Abstract

Phase-shifting interferometry based on the integrating-bucket technique with sinusoidal phase modulation is studied theoretically and demonstrated experimentally to obtain phase maps from double-beam interferometers. The method uses four frames obtained by integration of the time-varying intensity in an interference pattern during the four quarters of the modulation period. An optimum sinusoidal phase modulation is found to minimize the effect of the additive noise. The absolute accuracy of the phase measurements is discussed. Possible applications of the method are demonstrated with two interference microscopes with which the phase modulation is achieved by sinusoidal oscillation of a mirror attached to a piezoelectric transducer and by sinusoidal birefringence modulation with a photoelastic modulator. In both experimental arrangements, phase images can be produced in real time at a rate of several hertz. Noise measurements are reported and compared with theory.

© 2001 Optical Society of America

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References

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  1. K. Creath, “Phase-measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1988), Vol. 26, pp. 349–393.
  2. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1990), Vol. 28, pp. 271–359.
  3. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.
  4. R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.
  5. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
    [CrossRef]
  6. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  7. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculating algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  8. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  9. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 39, 3598–3600 (1993).
    [CrossRef]
  10. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  11. B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
    [CrossRef]
  12. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  13. J. Schmit, K. Creath, “Extended averaging technique for derivation of error compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef] [PubMed]
  14. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  15. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
    [CrossRef]
  16. K. D. Stumpf, “Real-time interferometer,” Opt. Eng. 18, 648–653 (1979).
    [CrossRef]
  17. B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurements of surfaces topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
    [CrossRef]
  18. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  19. O. Sasaki, H. Okazaki, “Sinusoidal phase modulating interferometry for surface profile measurement,” Appl. Opt. 25, 3137–3140 (1986).
    [CrossRef] [PubMed]
  20. O. Sasaki, H. Okazaki, “Analysis of measurement accuracy in sinusoidal phase modulating interferometry,” Appl. Opt. 25, 3152–3158 (1986).
    [CrossRef] [PubMed]
  21. O. Sasaki, H. Okazaki, M. Sakai, “Sinusoidal phase modulating interferometer using the integrating-bucket method,” Appl. Opt. 26, 1089–1093 (1987).
    [CrossRef] [PubMed]
  22. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  23. M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
    [CrossRef]
  24. G. S. Kino, S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef] [PubMed]
  25. A. Dubois, J. Selb, L. Vabre, A. C. Boccara, “Phase measurements with wide-aperture interferometers,” Appl. Opt. 39, 2326–2331 (1999).
    [CrossRef]
  26. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms, and Software (Wiley, New York, 1998).
  27. A. Dubois, A. C. Boccara, M. Lebec, “Real-time reflectivity and topography imagery of depth-resolved microscopic surfaces,” Opt. Lett. 24, 309–311 (1999).
    [CrossRef]
  28. P. Gleyzes, A. C. Boccara, H. Saint-Jalmes, “Multichannel Nomarski microscope with polarization modulation: performance and applications,” Opt. Lett. 22, 1529–1531 (1997).
    [CrossRef]
  29. G. de Villèle, V. Loriette, “Birefringence imaging with imperfect benches: application to large-scale birefringence measurements,” Appl. Opt. 39, 3864–3874 (2000).
    [CrossRef]
  30. D. Malacara, “Common-path interferometers,” in Optical Shop Testing, 2nd ed., J. W. Goodman, ed. (Wiley, New York, 1992), pp. 106–108.
  31. P. A. Flourney, R. W. McClure, G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. 11, 1907–1915 (1972).
    [CrossRef]
  32. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
    [CrossRef] [PubMed]

2000

1999

1997

1996

1995

1993

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 39, 3598–3600 (1993).
[CrossRef]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
[CrossRef] [PubMed]

1992

1990

1987

1986

1985

1979

K. D. Stumpf, “Real-time interferometer,” Opt. Eng. 18, 648–653 (1979).
[CrossRef]

1975

1972

1966

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Bhushan, B.

Boccara, A. C.

Brophy, C. P.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.

Caber, P. J.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Chim, S. C.

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

Creath, K.

J. Schmit, K. Creath, “Extended averaging technique for derivation of error compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

K. Creath, “Phase-measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1988), Vol. 26, pp. 349–393.

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

de Groot, P.

de Villèle, G.

Dubois, A.

Eiju, T.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Flourney, P. A.

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms, and Software (Wiley, New York, 1998).

Gleyzes, P.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.

Hariharan, P.

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

Kino, G. S.

Koliopoulos, C. L.

Kothiyal, M. P.

R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.

Larkin, K. G.

Lebec, M.

Loriette, V.

Malacara, D.

D. Malacara, “Common-path interferometers,” in Optical Shop Testing, 2nd ed., J. W. Goodman, ed. (Wiley, New York, 1992), pp. 106–108.

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

McClure, R. W.

Okazaki, H.

Oreb, B. F.

Phillion, D. W.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms, and Software (Wiley, New York, 1998).

Saint-Jalmes, H.

Sakai, M.

Sasaki, O.

Schmit, J.

Schreiber, H.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1990), Vol. 28, pp. 271–359.

Selb, J.

Sirohi, R. S.

R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.

Streibl, N.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Stumpf, K. D.

K. D. Stumpf, “Real-time interferometer,” Opt. Eng. 18, 648–653 (1979).
[CrossRef]

Surrel, Y.

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 39, 3598–3600 (1993).
[CrossRef]

Vabre, L.

Wyant, J. C.

Wyntjes, G.

Zhao, B.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Zoller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt.

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

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

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

J. Schmit, K. Creath, “Extended averaging technique for derivation of error compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurements of surfaces topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef]

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
[CrossRef] [PubMed]

O. Sasaki, H. Okazaki, “Sinusoidal phase modulating interferometry for surface profile measurement,” Appl. Opt. 25, 3137–3140 (1986).
[CrossRef] [PubMed]

O. Sasaki, H. Okazaki, “Analysis of measurement accuracy in sinusoidal phase modulating interferometry,” Appl. Opt. 25, 3152–3158 (1986).
[CrossRef] [PubMed]

O. Sasaki, H. Okazaki, M. Sakai, “Sinusoidal phase modulating interferometer using the integrating-bucket method,” Appl. Opt. 26, 1089–1093 (1987).
[CrossRef] [PubMed]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 39, 3598–3600 (1993).
[CrossRef]

G. S. Kino, S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
[CrossRef] [PubMed]

A. Dubois, J. Selb, L. Vabre, A. C. Boccara, “Phase measurements with wide-aperture interferometers,” Appl. Opt. 39, 2326–2331 (1999).
[CrossRef]

G. de Villèle, V. Loriette, “Birefringence imaging with imperfect benches: application to large-scale birefringence measurements,” Appl. Opt. 39, 3864–3874 (2000).
[CrossRef]

P. A. Flourney, R. W. McClure, G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. 11, 1907–1915 (1972).
[CrossRef]

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Metrologia

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng.

K. D. Stumpf, “Real-time interferometer,” Opt. Eng. 18, 648–653 (1979).
[CrossRef]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Opt. Lett.

Other

D. Malacara, “Common-path interferometers,” in Optical Shop Testing, 2nd ed., J. W. Goodman, ed. (Wiley, New York, 1992), pp. 106–108.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory, Algorithms, and Software (Wiley, New York, 1998).

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

K. Creath, “Phase-measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1988), Vol. 26, pp. 349–393.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1990), Vol. 28, pp. 271–359.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.

R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.

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Figures (6)

Fig. 1
Fig. 1

Simulation of the interferometric signal I(t) and how it is integrated over the four quarters of the modulation period (γ=2/3, ϕ=2, ψ=2.45, and θ=0.98).

Fig. 2
Fig. 2

Graphic representation of Γc and Γs: z=Γc(ψ, θ) and z=Γs(ψ, θ) for 0<ψ,θ<π. The intersection of the two maps z=Γc(ψ, θ)=Γs(ψ, θ) reaches its maximum value Γ=0.405 for ψ=2.45 and θ=0.98.

Fig. 3
Fig. 3

Sinusoidally phase-modulated Linnik-type microscope using a PZT to make the reference mirror oscillate.

Fig. 4
Fig. 4

Topographic image of details on a mask for integrated circuit manufacturing, obtained with our Linnik-type microscope.

Fig. 5
Fig. 5

Sinusoidally phase-modulated polarization Linnik-type microscope using a photoelastic birefringence modulator and stroboscopic illumination.

Fig. 6
Fig. 6

Noise measurements. Comparison with theory assuming our system to be shot-noise limited (N=100 000, γ=0.7).

Tables (2)

Tables Icon

Table 1 Phase Noise Resulting from the Quantization Error  a

Tables Icon

Table 2 Absolute Accuracy of Phase Measurements as a Function of the Error on the Phase-Modulation Parameters (ψ=ψ0+Δψ, θ=θ0+Δθ, with ψ0=2.45 and θ0=0.96 )

Equations (55)

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

I(x, y)=I¯(x, y)+A(x, y)cos ϕ(x, y),
I(x, y, t)=I¯(x, y)+A(x, y)cos[ϕ(x, y)+ψ sin(ωt+θ)].
Ep(x, y)=(p-1)T/4pT/4I(x, y, t)dt,p=1, 2, 3, 4.
I(t)=I¯+AJ0(ψ)cos ϕ+2A cos ϕ×n=1+J2n(ψ)cos[2n(ωt+θ)]-2A sin ϕ×n=0+J2n+1(ψ)sin[(2n+1)(ωt+θ)].
Ep=(T/4)(I¯+AJ0(ψ)cos ϕ)+(T/π)A cos ϕn=1+J2n(ψ)2n{sin[npπ+2nθ]-sin[n(p-1)π+2nθ]}-(T/π)A sin ϕn=0+J2n+1(ψ)2n+1×{cos[(2n+1)(p-1)π/2+(2n+1)θ]-cos[(2n+1)pπ/2+(2n+1)θ]}.
s=-E1+E2+E3-E4=(4T/π)Γs A sin ϕ,
c=-E1+E2-E3+E4=(4T/π)Γc A cos ϕ,
Γs=n=0+(-1)nJ2n+1(ψ)2n+1sin[(2n+1)θ],
Γc=n=0+J4n+2(ψ)2n+1sin[2(2n+1)θ].
tan ϕ=ΓcΓsΣsΣc.
np=0p=1, 2, 3, 4,
ninj=σ2,i=j0,ij,  |np|Ep.
η=tan(ϕ+ε)=ΓcΓsΣs+NsΣc+Nc=tan ϕ 1+Ns/Σs1+Nc/Σc,
Ns=-n1+n2+n3-n4,
Nc=-n1+n2-n3+n4.
ηtan ϕ1+NsΣs1-NcΣc+Nc2Σc2,
η2tan2 ϕ1+2 NsΣs+Ns2Σs2×1-2 NcΣc+3 Nc2Σc2.
η=tan ϕ1+4σ2Σc2,
η2=tan2 ϕ1+4σ21Σs2+3Σc2.
η=tan(ϕ+ε)a0+a1ε+a2ε2,
a0=tan ϕ,a1=1+tan2 ϕ,
a2=tan ϕ(1+tan2 ϕ).
η=a0+a1ε+a2ε2,
η2=a02+20a1ε+(a12+2a0a2)ε2.
ε=1a13 [-a0a12-a2a02+(a12+2a0a2)η-a2η2],
ε2=1a12 [a02-2a0η+η2].
ε=-sin ϕ cos ϕ-sin3 ϕ cos ϕ+cos2 ϕ(1+2 sin2 ϕ)×η-sin ϕ cos3 ϕη2,
ε2=cos4 ϕ[tan2 ϕ-2 tan ϕη+η2].
ε=4σ2sin ϕ cos ϕcos2 ϕΣc2-sin2 ϕΣs2
=σ2π24A2T2sin ϕ cos ϕ1Γc2-1Γs2,
ε2=4σ2sin2 ϕ cos2 ϕ1Σc2+1Σs2
=σ2π24T2A2sin2 ϕΓc2+cos2 ϕΓs2.
ε2min=σπ2TAΓ.
tan ϕ=ΣsΣc=E1-E2-E3+E4E1-E2+E3-E4.
N=Ep=I¯T/4.
σ=N.
γ=A/I¯,
ε2=π/(8ΓγN)1/(γN).
ϕ=4πz/λ,
Δz/λ=1/(32ΓγN)0.08/(γN).
σ=1/12.
L=Ep=I¯T/4.
ε2=π/(163ΓγL)0.28/(γL),
Δz/λ=1/(643ΓγL)0.025/(γL).
tan(ϕ+δ)=ΣsΣc=ΓsΓctan ϕ.
tan(ϕ+δ)tan ϕ+δ(1+tan2 ϕ).
δ=ρ-12sin(2ϕ),
δ˜=12π02πδ2(ϕ)dϕ1/2=|ρ-1|22.
δ˜=|0.30 Δψ-0.71 Δθ|,
ψ=ψ0+Δψ,θ=θ0+Δθ.
δ˜0.3|Δψ|+0.7|Δθ|,
Δz/λ0.03|Δψ|+0.06|Δθ|.
ε2=π/(4γN),
Δn=ψ0λ2πe=0.39 λe.

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