Abstract

The relationship between optical fractional Fourier transforms (OFRTs) obtained at different wavelengths is derived by use of the ABCD matrix formalism. It is shown that varying the wavelength while retaining the same optical system can be used to control the order of the OFRT. The advantage of this method of varying OFRT order is that no variation in the characteristics of the bulk optics is required. A general experimental verification of the theory is provided by showing the exact equivalence of two OFRT systems of different order when they are replayed using the same input function at different wavelengths.

© 2004 Optical Society of America

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References

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  1. D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  2. H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
    [CrossRef]
  3. A. W. Lohmann, J. Opt. Soc. Am. 10, 2181 (1993).
    [CrossRef]
  4. M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
    [CrossRef]
  5. S. A. Collins, J. Opt. Soc. Am. 60, 1168 (1970).
  6. A. Gerrad and J. M. Buch, Introduction to Matrix Optics (Wiley, New York, 1975).
  7. S. Abe and J. T. Sheridan, Opt. Lett. 19, 1801 (1994).
    [CrossRef]
  8. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
  9. L. M. Bernardo, Opt. Eng. 35, 732 (1996).
    [CrossRef]
  10. L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).
    [CrossRef]
  11. A. Sahin, H. M. Ozaktas, and D. Mendlovic, Appl. Opt. 37, 2130 (1998).
    [CrossRef]
  12. A. W. Lohmann, Opt. Commun. 115, 437 (1995).
    [CrossRef]
  13. A. W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, Opt. Commun. 146, 55 (1998).
    [CrossRef]
  14. J. T. Sheridan and R. Patten, Opt. Lett. 25, 448 (2000).
    [CrossRef]
  15. J. T. Sheridan, B. Hennelly, and D. Kelly, Opt. Lett. 28, 884 (2003).
    [CrossRef] [PubMed]
  16. W.-X. Cong, N.-X. Chen, and B.-Y. Gu, Appl. Opt. 37, 6906 (1998).
    [CrossRef]

2003 (1)

2002 (1)

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).
[CrossRef]

2000 (1)

1998 (3)

1997 (1)

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

1996 (1)

L. M. Bernardo, Opt. Eng. 35, 732 (1996).
[CrossRef]

1995 (1)

A. W. Lohmann, Opt. Commun. 115, 437 (1995).
[CrossRef]

1994 (1)

1993 (3)

1970 (1)

Abe, S.

Bernardo, L. M.

L. M. Bernardo, Opt. Eng. 35, 732 (1996).
[CrossRef]

Buch, J. M.

A. Gerrad and J. M. Buch, Introduction to Matrix Optics (Wiley, New York, 1975).

Cai, L. Z.

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).
[CrossRef]

Chen, N.-X.

Collins, S. A.

Cong, W.-X.

Dorsch, R. G.

A. W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, Opt. Commun. 146, 55 (1998).
[CrossRef]

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

Gerrad, A.

A. Gerrad and J. M. Buch, Introduction to Matrix Optics (Wiley, New York, 1975).

Gu, B.-Y.

Hennelly, B.

Kelly, D.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Lohmann, A. W.

A. W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, Opt. Commun. 146, 55 (1998).
[CrossRef]

A. W. Lohmann, Opt. Commun. 115, 437 (1995).
[CrossRef]

A. W. Lohmann, J. Opt. Soc. Am. 10, 2181 (1993).
[CrossRef]

Mendlovic, D.

A. W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, Opt. Commun. 146, 55 (1998).
[CrossRef]

A. Sahin, H. M. Ozaktas, and D. Mendlovic, Appl. Opt. 37, 2130 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
[CrossRef]

D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
[CrossRef]

Ozaktas, H. M.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, Appl. Opt. 37, 2130 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Patten, R.

Sahin, A.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, Appl. Opt. 37, 2130 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

Sheridan, J. T.

Wang, Y. Q.

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).
[CrossRef]

Zalevsky, Z.

A. W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, Opt. Commun. 146, 55 (1998).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

S. A. Collins, J. Opt. Soc. Am. 60, 1168 (1970).

A. W. Lohmann, J. Opt. Soc. Am. 10, 2181 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

A. W. Lohmann, Opt. Commun. 115, 437 (1995).
[CrossRef]

A. W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, Opt. Commun. 146, 55 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, Opt. Commun. 136, 52 (1997).
[CrossRef]

Opt. Eng. (1)

L. M. Bernardo, Opt. Eng. 35, 732 (1996).
[CrossRef]

Opt. Laser Technol. (1)

L. Z. Cai and Y. Q. Wang, Opt. Laser Technol. 34, 249 (2002).
[CrossRef]

Opt. Lett. (3)

Other (2)

A. Gerrad and J. M. Buch, Introduction to Matrix Optics (Wiley, New York, 1975).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

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Figures (2)

Fig. 1
Fig. 1

Relationship between a and a1 for different values of the parameter k=λ1/λ. The pair of values used in the experiment is shown -0.541,-0.453.

Fig. 2
Fig. 2

Output function from the Cai and Wang OFRT system with (a) a=1.459 and λ1=488 nm, (b) a1=1.541 and λ=633 nm.

Equations (10)

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

uaxa=Ia,f,λu0x0xa=1j2πλf sin ϕ1/2u0x0×expjπλfx02+xa2tan ϕ-2x0xasin ϕdx0.
cos ϕλf sin ϕ-sin ϕ/λfcos ϕ.
λ1f001/λ1fcos ϕsin ϕ-sin ϕcos ϕ×1/λ1f00λ1f.
λ1f001/λ1fcos ϕsin ϕ-sin ϕcos ϕ×λ/λ1001/λ1/λ1/λf00λf.
Ia1Mu0x0Mxa=1M1×expjπxa12qIa1u0x0xa1M1,
cos ϕsin ϕ-sin ϕcos ϕM001/M=10q1×M1001/M1cos ϕ1sin ϕ1-sin ϕ1cos ϕ1,
a1=2ϕ1π=2πtan-1tan ϕM2, M1=sin ϕM sin ϕ1, q=sin2 ϕ1-sin2 ϕsin ϕ cos ϕ.
λ1f001/λ1f10q1M1/λf00λf/M1×cos ϕ1λf sin ϕ1-sin ϕ1/λfcos ϕ1.
Ia,f,λ1u0x0xa=1M2×expjπq1xa2Ia1,f,λu0x0xa1M2,
a1=2πtan-1λ1 tan ϕλ=2πtan-1k tan aπ/2, M2=λ1fM1λf=k sin ϕsin ϕ1, q1=qλ1f2=sin2 ϕ1-sin2 ϕλ1f sin ϕ cos ϕ,

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