Abstract

Full-bandwidth phase-shifting methods as well as band-limited fringe carrier techniques are both problematic when testing high-NA spherical surfaces in Fizeau interferometers. Phase stepping is usually performed by moving a sample and reference sphere relative to each other along the optical axis. At a high NA the method suffers from phase-shift inhomogeneity across the sample surface. Fringe carrier techniques rely on a minimum fringe frequency and call for an off-axis position of the sample, which in turn introduces condenser aberrations. Distortion of the imaging optics generates further apparent aberrations. We propose to combine both principles. The phase shifts are replaced by a set of very low tilts such that the sample is virtually on axis. Initial wavefront estimates are generated by a fringe carrier method. An adaptive Misell-type algorithm combines the interferometric data and iteratively improves the reconstructed wavefront until full spatial bandwidth is achieved.

© 2006 Optical Society of America

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References

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  1. D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).
  2. K. Creath, "Temporal phase measurement methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 94-140 and references therein.
  3. K. G. Larkin and B. F. Oreb, "Design and assessment of symmetrical phase-shifting algorithm," J. Opt. Soc. Am. A 9, 1740-1748 (1992).
    [CrossRef]
  4. Y. Surrel, "Extended averaging and data windowing technique in phase-stepping measurements: an approach using the characteristic polynomial theory," Opt. Eng. 37, 2314-2319 (1998).
    [CrossRef]
  5. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts," J. Opt. Soc. Am. A 14, 918-930 (1997).
    [CrossRef]
  6. M. Chen, H. Guo, and C. Wei, "Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000).
    [CrossRef]
  7. J. Schmit and K. Creath, "Window function influence on phase error in phase-shifting algorithms," Appl. Opt. 35, 5642-5649 (1996).
    [CrossRef] [PubMed]
  8. P. J. de Groot, "Vibration in phase-shifting interferometry," J. Opt. Soc. Am. A 12, 354-365 (1995).
    [CrossRef]
  9. P. J. de Groot, "Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window," Appl. Opt. 34, 4723-4730 (1995).
    [CrossRef]
  10. P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2506 (1987).
    [CrossRef] [PubMed]
  11. D. W. Phillion, "General methods for generating phase-shifting interferometry algorithms," Appl. Opt. 36, 8098-8115 (1997).
    [CrossRef]
  12. M. Kujawinska, "Spatial phase measurement methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 141-193 and references therein.
  13. T. Yatagai, "Intensity based analysis methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 72-93 and references therein.
  14. D. J. Fischer, J. T. O'Brian, R. Lopez, and H. P. Stahl, "Vector formulation for interferogram surface fitting," Appl. Opt. 32, 4738-4743 (1993).
    [CrossRef] [PubMed]
  15. S. Vazquez-Montiel, J. J. Sánchez-Escobar, and O. Fuentes, "Obtaining the phase of an interferogram by use of an evolution strategy," Appl. Opt. 41, 3448-3452 (2002).
    [CrossRef] [PubMed]
  16. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  17. M. F. Küchel, "The new Zeiss interferometer," in Optical Testing and Metrology III: Recent Advances in Industrial Inspection, C.P.Grover, ed., Proc. SPIE 1332, 655-663 (1990).
  18. M. F. Küchel, "Advances in interferometric wavefront-measuring technology through the direct measuring interferometry (DMI) method," in Commercial Application of Precision Manufacturing at the Sub-Micron Level, L.R.Baker, ed., Proc. SPIE 1573159-162 (1991).
  19. M. Françon, Optical Interferometry (Academic 1966).
  20. D. Malacara, Optical Shop Testing (Wiley, 1978).
  21. J. Heil, J. Wesner, W. Müller, and Th. Sure, "Artificial star test by real-time video holography for the adjustment of high-numerical-aperture micro-objectives," Appl. Opt. 42, 5073-5085 (2003).
    [CrossRef] [PubMed]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).
  23. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).
  24. A. H. Nuttall, "On the quadrature approximation to the Hilbert transform of modulated signals," Proc. IEEE 54, 1458 (1966).
  25. A. W. Rihaczek, "Hilbert transforms and the complex representation of real signals," Proc. IEEE 54, 434 (1966).
  26. D. L. Misell, "A method for the solution of the phase problem in electron microscopy," J. Phys. D 6, L.6-L.9 (1973).
    [CrossRef]
  27. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  28. C. Roddier and F. Roddier, "Interferogram analysis using Fourier transform techniques," Appl. Opt. 26, 1668-1673 (1987).
    [CrossRef] [PubMed]
  29. J. H. Massig and J. Heppner, "Fringe pattern analysis with high accuracy by use of the Fourier transform method: theory and experimental tests," Appl. Opt. 40, 2081-2088 (2001).
    [CrossRef]
  30. M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, Vol. 1, Basics (Wiley, 1986).
    [CrossRef]
  31. F. James, "Monte Carlo theory and practice," Rep. Prog. Phy. 43,1145-1189 (1980).
  32. J. W. Goodman, Statistical Optics (Wiley, 1985).

2003 (1)

2002 (1)

2001 (1)

2000 (1)

1998 (1)

Y. Surrel, "Extended averaging and data windowing technique in phase-stepping measurements: an approach using the characteristic polynomial theory," Opt. Eng. 37, 2314-2319 (1998).
[CrossRef]

1997 (2)

1996 (1)

1995 (2)

1993 (1)

1992 (1)

1987 (2)

1984 (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

1973 (1)

D. L. Misell, "A method for the solution of the phase problem in electron microscopy," J. Phys. D 6, L.6-L.9 (1973).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).

Chen, M.

Creath, K.

J. Schmit and K. Creath, "Window function influence on phase error in phase-shifting algorithms," Appl. Opt. 35, 5642-5649 (1996).
[CrossRef] [PubMed]

K. Creath, "Temporal phase measurement methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 94-140 and references therein.

de Groot, P. J.

Eiju, T.

Farrant, D. I.

Fischer, D. J.

Françon, M.

M. Françon, Optical Interferometry (Academic 1966).

Fuentes, O.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Guo, H.

Hariharan, P.

Heil, J.

Heppner, J.

Hibino, K.

James, F.

F. James, "Monte Carlo theory and practice," Rep. Prog. Phy. 43,1145-1189 (1980).

Kalos, M. H.

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, Vol. 1, Basics (Wiley, 1986).
[CrossRef]

Küchel, M. F.

M. F. Küchel, "The new Zeiss interferometer," in Optical Testing and Metrology III: Recent Advances in Industrial Inspection, C.P.Grover, ed., Proc. SPIE 1332, 655-663 (1990).

M. F. Küchel, "Advances in interferometric wavefront-measuring technology through the direct measuring interferometry (DMI) method," in Commercial Application of Precision Manufacturing at the Sub-Micron Level, L.R.Baker, ed., Proc. SPIE 1573159-162 (1991).

Kujawinska, M.

M. Kujawinska, "Spatial phase measurement methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 141-193 and references therein.

Larkin, K. G.

Lopez, R.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1978).

Massig, J. H.

Misell, D. L.

D. L. Misell, "A method for the solution of the phase problem in electron microscopy," J. Phys. D 6, L.6-L.9 (1973).
[CrossRef]

Müller, W.

Nuttall, A. H.

A. H. Nuttall, "On the quadrature approximation to the Hilbert transform of modulated signals," Proc. IEEE 54, 1458 (1966).

O'Brian, J. T.

Oreb, B. F.

Phillion, D. W.

Reid, G. T.

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).

Rihaczek, A. W.

A. W. Rihaczek, "Hilbert transforms and the complex representation of real signals," Proc. IEEE 54, 434 (1966).

Robinson, D. W.

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).

Roddier, C.

Roddier, F.

Sánchez-Escobar, J. J.

Schmit, J.

Stahl, H. P.

Sure, Th.

Surrel, Y.

Y. Surrel, "Extended averaging and data windowing technique in phase-stepping measurements: an approach using the characteristic polynomial theory," Opt. Eng. 37, 2314-2319 (1998).
[CrossRef]

Vazquez-Montiel, S.

Wei, C.

Wesner, J.

Whitlock, P. A.

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, Vol. 1, Basics (Wiley, 1986).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Womack, K. H.

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Yatagai, T.

T. Yatagai, "Intensity based analysis methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 72-93 and references therein.

Appl. Opt. (10)

P. J. de Groot, "Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window," Appl. Opt. 34, 4723-4730 (1995).
[CrossRef]

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2506 (1987).
[CrossRef] [PubMed]

D. W. Phillion, "General methods for generating phase-shifting interferometry algorithms," Appl. Opt. 36, 8098-8115 (1997).
[CrossRef]

D. J. Fischer, J. T. O'Brian, R. Lopez, and H. P. Stahl, "Vector formulation for interferogram surface fitting," Appl. Opt. 32, 4738-4743 (1993).
[CrossRef] [PubMed]

S. Vazquez-Montiel, J. J. Sánchez-Escobar, and O. Fuentes, "Obtaining the phase of an interferogram by use of an evolution strategy," Appl. Opt. 41, 3448-3452 (2002).
[CrossRef] [PubMed]

J. Heil, J. Wesner, W. Müller, and Th. Sure, "Artificial star test by real-time video holography for the adjustment of high-numerical-aperture micro-objectives," Appl. Opt. 42, 5073-5085 (2003).
[CrossRef] [PubMed]

M. Chen, H. Guo, and C. Wei, "Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000).
[CrossRef]

J. Schmit and K. Creath, "Window function influence on phase error in phase-shifting algorithms," Appl. Opt. 35, 5642-5649 (1996).
[CrossRef] [PubMed]

C. Roddier and F. Roddier, "Interferogram analysis using Fourier transform techniques," Appl. Opt. 26, 1668-1673 (1987).
[CrossRef] [PubMed]

J. H. Massig and J. Heppner, "Fringe pattern analysis with high accuracy by use of the Fourier transform method: theory and experimental tests," Appl. Opt. 40, 2081-2088 (2001).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Phys. D (1)

D. L. Misell, "A method for the solution of the phase problem in electron microscopy," J. Phys. D 6, L.6-L.9 (1973).
[CrossRef]

Opt. Eng. (2)

Y. Surrel, "Extended averaging and data windowing technique in phase-stepping measurements: an approach using the characteristic polynomial theory," Opt. Eng. 37, 2314-2319 (1998).
[CrossRef]

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Other (16)

M. F. Küchel, "The new Zeiss interferometer," in Optical Testing and Metrology III: Recent Advances in Industrial Inspection, C.P.Grover, ed., Proc. SPIE 1332, 655-663 (1990).

M. F. Küchel, "Advances in interferometric wavefront-measuring technology through the direct measuring interferometry (DMI) method," in Commercial Application of Precision Manufacturing at the Sub-Micron Level, L.R.Baker, ed., Proc. SPIE 1573159-162 (1991).

M. Françon, Optical Interferometry (Academic 1966).

D. Malacara, Optical Shop Testing (Wiley, 1978).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).

A. H. Nuttall, "On the quadrature approximation to the Hilbert transform of modulated signals," Proc. IEEE 54, 1458 (1966).

A. W. Rihaczek, "Hilbert transforms and the complex representation of real signals," Proc. IEEE 54, 434 (1966).

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).

K. Creath, "Temporal phase measurement methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 94-140 and references therein.

M. Kujawinska, "Spatial phase measurement methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 141-193 and references therein.

T. Yatagai, "Intensity based analysis methods," in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Ref. 1), pp. 72-93 and references therein.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, Vol. 1, Basics (Wiley, 1986).
[CrossRef]

F. James, "Monte Carlo theory and practice," Rep. Prog. Phy. 43,1145-1189 (1980).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (20)

Fig. 1
Fig. 1

Scheme of a Fizeau interferometer for testing spherical surfaces. (i) “Illumination” axis, (v) “view” axis for interferogram observation, (a) “align” axis for condenser and test piece adjustment. (1) Laser, (2) focusing lens, (3) pinhole, (4) beam splitter, (5) collimating lens, (6) Fizeau objective, (7) reference surface, (8) test piece. A stop (9) with an adjustable lens system (10) and the collimating lens (5) form an afocal system imaging the test piece onto a rotating diffuser (11). A zoom lens (12) and a lens with fixed focal length (13) image the interferogram onto a CCD (14). A mirror (15) steers the focused object and reference waves onto a screen with a crosshair (16). An additional lens system (17) images them onto a CCD (14), when properly aligned, both focal spots coincide with the crosshair.

Fig. 2
Fig. 2

Fizeau testing of a spherical surface with radius R and diameter 2rp,max . The OPD of a spherical wave with respect to the lens surface introduced by a displacement d of the Fizeau condenser along the optical axis z varies like OPD = d cos(θ) and is effectively a defocus. θ is the angle measured against z, and u = θmax is the opening angle of the surface.

Fig. 3
Fig. 3

(a) Interferogram of a nearly perfect wavefront. The tilt between object and reference wave introduces an average fringe period p. (b) Spatial frequency amplitude spectrum of (a). The low-frequency components (labeled DC) are filtered out, because they would otherwise dominate the image. Two carrier peaks with side lobe structure appear centered at ±k r with |k r | = 2π∕p.

Fig. 4
Fig. 4

(a) Interferogram Ii exhibiting pronounced coma wavefront error. (b) Full-bandwidth real part ℜ(Ao ) computed from Eq. (10); the amplitude outside the area of interest is set to zero (Io , Ir not shown). (c), (d) Real part; (e), (f) imaginary part; and (g), (h) amplitude of the spatial frequency spectrum of (a), (b), respectively. The dc peak is clipped in (c) and the grayscale is nonlinear in (g) to bring out the low signal features. The two PSF structures and the dc peak overlap and interfere.

Fig. 5
Fig. 5

Flowchart of the N-step Misell algorithm described in the text.

Fig. 6
Fig. 6

First estimate for ℑ(Ao ) according to Eq. (15). (a) Real part ℜ(Ao ) from Eq. (10); the data are the same as for Fig. 4. (b) Spatial frequency amplitude spectrum of (a), the edge of the bandpass filter BP is indicated as a white circle. (c) Filtered spectrum. (d) Band-limited estimate for ℑ(Ao start).

Fig. 7
Fig. 7

Reconstruction from N = 2 simulated noiseless 2D interferograms after i = 200 iterations. (a), (c) Real parts ( A o , 1 rec ) , ( A o ,2 rec ) . (b), (d) Imaginary parts ℑ ( A o , 1 rec ) , ℑ ( A o ,2 rec ) . The arrow points to an erroneous isolated phase discontinuity near the rim (parameters: Px, 1 = 0.05, Px, 2 = −0.05, Tx, 1, Tx, 2, Ty, 1 = 1, Ty, 2 = −1, Dx, 1 = 0.02, Dx, 2 = −0.03).

Fig. 8
Fig. 8

Same as Fig. 7, except Ao is not forced to zero resulting in a pupil extrapolation. (a), (c) Real parts ( A o , 1 rec ) , ( A o ,2 rec ) . (b), (d) Imaginary parts ℑ ( A o , 1 rec ) , ℑ ( A o , 2 rec ) . The error in Fig. 7 has been suppressed.

Fig. 9
Fig. 9

Full-bandwidth reconstructions from N = 2 simulated interferograms for different levels of Gaussian noise after i = 1000 iterations. The wavefront is the same as in Figs. 7 and 8. (a)–(d) Real part ℜ(Ao ); (e)–(h) imaginary part ℑ(Ao ) with 1∕SNR = 0.005, 0.02, 0.04, 0.07, respectively. See Eq. (29) for SNR.

Fig. 10
Fig. 10

Reconstruction errors versus iteration number i for different noise levels. (a) rms OPD error of the noisy reconstruction Ao rec against the original noiseless wavefront Ao org, the five curves correspond to Gaussian noise with 1∕SNR = 0.001, 0.005, 0.02, 0.04, 0.07. Minimum error position and value depend on the noise level. (b) rms error of ℜ(Ao rec), ℑ(Ao rec) against the original ℜ(Ao org), ℑ(Ao org) (i.e., without noise) for 1∕SNR = 0.02; ℑ(Ao rec) contains more error than ℜ(Ao rec). (c) rms error of the reconstruction |Ao rec| against the original amplitude Io 1∕2 (i.e., with noise). (d) rms error of ℜ(Ao rec) against the real part from Eq. (10), (i.e., with noise).

Fig. 11
Fig. 11

(a) Minimum rms error of the reconstructed real part ℜ(Ao rec) against the actual real part according to Eq. (10) (i.e., with noise); absolute value of the reconstructed amplitude |Ao rec| against the actual amplitude Io 1∕2 and of the reconstructed OPDrec including noise against the original OPDorg without noise. All three quantities depend linear on lg(1∕SNR). (b) The iteration index i min corresponding to the minimum rms errors of the three quantities versus log(1∕SNR) shows a roughly linear behavior.

Fig. 12
Fig. 12

Same as Fig. 9, but the iteration stopped at i min = 370, 250, 180, 150, corresponding to 37%, 25%, 18%, 15% of the full bandwidth for (a), (b), (c), (d) [or (e), (f), (g), (h)], respectively.

Fig. 13
Fig. 13

OPD of the optimum reconstructions in Fig. 12 against the original wavefront. The PV differences are ≈0.08, 0.2, 0.65, 1 for (a), (b), (c), (d), respectively. See Fig. 11 for rms errors. All graphs are shown on the same scale. Within rp ≤ 0.95 rp,max the PV error is ≈0.015, 0.03, 0.05, 0.15 (≈λ∕60, λ∕30, λ∕16, λ∕6) for (a), (b), (c), (d), respectively. The rms errors are ≈λ10−3 (see Fig. 11).

Fig. 14
Fig. 14

Experimental data set #1. (a)–(d) (Left column) Full-bandwidth real part ℜ(Ao ) computed from Eq. (10). (e)–(h) (Right column) Corresponding amplitude spectra. The spectra are presented on a nonlinear grayscale to enhance low signal details. Note how the overlapping “coma tails” make it impossible to separate the PSF structures by simple filtering.

Fig. 15
Fig. 15

Data set #2. The wavefront is the same as in Fig. 14 with a different set of tilts applied, including Tx , T y ≈ 0 in (c), (g).

Fig. 16
Fig. 16

Data set #3. The wavefront is again the same as in Figs. 14 and 15. Yet another set of tilts is applied.

Fig. 17
Fig. 17

(a) Real part ℜ(Ao ) and (b) imaginary part ℑ(Ao ) of the wavefront recovered from data set #1 in Fig. 14. (c) Amplitude spectrum of the full-bandwidth real part [see Fig. 14(e)] and (d) isolated PSF. The iteration was stopped at i min = 350, k opt = 0.55k max. The spectra are presented on a nonlinear gray scale to enhance the low signal details. (e) and (f) show the real and imaginary part of the recovered spectrum.

Fig. 18
Fig. 18

Similar to Fig. 17 for data set #2 with i min = 250, k opt = 0.38k max.

Fig. 19
Fig. 19

Similar to Fig. 17 for data set #3 with i min = 300, k opt = 0.49k max.

Fig. 20
Fig. 20

Difference between the reconstructions with the relative piston ΔP, tilt ΔTx , ΔTy , and defocus ΔD removed. (a) ΔOPD#2,#1, the OPD difference of the reconstruction of data set #2 and #1, (b) ΔOPD#3,#1, (c) ΔOPD#3,#2, see Table 1 for numerical values. All graphs are on the same scale; the height of the peak in (a), (b) is ≈0.5.

Tables (1)

Tables Icon

Table 1 Offset, Tilt, Defocus, OPD of Reconstructions #1, #2, #3

Equations (36)

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I i = I r + I o + 2 ( I o I r ) 1 / 2 cos ( ϕ o , r ) ,
I i , n ( r ) = I r ( r ) + I o ( r ) + 2 { I o ( r ) I r ( r ) } 1 / 2 ×  cos { ϕ o , r ( r ) + ϕ n } ,
C = n = 0 N 1 I i , n cos ( ϕ n ) = 2 ( I o I r ) 1 / 2 cos ( ϕ o , r ) k ,
S = n = 0 N 1 I i , n sin ( ϕ n ) = 2 ( I o I r ) 1 / 2 sin ( ϕ o , r ) k ,
ϕ o , r = arctan ( S / C ) .
Δ ϕ ( θ ) = ( 2 π d / λ ) cos ( θ ) ,
I i = I r + I o + ( I o I r ) 1 / 2 { exp ( i ϕ o , r ) + exp ( i ϕ o , r ) } ,
   A o = I o     1 / 2 exp ( i ϕ o , r ) = I o     1 / 2 { cos ( ϕ o , r ) + i sin ( ϕ o , r ) } .
I i , HF A o exp ( i k r r ) + A ¯ o exp ( i k r r ) ,
( A o ) = ( I i I r I o ) / ( 2 I r     1 / 2 ) ,
A o = 2 F 1 [ step ( k e r ) F { ( A o ) } ] ,
( A o ) = H { ( A o ) } = F 1 [ ( i ) sign ( k e r ) F { ( A o ) } ] ,
k loc ( r ) e r = [ k r + { ϕ ( r ) } ] e r 0 ,
ϕ o , r arctan [ ( A o ) / ( A o ) ] ,
( A o start ) 2 ( F 1 [ B P ( F { ( A o ) } ) ] ) ,
A o , 1 1 = A o , 1 start .
A o , 1 2 = ( A o , 1 ) + i ( A o , 1 1 ) .
A o , 1 3 = ( A o , 1 2 / | A o , 1 2 | ) I o 1 / 2 .
A o , 2 1 = A o , 1 3 exp ( i Δ ϕ 2 , 1 ) .
Δ ϕ 1 , N = ( 1 ) { Δ ϕ 2 , 1 + Δ ϕ 3 , 2 + + Δ ϕ N , ( N 1 ) } = ( 1 ) n = 1 N 1 Δ ϕ ( n + 1 ) , n .
A o , 1 rec = A o , 1 3 .
ϕ p ( r , φ ) = 2 π OPD p ( r , φ ) = 2 π P Z 0 = 2 π P 1 ,
ϕ t x ( r , φ ) = 2 π OPD t x ( r , φ ) = 2 π T x Z 1
= 2 π T x { r cos ( φ ) } ,
ϕ t y ( r , φ ) = 2 π OPD t y ( r , φ ) = 2 π T y Z 2
= 2 π T y { r sin ( φ ) } ,
ϕ d ( r , φ ) = 2 π OPD d ( r , φ ) = 2 π D Z 3
= 2 π D { 2 r 2 1 } .
OPD d = d { 1 ( r NA ) 2 } 1 / 2 ,
Δ ϕ n + 1 , n = 2 π { Δ P n + 1 , n Z 0 + Δ T x , n + 1 , n Z 1 + Δ T y , n + 1 , n Z 2 + Δ D n + 1 , n Z 3 } .
δ A o = α o exp ( i δ ϕ n + 1 , n ) = { A o , n 3 exp ( i Δ ϕ n + 1 , n ) } / A o , n + 1 3 .
δ P n + 1 , n = S 1 S δ ϕ n + 1 , n / ( 2 π ) OPD p d S ,
Δ P ( n + 1 , n ) , ( i + 1 ) = Δ P ( n + 1 , n ) , i + δ P n + 1 , n ,
N G I noise ( 6 / K ) 1 / 2 k = 0 K { rnd 1 / 2 } ,
SNR = I o , max / I noise ,
Δ P = Δ P ( # 2 , # 1 ) + Δ P ( # 3 , # 2 ) Δ P ( # 3 , # 1 ) ,

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