Abstract

An optical transparency that can produce a phase-only Fourier transform is described. One can use the transparency for finding the best focus, testing optical systems, and making holograms.

© 1994 Optical Society of America

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References

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  1. F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  4. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  5. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).
  6. M. D. Levenson, “Wave-front engineering for photolithography,” Phys. Today 46 (7), 28–36 (1993).
    [CrossRef]
  7. M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron. Devices ED-29, 1828–1836(1982).
    [CrossRef]
  8. J. L. Horner, “Collimation invariant technique for measuring the focal length of a lens,” Appl. Opt. 28, 1047–1048 (1989).
    [CrossRef] [PubMed]
  9. J. M. Burch, “Scatter-fringe interferometry,” J. Opt. Soc. Am. 52, 600 (1962).
  10. E. N. Leith, J. Upatnieks, “Wave-front reconstruction with diffused illumination and three-dimensional objects,” J. Opt. Soc. Am. 54, 1295–1301 (1964).
    [CrossRef]

1993 (1)

M. D. Levenson, “Wave-front engineering for photolithography,” Phys. Today 46 (7), 28–36 (1993).
[CrossRef]

1989 (2)

1984 (1)

1982 (1)

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron. Devices ED-29, 1828–1836(1982).
[CrossRef]

1964 (1)

1962 (1)

J. M. Burch, “Scatter-fringe interferometry,” J. Opt. Soc. Am. 52, 600 (1962).

Bahri, Z.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

Burch, J. M.

J. M. Burch, “Scatter-fringe interferometry,” J. Opt. Soc. Am. 52, 600 (1962).

Gianino, P. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Horner, J. L.

Jutamulia, S.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

Leith, E. N.

Levenson, M. D.

M. D. Levenson, “Wave-front engineering for photolithography,” Phys. Today 46 (7), 28–36 (1993).
[CrossRef]

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron. Devices ED-29, 1828–1836(1982).
[CrossRef]

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron. Devices ED-29, 1828–1836(1982).
[CrossRef]

Upatnieks, J.

Vijaya Kumar, B. V. K.

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron. Devices ED-29, 1828–1836(1982).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

Appl. Opt. (3)

IEEE Trans. Electron. Devices (1)

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron. Devices ED-29, 1828–1836(1982).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Today (1)

M. D. Levenson, “Wave-front engineering for photolithography,” Phys. Today 46 (7), 28–36 (1993).
[CrossRef]

Other (3)

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram showing that an optical transparency generates a phase-only Fourier transform (F.T.).

Fig. 2
Fig. 2

(a) Simplest phase function of the phase-only Fourier transform that is generated by a transparency, (b) random phase function of the phase-only Fourier transform that can be generated by a transparency.

Equations (19)

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

F ( s ) = f ( x ) exp ( i 2 π x s ) d x ,
f ( x ) = F ( s ) exp ( i 2 π x s ) d s .
f ( x ) F ( s ) .
f * ( x ) exp ( i 2 π x s ) d x ,
= { f ( x ) exp [ i 2 π x ( s ) ] d x } * ,
= F * ( s ) ,
f * ( x ) F * ( s ) .
f ( x ) = f * ( x ) ,
F ( s ) = F * ( s ) .
F ( s ) = exp [ i α ( s ) ] ,
exp [ i α ( s ) ] = exp [ i α ( s ) ] ,
α ( s ) = α ( s ) ,
α ( s ) = 2 π s ,
F ( s ) = exp ( i 2 π s ) ,
f ( x ) = exp ( i 2 π s ) exp ( i 2 π x s ) d s ,
f ( x ) = δ ( x + 1 ) .
for 0 s , assign α ( s ) = 2 π Rnd ( s ) ,
for s 0 , assign α ( s ) = α ( s ) ,
f ( x ) = exp [ i α ( s ) ] exp ( i 2 π x s ) d s .

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