Abstract

We have developed an analytical model for the design and optimization of common-path interferometers (CPI’s) based on spatial filtering. We describe the mathematical analysis in detail and show how its application to the optimization of a range of different CPI’s results in the development of a graphical framework to characterize quantitatively CPI performance. A detailed analytical treatment of the effect of curvature in the synthetic reference wave is undertaken. We show that it is possible to improve the linearity and fringe accuracy of certain standard interferometers by a modification of the Fourier filter, and we propose and analyze a dual CPI system for the unambiguous mapping of phase to intensity over the complete input phase range.

© 2001 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, Calif., 1996).
  2. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [CrossRef] [PubMed]
  3. H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London Sect. B. 66, 331–333 (1953).
    [CrossRef]
  4. S. F. Paul, “Dark-ground illumination as a quantitative diagnostic for plasma density,” Appl. Opt. 21, 2531–2537 (1982).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  7. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 302–305.
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    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 426–427.
  10. C. A. Mack, “Phase contrast lithography,” in Optical/Laser Microlithography VI, J. D. Cuthbert, ed., Proc. SPIE1927, 512–520 (1993).
    [CrossRef]
  11. Y. Arieli, N. Eisenberg, A. Lewis, “Pattern generation by inverse phase contrast,” Opt. Commun. 138, 284–286 (1997).
    [CrossRef]
  12. J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996).
    [CrossRef]
  13. J. Glückstad, L. Lading, H. Toyoda, T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997).
    [CrossRef]
  14. J. Glückstad, “Pattern generation by inverse phase contrast—comment,” Opt. Commun. 147, 16–19 (1998).
    [CrossRef]
  15. J. Glückstad, “Graphic method for analyzing common path interferometers,” Appl. Opt. 37, 8151–8152 (1998).
    [CrossRef]
  16. J. Glückstad, P. C. Mogensen, “Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method,” Opt. Commun. 173, 169–175 (2000).
    [CrossRef]
  17. P. C. Mogensen, J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
    [CrossRef]
  18. C. S. Anderson, “Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances,” Appl. Opt. 34, 7474–7485 (1995).
    [CrossRef] [PubMed]
  19. H. B. Henning, “A new scheme for viewing phase contrast images,” Electro-Opt. Syst. Des. 6, 30–34 (1974).
  20. G. O. Reynolds, J. B. Develis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1989), Chap. 35.
    [CrossRef]
  21. J. Glückstad, “Image decrypting common path interferometer,” in Optical Pattern Recognition X, D. P. Casasent, T. Chao, eds., Proc. SPIE3715, 152–159 (1999).
    [CrossRef]
  22. C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35, 1633–1642 (1996).
    [CrossRef] [PubMed]
  23. C. Koliopoulus, O. Kwon, R. Shagam, J. C. Wyant, C. R. Hayslett, “Infrared point-diffraction interferometer,” Opt. Lett. 3, 118–120 (1978).
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  24. H. Kadano, M. Ogusu, T. Asakura, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
    [CrossRef]
  25. R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
  26. W. P. Linnik, “Ein einfaches interferometer zur prüfung von optischen systemen,” Proc. Acad. Sci. USSR Sect. Appl. Phys. 1, 208–211 (1933).
  27. P. M. Birch, J. Gourlay, G. D. Love, A. Purvis, “Real-time optical aberration correction with a ferroelectric liquid-crystal spatial light modulator,” Appl. Opt. 37, 2164–2169 (1998).
    [CrossRef]
  28. C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
    [CrossRef] [PubMed]
  29. A. van den Bos, “Aberration and the Strehl ratio,” J. Opt. Soc. Am. A 17, 356–358 (2000).
    [CrossRef]
  30. G. D. Love, N. Andrews, P. M. Birch, D. Buscher, P. Doel, C. Dunlop, J. Major, R. Myers, A. Purvis, R. Sharples, A. Vick, A. Zadrozny, S. R. Restaino, A. Glindemann, “Binary adaptive optics: atmospheric wave-front correction with a half-wave phase shifter,” Appl. Opt. 34, 6058–6066 (1995); addenda 35, 347–350 (1996).

2000 (3)

J. Glückstad, P. C. Mogensen, “Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method,” Opt. Commun. 173, 169–175 (2000).
[CrossRef]

P. C. Mogensen, J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

A. van den Bos, “Aberration and the Strehl ratio,” J. Opt. Soc. Am. A 17, 356–358 (2000).
[CrossRef]

1998 (3)

1997 (2)

J. Glückstad, L. Lading, H. Toyoda, T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997).
[CrossRef]

Y. Arieli, N. Eisenberg, A. Lewis, “Pattern generation by inverse phase contrast,” Opt. Commun. 138, 284–286 (1997).
[CrossRef]

1996 (2)

1995 (2)

1994 (2)

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
[CrossRef] [PubMed]

H. Kadano, M. Ogusu, T. Asakura, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

1986 (1)

A. K. Aggarwal, S. K. Kaura, “Further applications of point diffraction interferometer,” J. Opt. (Paris) 17, 135–138 (1986).
[CrossRef]

1985 (1)

1982 (1)

1978 (1)

1975 (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

1974 (1)

H. B. Henning, “A new scheme for viewing phase contrast images,” Electro-Opt. Syst. Des. 6, 30–34 (1974).

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

1953 (1)

H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London Sect. B. 66, 331–333 (1953).
[CrossRef]

1933 (1)

W. P. Linnik, “Ein einfaches interferometer zur prüfung von optischen systemen,” Proc. Acad. Sci. USSR Sect. Appl. Phys. 1, 208–211 (1933).

Aggarwal, A. K.

A. K. Aggarwal, S. K. Kaura, “Further applications of point diffraction interferometer,” J. Opt. (Paris) 17, 135–138 (1986).
[CrossRef]

Anderson, C. S.

Anderson, R. C.

Andrews, N.

Arieli, Y.

Y. Arieli, N. Eisenberg, A. Lewis, “Pattern generation by inverse phase contrast,” Opt. Commun. 138, 284–286 (1997).
[CrossRef]

Asakura, T.

H. Kadano, M. Ogusu, T. Asakura, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Birch, P. M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 426–427.

Buscher, D.

Chapman, G. T.

M. P. Loomis, M. Holt, G. T. Chapman, M. Coon, “Applications of dark central ground interferometry,” paper AIAA 91-0565, presented at the Twenty-Ninth Aerospace Sciences Meeting, Reno, Nev., January 1991 (American Institute of Aeronautics and Astronautics, New York, 1991), pp. 1–8.

Coon, M.

M. P. Loomis, M. Holt, G. T. Chapman, M. Coon, “Applications of dark central ground interferometry,” paper AIAA 91-0565, presented at the Twenty-Ninth Aerospace Sciences Meeting, Reno, Nev., January 1991 (American Institute of Aeronautics and Astronautics, New York, 1991), pp. 1–8.

Creath, K.

Develis, J. B.

G. O. Reynolds, J. B. Develis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1989), Chap. 35.
[CrossRef]

Doel, P.

Dunlop, C.

Eisenberg, N.

Y. Arieli, N. Eisenberg, A. Lewis, “Pattern generation by inverse phase contrast,” Opt. Commun. 138, 284–286 (1997).
[CrossRef]

Glindemann, A.

Glückstad, J.

J. Glückstad, P. C. Mogensen, “Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method,” Opt. Commun. 173, 169–175 (2000).
[CrossRef]

P. C. Mogensen, J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

J. Glückstad, “Pattern generation by inverse phase contrast—comment,” Opt. Commun. 147, 16–19 (1998).
[CrossRef]

J. Glückstad, “Graphic method for analyzing common path interferometers,” Appl. Opt. 37, 8151–8152 (1998).
[CrossRef]

J. Glückstad, L. Lading, H. Toyoda, T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997).
[CrossRef]

J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996).
[CrossRef]

J. Glückstad, “Image decrypting common path interferometer,” in Optical Pattern Recognition X, D. P. Casasent, T. Chao, eds., Proc. SPIE3715, 152–159 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, Calif., 1996).

Gourlay, J.

Hara, T.

Hayslett, C. R.

Henning, H. B.

H. B. Henning, “A new scheme for viewing phase contrast images,” Electro-Opt. Syst. Des. 6, 30–34 (1974).

Holt, M.

M. P. Loomis, M. Holt, G. T. Chapman, M. Coon, “Applications of dark central ground interferometry,” paper AIAA 91-0565, presented at the Twenty-Ninth Aerospace Sciences Meeting, Reno, Nev., January 1991 (American Institute of Aeronautics and Astronautics, New York, 1991), pp. 1–8.

Hopkins, H. H.

H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London Sect. B. 66, 331–333 (1953).
[CrossRef]

Kadano, H.

H. Kadano, M. Ogusu, T. Asakura, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Kaura, S. K.

A. K. Aggarwal, S. K. Kaura, “Further applications of point diffraction interferometer,” J. Opt. (Paris) 17, 135–138 (1986).
[CrossRef]

Koliopoulus, C.

Kwon, O.

Lading, L.

Lewis, A.

Y. Arieli, N. Eisenberg, A. Lewis, “Pattern generation by inverse phase contrast,” Opt. Commun. 138, 284–286 (1997).
[CrossRef]

Lewis, S.

Linnik, W. P.

W. P. Linnik, “Ein einfaches interferometer zur prüfung von optischen systemen,” Proc. Acad. Sci. USSR Sect. Appl. Phys. 1, 208–211 (1933).

Loomis, M. P.

M. P. Loomis, M. Holt, G. T. Chapman, M. Coon, “Applications of dark central ground interferometry,” paper AIAA 91-0565, presented at the Twenty-Ninth Aerospace Sciences Meeting, Reno, Nev., January 1991 (American Institute of Aeronautics and Astronautics, New York, 1991), pp. 1–8.

Love, G. D.

Mack, C. A.

C. A. Mack, “Phase contrast lithography,” in Optical/Laser Microlithography VI, J. D. Cuthbert, ed., Proc. SPIE1927, 512–520 (1993).
[CrossRef]

Major, J.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 302–305.

Mercer, C. R.

Mogensen, P. C.

J. Glückstad, P. C. Mogensen, “Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method,” Opt. Commun. 173, 169–175 (2000).
[CrossRef]

P. C. Mogensen, J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

Myers, R.

Ogusu, M.

H. Kadano, M. Ogusu, T. Asakura, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Parrent, G. B.

G. O. Reynolds, J. B. Develis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1989), Chap. 35.
[CrossRef]

Paul, S. F.

Purvis, A.

Restaino, S. R.

Reynolds, G. O.

G. O. Reynolds, J. B. Develis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1989), Chap. 35.
[CrossRef]

Shagam, R.

Sharples, R.

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Thompson, B. J.

G. O. Reynolds, J. B. Develis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1989), Chap. 35.
[CrossRef]

Toyoda, H.

van den Bos, A.

Vick, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 426–427.

Wyant, J. C.

Zadrozny, A.

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Appl. Opt. (7)

Electro-Opt. Syst. Des. (1)

H. B. Henning, “A new scheme for viewing phase contrast images,” Electro-Opt. Syst. Des. 6, 30–34 (1974).

J. Opt. (Paris) (1)

A. K. Aggarwal, S. K. Kaura, “Further applications of point diffraction interferometer,” J. Opt. (Paris) 17, 135–138 (1986).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Opt. Commun. (6)

J. Glückstad, P. C. Mogensen, “Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method,” Opt. Commun. 173, 169–175 (2000).
[CrossRef]

P. C. Mogensen, J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000).
[CrossRef]

H. Kadano, M. Ogusu, T. Asakura, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Y. Arieli, N. Eisenberg, A. Lewis, “Pattern generation by inverse phase contrast,” Opt. Commun. 138, 284–286 (1997).
[CrossRef]

J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996).
[CrossRef]

J. Glückstad, “Pattern generation by inverse phase contrast—comment,” Opt. Commun. 147, 16–19 (1998).
[CrossRef]

Opt. Lett. (3)

Proc. Acad. Sci. USSR Sect. Appl. Phys. (1)

W. P. Linnik, “Ein einfaches interferometer zur prüfung von optischen systemen,” Proc. Acad. Sci. USSR Sect. Appl. Phys. 1, 208–211 (1933).

Proc. Phys. Soc. London Sect. B. (1)

H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London Sect. B. 66, 331–333 (1953).
[CrossRef]

Science (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Other (7)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, Calif., 1996).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 426–427.

C. A. Mack, “Phase contrast lithography,” in Optical/Laser Microlithography VI, J. D. Cuthbert, ed., Proc. SPIE1927, 512–520 (1993).
[CrossRef]

M. P. Loomis, M. Holt, G. T. Chapman, M. Coon, “Applications of dark central ground interferometry,” paper AIAA 91-0565, presented at the Twenty-Ninth Aerospace Sciences Meeting, Reno, Nev., January 1991 (American Institute of Aeronautics and Astronautics, New York, 1991), pp. 1–8.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 302–305.

G. O. Reynolds, J. B. Develis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1989), Chap. 35.
[CrossRef]

J. Glückstad, “Image decrypting common path interferometer,” in Optical Pattern Recognition X, D. P. Casasent, T. Chao, eds., Proc. SPIE3715, 152–159 (1999).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Generic CPI based on a 4-f optical system (lenses L1 and L2). The input phase disturbance is shown as an aperture-truncated phase function ϕ(x, y), which generates an intensity distribution I(x′, y′) in the observation plane by a filtering operation in the Fourier plane. The values of the filter parameters (A, B, θ) determine the type of filtering operation.

Fig. 2
Fig. 2

Schematic representation of the relationship between the profile of the input aperture-diffracted Airy function and the spatial profile of a generic Fourier filter (see Fig. 1). The term η is defined as the ratio of the radii of the filter R 1 and the mainlobe of the Airy function R 2.

Fig. 3
Fig. 3

Plot of the spatial variation of the normalized SRW amplitude g(r′) as a function of the normalized CPI observation plane radius for a range of η values from 0.2 to 0.627. This plot shows that a large value of η produces significant curvature in the SRW across the aperture, which will cause a distortion of the output interference pattern. In contrast, a low value of η generates a flat SRW, but at the cost of a reduction in the SRW amplitude.

Fig. 4
Fig. 4

Complex filter space plot of the modulus of the combined filter parameter |C| versus the phase ψ C over the complete 2π phase region. Use of these combined parameters [defined in Eqs. (14) and (16)] allows us to visualize simultaneously all the available combinations of the terms A, B, and θ. The bold curve is the operating curve for a phase-only (lossless) filter, whereas the fine grid represents the operating curves for different values of the filter terms A, B, and θ. The operating regimes are marked for a number of CPI architectures, including (A), Zernike; (B), Henning; (C), generalized phase-contrast methods; (D), dark central ground; (E), field absorption filtering; (F, G), point-diffraction. Full filter details for these techniques are summarized in Table 1.

Fig. 5
Fig. 5

Set of complex filter space plots for the optimization of the visibility and the peak irradiance with different input phase distributions. The four pairs of plots illustrate the variation in the visibility and the peak irradiance at the output of a CPI for a uniform random-phase distribution of dynamic ranges of (a) π/8, (b) π/3, (c) π, and (d) 3π/2, for a constant value of K = 1/2. The gray scale of the contour plots defines the relative visibility or peak irradiance (with the darkest regions representing the maximum value) that can be achieved when the Fourier filter parameters are varied. The operating curve for a lossless filter is shown on all the plots.

Fig. 6
Fig. 6

Optimization of the visibility and the peak irradiance for an unambiguous direct mapping of the input phase to the output intensity for a range of uniform random-phase distributions of dynamic ranges of (a) π/8, (b) π/3, (c) π/2, and (d) 3π/4. A truncation of the allowed values for ψ C arises from the unambiguous phase-to-intensity mapping requirement. As the depth of the input phase distribution increases, the allowed values of the combined filter parameter are further constrained.

Fig. 7
Fig. 7

Graphical phasor chart analysis for the generalized Henning method. The zero point of the quadratic intensity scale I is fixed at the point where ϕ̃ = 3π/2 on the ϕ̃ unit phase circle. The intersection of I with the ϕ̃ unit circle directly yields the output intensity value, which corresponds to the input phase of the CPI. This mapping is approximately linear for the range ±π/3 about the points ϕ̃ = 0 and ϕ̃ = π (indicated by the dashed curves).

Fig. 8
Fig. 8

Graphical phasor chart for the CPI with filter parameters given by Eqs. (46) and (47). The zero point of the quadratic intensity scale I is fixed at the point where ϕ̃ = 0 on the ϕ̃ unit phase circle. As in Fig. 7, the intersection of I with the ϕ̃ unit circle yields the output intensity for a given input phase. This phase-to-intensity mapping is approximately linear for the range ±π/3 about the points ϕ̃ = π/2 and ϕ̃ = 3π/2 (indicated by the dashed curves).

Fig. 9
Fig. 9

Graphical phasor chart for a symmetric dual CPI configuration with both CPI’s shown on the same chart. The zero point of the quadratic intensity scales I for each of the two CPI’s are fixed at the points ϕ̃ = 5π/4 and ϕ̃ = π/4 on the ϕ̃ unit circle. The intensity scales are labeled to indicate the region for which a linear mapping is provided (i.e., 1Q corresponds to the first quadrant, 2Q is the second quadrant, and so forth).

Fig. 10
Fig. 10

Basic layout of the graphical phasor chart for mapping an input phase ϕ̃ to an output intensity I for a CPI with the Fourier filter parameters (A, B, θ). The key elements of the chart are labeled for clarity.

Fig. 11
Fig. 11

Application of the phasor chart for K|̅α| = 1/2 and θ = π/2 with a radial scaling for the zero point of the quadratic intensity scale determined by the amplitude transmission terms A = 1/2 and B = 1. When the zero point of the intensity scale is fixed at the desired point, the phase-to-intensity mapping is determined from the intersection of the quadratic intensity scale with the unity phase circle.

Fig. 12
Fig. 12

Modified phasor chart that was adapted to work directly with the two components of the combined filter parameter: |C| and ψ C .

Fig. 13
Fig. 13

Application of the modified phasor chart to a CPI with the same zero intensity point as that described by the phasor chart in Fig. 11.

Tables (2)

Tables Icon

Table 1 Comparison of Filter Parameters for the Different CPI Types Highlighted in Fig. 4

Tables Icon

Table 2 Dual CPI Configuration for Linear Phase-to-Intensity Mapping over the 2π Phase Circle

Equations (48)

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

expiϕx, y1+iϕx, y.
Ix, y1+2ϕx, y.
expiϕx, y=1+iϕx, y-12 ϕ2x, y-16 iϕ3x, y+124 ϕ4x, y+.
expiϕx, y=Ωdxdy-1Ωexpiϕx, ydxdy+higher-frequency terms.
ax, y=circr/Δrexpiϕx, y
Hfx, fy=A1+BA-1 expiθ-1circfr/Δfr,
Ix, y=A2|expiϕ˜x, ycircr/Δr+|α¯|BA-1 expiθ-1gr|2,
α¯=πΔr2-1x2+y2Δrexpiϕx, ydxdy=|α¯|expiϕα¯,ϕ˜=ϕ-ϕα¯.
gr=2πΔr 0Δfr J12πΔrfrJ02πrfrdfr.
η=R1/R2=0.61-1ΔrΔfr,
gr=1-J01.22πη-0.61πη2J21.22πηr/Δr2+0.61πη3/42J31.22πη-0.61πηJ41.22πηr/Δr4.
grcentral region1-J01.22πη.
Ix, yA2|expiϕ˜x, y+K|α¯|BA-1 expiθ-1|2,
C=|C|expiψC=BA-1 expiθ-1.
Ix, y=A2|expiϕ˜x, y-iψC+K|α¯C2,
BA-1=1+2|C|cosψC+|C|21/2,θ=sin-1BA-1-1|C|sinψC.
BA-1<1  A=1, B=|C+1|,BA-1=1  A=1, B=1,BA-1>1  B=1, A=|C+1|-1.
expiθ-1=2|sinθ/2|expiθ+π/2,
|C|=2|cosψC| for ψCπ/2; 3π/2,|C|=0 for ψCπ/2; 3π/2.
V=Imax-IminImax+Imin.
V=max cos-min cosK|α¯C|-1+K|α¯C|+max cos+min cos-1,
Imax=A21+K|α¯C|2+2K|α¯C|max cos,
min cos=mincosϕ˜-ψC,max cos=maxcosϕ˜-ψC, ϕ˜-Δϕ˜/2; Δϕ˜/2.
|α¯|uniform=|sincΔϕ˜/2|.
CV=1=K|α¯|-1 expiπ+iϕ˜,  ϕ˜-Δϕ˜/2; Δϕ˜/2.
CV=1=|CV=1|=K|α¯|-1,  Δϕ˜2π.
V=2K|α¯C|-1+K|α¯C|-1,
Imax=A21+K|α¯C|2+2K|α¯C|.
V=IΔϕ˜/2-I-Δϕ˜/2IΔϕ˜/2+I-Δϕ˜/2-1=2 sinΔϕ˜/2sinψCK|α¯C|-1+K|α¯C|+2 cosΔϕ˜/2cosψC-1,
Imax=IΔϕ˜/2=A2|expiΔϕ˜/2-iψC+K|α¯C2.
CV=1=K|α¯|-1 expiπ-iΔϕ˜/2.
BA-1=1-2K|α¯|-1 cosΔϕ˜/2+K|α¯|-21/2, θ=sin-1BA-1K|α¯|-1 sinÜΔϕ˜/2).
ψCΔϕ˜/2; π-Δϕ˜/2,
IZernikex, y=|expiϕ˜x, y+i-1|2=3+2sinϕ˜x, y-cosϕ˜x, y.
IZernike=3+2sinϕ˜-cosϕ˜ 1+2ϕ˜+ϕ˜2 for ϕ˜  0.
IHenningx, y=1/2|expiϕ˜x, y+2expiπ/4-1|2=1+sinϕ˜x, y.
IHenning=1+sinϕ˜  1+ϕ˜ for ϕ˜  0.
Ix, y=A21+K|α¯C|2+2K|α¯C|cosϕ˜x, y-ψC,
K|α¯|=1/2  C=2 expiπ/2.
BA-1=5  A=1/5,  B=1,θ=sin-12/5.
Ix, yHenningK=1/2=2/51+sinϕ˜x, y.
C=K|α¯|-1 expiπ/2.
BA-1=1+K|α¯|-21/2  A=1+K|α¯|-2-1/2,B=1, θ=sin-11+K|α¯|2-1/2.
Ix, ygeneralizedHenning=21+K|α¯|-2-11+sinϕ˜x, y.
L2=cos2ϕ˜+sinϕ˜+12  1+sinϕ˜.
C=K|α¯|-1 expiπ,
θ=π,BA-1=1-2K|α¯|-1+K|α¯|-21/2,
Ix, y=2A21-cosϕ˜x, y.

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