Abstract

A method of spatial modal decomposition for optical beams by fractional Fourier transform, and its practical implementation with reduced complexity by use of modal interleavers, are discussed.

© 2001 Optical Society of America

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References

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  1. C. J. Chang-Hasnain, “Vertical-cavity sufrace-emitting lasers,” in Semiconductor Lasers: Past, Present, and Future, G. P. Agrawal, ed. (AIP Press, Woodbury, N.Y., 1995), p. 145.
  2. E. G. Johnson, “Direct measurement of the spatial modes of a laser pulse: theory,” Appl. Opt. 25, 2967 (1986).
    [CrossRef]
  3. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212 (1993).
    [CrossRef]
  4. Y. Satuby and M. Orenstein, “Small-signal modulation of multitransverse modes vertical-cavity surface-emitting semiconductor lasers,” IEEE Photon. Technol. Lett. 10, 757 (1998).
    [CrossRef]
  5. F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “Intensity-based modal analysis of partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989 (1998).
    [CrossRef]
  6. M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite–Gaussian modes,” Appl. Opt. 38, 5272 (1999).
    [CrossRef]
  7. X. Xue, H. Wei, and A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite–Gaussian functions,” J. Opt. Soc. Am. A 17, 1086 (2000).
    [CrossRef]
  8. M. A. Golub, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 12, 1208 (1982).
    [CrossRef]
  9. H. O. Bartelt, A. W. Lohmann, W. Freude, and G. K. Grau, “Mode analysis of optical fibers using computer-generated matched filters,” Electron. Lett. 19, 247 (1983).
    [CrossRef]
  10. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1998), Vol. 38, p. 263.
    [CrossRef]
  11. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]

2000 (1)

1999 (1)

1998 (2)

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “Intensity-based modal analysis of partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989 (1998).
[CrossRef]

Y. Satuby and M. Orenstein, “Small-signal modulation of multitransverse modes vertical-cavity surface-emitting semiconductor lasers,” IEEE Photon. Technol. Lett. 10, 757 (1998).
[CrossRef]

1993 (2)

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

1986 (1)

1983 (1)

H. O. Bartelt, A. W. Lohmann, W. Freude, and G. K. Grau, “Mode analysis of optical fibers using computer-generated matched filters,” Electron. Lett. 19, 247 (1983).
[CrossRef]

1982 (1)

M. A. Golub, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 12, 1208 (1982).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, A. W. Lohmann, W. Freude, and G. K. Grau, “Mode analysis of optical fibers using computer-generated matched filters,” Electron. Lett. 19, 247 (1983).
[CrossRef]

Borghi, R.

Chang-Hasnain, C. J.

C. J. Chang-Hasnain, “Vertical-cavity sufrace-emitting lasers,” in Semiconductor Lasers: Past, Present, and Future, G. P. Agrawal, ed. (AIP Press, Woodbury, N.Y., 1995), p. 145.

Freude, W.

H. O. Bartelt, A. W. Lohmann, W. Freude, and G. K. Grau, “Mode analysis of optical fibers using computer-generated matched filters,” Electron. Lett. 19, 247 (1983).
[CrossRef]

Golub, M. A.

M. A. Golub, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 12, 1208 (1982).
[CrossRef]

Gori, F.

Grau, G. K.

H. O. Bartelt, A. W. Lohmann, W. Freude, and G. K. Grau, “Mode analysis of optical fibers using computer-generated matched filters,” Electron. Lett. 19, 247 (1983).
[CrossRef]

Guattari, G.

Johnson, E. G.

Kirk, A. G.

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

H. O. Bartelt, A. W. Lohmann, W. Freude, and G. K. Grau, “Mode analysis of optical fibers using computer-generated matched filters,” Electron. Lett. 19, 247 (1983).
[CrossRef]

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1998), Vol. 38, p. 263.
[CrossRef]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1998), Vol. 38, p. 263.
[CrossRef]

Orenstein, M.

Y. Satuby and M. Orenstein, “Small-signal modulation of multitransverse modes vertical-cavity surface-emitting semiconductor lasers,” IEEE Photon. Technol. Lett. 10, 757 (1998).
[CrossRef]

Prokhorov, A. M.

M. A. Golub, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 12, 1208 (1982).
[CrossRef]

Santarsiero, M.

Satuby, Y.

Y. Satuby and M. Orenstein, “Small-signal modulation of multitransverse modes vertical-cavity surface-emitting semiconductor lasers,” IEEE Photon. Technol. Lett. 10, 757 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212 (1993).
[CrossRef]

Sisakyan, I. N.

M. A. Golub, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 12, 1208 (1982).
[CrossRef]

Soifer, V. A.

M. A. Golub, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 12, 1208 (1982).
[CrossRef]

Townsend, S. W.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212 (1993).
[CrossRef]

Wei, H.

Xue, X.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1998), Vol. 38, p. 263.
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (1)

H. O. Bartelt, A. W. Lohmann, W. Freude, and G. K. Grau, “Mode analysis of optical fibers using computer-generated matched filters,” Electron. Lett. 19, 247 (1983).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y. Satuby and M. Orenstein, “Small-signal modulation of multitransverse modes vertical-cavity surface-emitting semiconductor lasers,” IEEE Photon. Technol. Lett. 10, 757 (1998).
[CrossRef]

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

Opt. Lett. (1)

Sov. J. Quantum Electron. (1)

M. A. Golub, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 12, 1208 (1982).
[CrossRef]

Other (2)

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1998), Vol. 38, p. 263.
[CrossRef]

C. J. Chang-Hasnain, “Vertical-cavity sufrace-emitting lasers,” in Semiconductor Lasers: Past, Present, and Future, G. P. Agrawal, ed. (AIP Press, Woodbury, N.Y., 1995), p. 145.

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Figures (3)

Fig. 1
Fig. 1

1×4 modal decomposer. The setups on the left and right sides of the dashed line produce the linear transforms of Eqs.  (7) and (8), respectively.

Fig. 2
Fig. 2

Binary tree for modal decomposition; each node represents a modal interleaver.

Fig. 3
Fig. 3

SMI, which is a Mach–Zehnder interferometer with a FRFT operator inserted into one of the two arms and a delay compensator inserted into the other.

Equations (8)

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

amn=umn*x,yEx,ydxdy,m0,n0
ψmx=Amw0Hmx2w0exp-x2w02
Ex=m=0M-1amψmx,
FθHmx=expimθHmx,m0,
Ex=FθEx=m=0M-1amψmxexpimθ,
θk=2πkM,k=0,1,,M-1,
Ekx=1Mm=0M-1amψmxexpimθk=1Mm=0M-1amψmxexpi2πkm/M
amψmx=1Mk=0M-1Ekxexp-i2πkm/M

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