7. J. Ebbeni, “Noveaux aspects du phenomene de moire. II Formation des franges en eclairement coherent,” Nouv. Rev. Opt. Appl. 1, 353–358 (1967).
    [CrossRef]
  8. P. Szwaykowski, K. Patorski, “Properties of the Fresnel field of a double-diffraction system,” J. Opt. 16, 95–103 (1985).
    [CrossRef]
  9. C. C. Iemmi, J. M. Simon, J. O. Ratto, “Synthesis of asymmetric profiles from a double grating interferometer,” Appl. Opt. 25, 3171–3178 (1986).
    [CrossRef] [PubMed]
  10. D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
    [CrossRef] [PubMed]
  11. K. Patorski, P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23–31 (1984), and references therein.
    [CrossRef]
  12. M. Kujawinska, “Development of the theory of quasi-periodic diffraction grating systems,” J. Opt. Soc. Am. A 5, 206–213 (1988).
    [CrossRef]
  13. J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).
  14. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  15. T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
    [CrossRef] [PubMed]
  16. O. Y. Kwon, D. M. Shough, “Multichannel grating phase-shift interferometers,” in Optics in Engineering MeasurementW. F. Fagan, ed., Proc. Soc. Phot-Opt. Instrum. Eng.599, 273–279 (1985).
    [CrossRef]
  17. K. Patorski, “Modified double grating shearing interferometer,” Opt. Appl. 14, 149–152 (1984).
  18. K. Patorski, L. Satbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry: further considerations,” Opt. Acta 32, 1323–1331 (1985).
    [CrossRef]
  19. R. Jozwicki, “Fresnel field of a multiple diffraction grating system illuminated by a point source,” Opt. Appl. 17, 13–311987).
  20. G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1987).
    [CrossRef]
  21. D. W. Robinson, D. C. Williams, “Automatic fringe analysis in double exposure and live fringe holographic interferometry,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 134–140 (1985).
    [CrossRef]
  22. J. Schwider, R. Burow, K. -E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  23. Y. Y-Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef]

1988 (1)

1987 (2)

R. Jozwicki, “Fresnel field of a multiple diffraction grating system illuminated by a point source,” Opt. Appl. 17, 13–311987).

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1987).
[CrossRef]

1986 (2)

1985 (3)

P. Szwaykowski, K. Patorski, “Properties of the Fresnel field of a double-diffraction system,” J. Opt. 16, 95–103 (1985).
[CrossRef]

K. Patorski, L. Satbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry: further considerations,” Opt. Acta 32, 1323–1331 (1985).
[CrossRef]

Y. Y-Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef]

1984 (3)

K. Patorski, P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23–31 (1984), and references therein.
[CrossRef]

T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
[CrossRef] [PubMed]

K. Patorski, “Modified double grating shearing interferometer,” Opt. Appl. 14, 149–152 (1984).

1983 (1)

1982 (1)

1981 (1)

1974 (1)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–322 (1974).
[CrossRef]

1972 (1)

1971 (1)

N. M. Spornik, V. I. Yanichkin, “Grating interferometer with variable shearing wavefronts and arbitrary fringe detection,” Sov. J. Opt. Technol. 38, 487–489 (1971).

1967 (2)

A. Lohmann, O. Bryngdahl, “A lateral wavefront shearing interferometer with variable shear,” Appl. Opt. 6, 1934–1937 (1967).
[CrossRef] [PubMed]

J. Ebbeni, “Noveaux aspects du phenomene de moire. II Formation des franges en eclairement coherent,” Nouv. Rev. Opt. Appl. 1, 353–358 (1967).
[CrossRef]

1964 (1)

V. A. Komissaruk, “Investigation of wave front aberration of optical systems using three-beam interference,” Opt. Spek-trosk. 16, 571–575 (1964).

Bruning, J. H.

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

Bryngdahl, O.

Burow, R.

Ebbeni, J.

J. Ebbeni, “Noveaux aspects du phenomene de moire. II Formation des franges en eclairement coherent,” Nouv. Rev. Opt. Appl. 1, 353–358 (1967).
[CrossRef]

Elssner, K. -E.

Grzanna, J.

Hariharan, P.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–322 (1974).
[CrossRef]

Iemmi, C. C.

Ina, H.

Jozwicki, R.

R. Jozwicki, “Fresnel field of a multiple diffraction grating system illuminated by a point source,” Opt. Appl. 17, 13–311987).

Kobayashi, S.

Komissaruk, V. A.

V. A. Komissaruk, “Investigation of wave front aberration of optical systems using three-beam interference,” Opt. Spek-trosk. 16, 571–575 (1964).

Kujawinska, M.

Kwon, O. Y.

O. Y. Kwon, D. M. Shough, “Multichannel grating phase-shift interferometers,” in Optics in Engineering MeasurementW. F. Fagan, ed., Proc. Soc. Phot-Opt. Instrum. Eng.599, 273–279 (1985).
[CrossRef]

Lohmann, A.

Merkel, K.

Patorski, K.

K. Patorski, “Grating shearing interferometer with variable shear and fringe orientation,” Appl. Opt. 25, 4192–4198 (1986), and references therein.
[CrossRef] [PubMed]

K. Patorski, L. Satbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry: further considerations,” Opt. Acta 32, 1323–1331 (1985).
[CrossRef]

P. Szwaykowski, K. Patorski, “Properties of the Fresnel field of a double-diffraction system,” J. Opt. 16, 95–103 (1985).
[CrossRef]

K. Patorski, P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23–31 (1984), and references therein.
[CrossRef]

K. Patorski, “Modified double grating shearing interferometer,” Opt. Appl. 14, 149–152 (1984).

Ratto, J. O.

Reid, G. T.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1987).
[CrossRef]

Robinson, D. W.

D. W. Robinson, D. C. Williams, “Automatic fringe analysis in double exposure and live fringe holographic interferometry,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 134–140 (1985).
[CrossRef]

Satbut, L.

K. Patorski, L. Satbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry: further considerations,” Opt. Acta 32, 1323–1331 (1985).
[CrossRef]

Schwider, J.

Shough, D. M.

O. Y. Kwon, D. M. Shough, “Multichannel grating phase-shift interferometers,” in Optics in Engineering MeasurementW. F. Fagan, ed., Proc. Soc. Phot-Opt. Instrum. Eng.599, 273–279 (1985).
[CrossRef]

Silva, D. E.

Simon, J. M.

Spolaczyk, R.

Spornik, N. M.

N. M. Spornik, V. I. Yanichkin, “Grating interferometer with variable shearing wavefronts and arbitrary fringe detection,” Sov. J. Opt. Technol. 38, 487–489 (1971).

Steel, W. H.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–322 (1974).
[CrossRef]

Szwaykowski, P.

P. Szwaykowski, K. Patorski, “Properties of the Fresnel field of a double-diffraction system,” J. Opt. 16, 95–103 (1985).
[CrossRef]

K. Patorski, P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23–31 (1984), and references therein.
[CrossRef]

Takeda, M.

Williams, D. C.

D. W. Robinson, D. C. Williams, “Automatic fringe analysis in double exposure and live fringe holographic interferometry,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 134–140 (1985).
[CrossRef]

Wyant, J. C.

Y. Y-Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef]

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–322 (1974).
[CrossRef]

Yanichkin, V. I.

N. M. Spornik, V. I. Yanichkin, “Grating interferometer with variable shearing wavefronts and arbitrary fringe detection,” Sov. J. Opt. Technol. 38, 487–489 (1971).

Yatagai, T.

Y-Cheng, Y.

Appl. Opt. (8)

J. Opt. (1)

P. Szwaykowski, K. Patorski, “Properties of the Fresnel field of a double-diffraction system,” J. Opt. 16, 95–103 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nouv. Rev. Opt. Appl. (1)

J. Ebbeni, “Noveaux aspects du phenomene de moire. II Formation des franges en eclairement coherent,” Nouv. Rev. Opt. Appl. 1, 353–358 (1967).
[CrossRef]

Opt. Acta (2)

K. Patorski, L. Satbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry: further considerations,” Opt. Acta 32, 1323–1331 (1985).
[CrossRef]

K. Patorski, P. Szwaykowski, “Optical differentiation of quasi-periodic patterns using Talbot interferometry,” Opt. Acta 31, 23–31 (1984), and references therein.
[CrossRef]

Opt. Appl. (2)

K. Patorski, “Modified double grating shearing interferometer,” Opt. Appl. 14, 149–152 (1984).

R. Jozwicki, “Fresnel field of a multiple diffraction grating system illuminated by a point source,” Opt. Appl. 17, 13–311987).

Opt. Commun. (1)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–322 (1974).
[CrossRef]

Opt. Lasers Eng. (1)

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1987).
[CrossRef]

Opt. Spek-trosk. (1)

V. A. Komissaruk, “Investigation of wave front aberration of optical systems using three-beam interference,” Opt. Spek-trosk. 16, 571–575 (1964).

Sov. J. Opt. Technol. (1)

N. M. Spornik, V. I. Yanichkin, “Grating interferometer with variable shearing wavefronts and arbitrary fringe detection,” Sov. J. Opt. Technol. 38, 487–489 (1971).

Other (3)

D. W. Robinson, D. C. Williams, “Automatic fringe analysis in double exposure and live fringe holographic interferometry,” in Optics in Engineering Measurement, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.599, 134–140 (1985).
[CrossRef]

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

O. Y. Kwon, D. M. Shough, “Multichannel grating phase-shift interferometers,” in Optics in Engineering MeasurementW. F. Fagan, ed., Proc. Soc. Phot-Opt. Instrum. Eng.599, 273–279 (1985).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Field propagation through a double-grating system: G1 and G2, diffraction gratings; S, a coherent source; π, an observation plane.

Fig. 2
Fig. 2

Double-grating system with spatial filtering: (a) the grating-shearing interferometer; (b) the Talbot interferometer configuration. S, a coherent source; L1 and L2, lenses; G1 and G2, diffraction gratings (G1 ⊥ G2); SF, a spatial filter; π, an observation plane.

Fig. 3
Fig. 3

Double-grating system under plane-wave-front (or quasi-plane-wave-front) illumination: G1 and G2, diffraction gratings; L1, L2, and L3, lenses; SF, a spatial filter; SP, a shearing plate.

Fig. 4
Fig. 4

Interferograms obtained in the setup shown in Fig. 3, with the quasi-linear gratings illuminated by a plane-wave front: filtering of the (a) (0, 0)th, (b) (1, 0)th, (c) (0, 1)th, and (d) (1, 1)th diffraction orders.

Fig. 5
Fig. 5

Fourier spectrum of a double-grating system with distorted gratings and under quasi-plane-wave-front illumination.

Fig. 6
Fig. 6

Interferograms obtained in the setup shown in Fig. 3, with the quasi-linear gratings illuminated by a quasi-plane-wave front with comatic aberration: filtering of the diffraction orders as indicated in the figure.

Fig. 7
Fig. 7

Double-grating system under plane (or quasi-plane) illumination. Experimental setup for fringe pattern analysis by the phase-stepping method: S, a coherent source; G1 and G2, diffraction gratings; L1, L2, and L3, lenses; SF, a spatial filter; π, an observation plane.

Fig. 8
Fig. 8

(a), (b), (c) Intensity patterns for three positions of the second grating related to phase differences of −120, 0, and 120 deg, respectively. Linear phase fringes with discontinuities every 2π radians. (e) Result of automatically counting the fringes and removing the discontinuities, (f) Three-dimensional plot of the phase (the distortion of the grating G1).

Fig. 9
Fig. 9

A continuous phase map as computed from interferograms obtained in a grating-shearing interferometer for testing the first derivative of the aberration of the illuminating wave front. Continuous intensity variation maps obtained for different measurements with the phase shifts introduced by (a), (b) a lateral translation of G2, (d), (e) a longitudinal translation of G1. (c), (f) Phase-difference distributions obtained by comparisons of the phase distributions in (a) and (b) and in (d) and (e), respectively.

Equations (47)

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

V F 13 ( ρ m 1 m 2 ) = λ z 1 m 1 m 2 c m 1 c m 2 exp ( i m 2 ω 2 A 0 ) × exp ( i P 12 m 1 2 ω 1 2 ) exp { i P 23 [ ρ m 1 m 2 ( m 1 ω 1 + m 2 ω 2 ) ] 2 } G F m 1 m 2 ( ρ m 1 m 2 ) ,
G F m 1 m 2 ( ρ m 1 m 2 ) = FT A ρ m 1 m 2 { U m 2 ( A A 0 ) H m 1 [ A + 2 P 12 m 1 ω 1 ] }
H m 1 ( A ) = FT A ρ m 1 { FT [ U m 1 ( A ) U Σ ( A ) ] exp ( i P 12 ρ m 1 2 ) }
P i j = k 2 ( 1 z i 1 z j ) , k = 2 π / λ , i , j = 1 , 2 , 3 ;
V 3 ( A ) = m 1 m 2 K m 1 m 2 ( A ) H m 1 m 2 × [ A A 0 + 2 P 23 ( m 1 ω 1 + m 2 ω 2 ) ] ,
K m 1 m 2 ( A ) = z 1 z 3 c m 1 c m 2 exp ( i m 2 ω 2 A 0 ) exp ( i P 12 m 1 2 ω 1 2 ) × exp [ i P 23 ( m 1 ω 1 + m 2 ω 2 ) 2 ] × exp [ i A ( m 1 ω 1 + m 2 ω 2 ) ]
H m 1 m 2 ( A ) = FT A ρ m 1 m 2 [ G F m 1 m 2 ( ρ m 1 m 2 ) exp ( i P 23 ρ m 1 m 2 2 ) ] .
V F , 3 ( ρ Σ ) = λ z 1 m 1 m 2 c m 1 c m 2 × exp ( i m 2 ω 2 A 0 ) exp ( i P 12 m 1 2 ω 1 2 ) × exp { i P 23 [ ρ Σ ( m 1 ω 1 + m 2 ω 2 ) ] 2 } × FT A ρ Σ [ H m 1 ( A + 2 P 12 m 1 ω 1 ) ] ,
ρ Σ = ρ + m 1 ω 1 + m 2 ω 2
H m 1 ( A ) = FT A ρ Σ [ V F Σ ( ρ Σ ) exp ( i P 12 ρ Σ 2 ) ] .
V 3 ( A ) = V 0 m 1 m 2 K m 1 m 2 ( A ) H Σ { A + 2 [ P 12 m 1 ω 1 + P 23 ( m 1 ω 1 + m 2 ω 2 ) ] } ,
H Σ ( A ) = FT A ρ Σ [ V F Σ ( ρ Σ ) exp ( i P 13 ρ Σ 2 ) ]
P 13 = k 2 ( 1 z 1 1 z 3 ) .
V 3 ( A ) = z 1 z 3 V 0 exp ( i P 23 ω 2 ) exp ( i A ω ) { c 1 1 c 0 2 exp ( i P 12 ω 2 ) × H Σ [ A + 2 ω ( P 12 + P 23 ) ] + c 0 1 c 1 2 exp ( i ω A 0 ) × H Σ ( A + 2 P 23 ω ) } ,
δ = 2 π a 0 d ,
I 3 ( A ) = V 3 ( A ) V 3 * ( A ) c 0 + c 1 cos [ P 12 ω 2 δ + Φ ( A ) Φ ( A A Δ ) ] ,
Φ ( A ) Φ ( A A Δ ) + A Δ Φ ( A ) A ,
I 3 ( A ) = c 0 + c 1 cos [ P 12 ω 2 δ + Φ ( A ) A A Δ ] .
A Δ = 2 P 12 ω m = k ( 1 z 1 1 z 2 ) 2 π z 1 k d m = 2 π 2 Δ z d z 2 m ,
δ r + 1 = P 12 ω 2 ω A 0 ,
δ r 1 = P 12 ω 2 ω A 0 .
U 1 ( A ) 1 , U 2 ( A ) = 1 , V F 3 ( ρ m 1 ) = λ z 1 m 1 m 2 c m 1 c m 2 exp ( i m 2 ω 2 A 0 ) × exp [ i P 12 ( m 1 ω 1 ) 2 ] exp { i P 23 [ ρ m 1 ( m 1 ω 1 + m 2 ω 2 ) ] 2 } FT A ρ m 1 [ H m 1 ( A + 2 P 12 m 1 ω 1 ) ] ,
H m 1 ( A ) = FT A ρ m 1 [ U Fm 1 ( ρ m 1 ) exp ( i P 12 ρ m 1 2 ) ] , V 3 ( A ) = V 0 m 1 m 2 K m 1 m 2 ( A ) H m 1 m 2 × { A + 2 [ P 12 ( m 1 ω 1 ) + P 23 ( m 1 ω 1 + m 2 ω 2 ) ] } ,
H m 1 m 2 ( A ) = FT A ρ m 1 [ U F m 1 ( ρ m 1 ) exp ( i P 23 ρ m 1 2 ) ] .
U 1 ( A ) = 1 , U 2 ( A ) 1 , V F 3 ( ρ m 1 ) = λ z 1 m 1 m 2 c m 1 c m 2 exp ( i m 1 ω 2 A 0 ) × exp ( i P 12 ( m 1 ω 1 ) 2 exp { i P 23 [ ρ m 2 ( m 1 ω 1 + m 2 ω 2 ) ] 2 } FT A ρ m 2 [ U 2 ( A A 0 ) ] ,
V 3 ( A ) = V 0 K ( A ) H m 1 m 2 [ A A 0 + 2 P 23 ( m 1 ω 1 + m 2 ω 2 ) ] ,
H m 1 m 2 ( A ) = FT A ρ m 2 [ ( U F m 2 ( ρ m 2 ) exp ( i P 23 ρ m 2 2 ) ] .
U 1 ( A ) 1 , U 2 ( A ) 1 , V F , 3 ( ρ m 1 m 2 ) = λ z 1 m 1 m 2 c m 1 c m 2 exp ( i m 2 ω 2 A 0 ) × exp [ i P 12 ( m 1 ω 1 ) 2 ] exp { i P 23 [ ρ m 1 m 2 ( m 1 ω 1 + m 2 ω 2 ) ] 2 } G F m 1 m 2 ( ρ m 1 m 2 ) ,
G F m 1 m 2 ( ρ m 1 m 2 ) = FT A ρ m 1 m 2 [ U m 2 ( A A 0 ) × H m 1 ( A + 2 P 12 m 1 ω 1 ) ] , ρ m 1 m 2 = ρ m 1 + ρ m 2 ,
H m 1 ( A ) = FT A ρ m 1 [ U F m 1 ( ρ m 1 ) exp ( i P 12 ρ m 1 2 ) ] , V 3 ( A ) = V 0 m 1 m 2 K m 1 m 2 ( A ) H m 1 m 2 [ A A 0 + 2 P 23 ( m 1 ω 1 + m 2 ω 2 ) ] ,
H m 1 m 2 ( A ) = FT A ρ m 1 m 2 [ G F m 1 m 2 ( ρ m 1 m 2 ) exp ( i P 23 ρ m 1 m 2 2 ) ] .
V 2 ( A ) = z 1 z 2 V 0 exp ( i A ω ) [ c 1 1 c 0 2 exp ( i P 12 ω 2 ) H 10 ( A + 2 P 12 ω ) + c 0 1 c 1 2 exp ( i ω A 0 ) H 01 ( A A 0 ) ] ,
H 10 ( A ) = FT A ρ m 1 [ U F 1 ( ρ m 1 ) exp ( i P 12 ρ m 1 2 ) δ ( ρ m ) ] U 1 ( A ) , H 01 ( A ) = FT A ρ m 1 [ δ ( ρ m ) U F 1 ( ρ m 1 ) ] = U 1 ( A ) .
I 2 ( A ) = V 2 ( A ) · V 2 * ( A ) c 0 + c 1 cos [ P 12 ω 2 A 0 ω + u ( A A 0 ) u ( A A Δ ) ] ,
A Δ = 2 P 12 ω , c 0 = c 1 1 2 c 0 2 2 + c 0 1 2 c 1 2 2 , c 1 = c 0 1 c 1 1 c 0 2 c 1 2 .
I ( A ) c 0 + c 1 cos ( P 12 ω 2 A 0 ω + U ( A ) A A Δ ) ,
H 00 ( a ) = FT aR [ ( R ) ( R ) ] = 1 ,
H 10 ( a ) = FT a R 1 [ U F 1 ( R 1 ) exp ( i Δ z 12 k ω 1 R 1 ) × exp ( i Δ z 12 2 k R 1 2 ) ( R 2 ) ] U 1 ( a Δ z 12 k ω 1 ) ,
U 1 ( x , y ) = exp [ i u ( x , y ) ] = exp [ i m 1 4 ω 40 x ( x 2 + y 2 ) ] .
4 m 1 w 40 ( 3 x 2 + y 2 ) k λ = 0 .
H 01 ( a ) = FT a R 2 [ δ ( R ) U F m 2 ( R 2 ) ] = U 2 ( a ) .
U 2 ( x , y ) = exp [ i m 2 4 w 40 y ( x 2 + y 2 ) ] .
8 m 2 w 40 x y k λ = 0 ,
H 00 ( a ) = FT a R Σ [ V F Σ ( R Σ ) exp ( i Δ z 12 k ω 1 R Σ ) exp ( i Δ z 12 2 k R Σ 2 ) δ ( R ) ] V Σ ( a Δ z 12 k ω ) .
V Σ ( x , y ) = exp [ i Φ ( x , y ) ] = exp [ i w 31 ( x 2 + y 2 ) x ] ,
w 31 ( 3 x 2 + y 2 ) k λ = 0 .
Φ = tan 1 3 ( I 2 I 3 ) / ( 2 I 1 I 2 I 3 ) .

Metrics