Abstract

We derive a diffraction integral to describe the paraxial propagation of an optical beam in a graded index medium with the permittivity linearly varying with the transverse coordinate. This integral transformation is irreducible to the familiar ABCD transformation. The form of the integral transformation suggests that, unlike a straight path in a homogeneous space, any paraxial optical beam will travel on a parabola bent toward the denser medium. By way of illustration, an explicit expression for the complex amplitude of a Hermite–Gaussian beam in the linear index medium is derived.

© 2014 Optical Society of America

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2011

2010

2009

S. N. Khonina, A. S. Striletz, A. A. Kovalev, and V. V. Kotlyar, “Propagation of laser vortex beams in a parabolic optical fiber,” Proc. SPIE 7523, 7523B (2009).

2007

2006

2005

1999

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The general wavefunction for a particle under uniform force,” Eur. J. Phys. 20, 477–482 (1999).
[CrossRef]

1995

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

1979

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

1974

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt +Uxx−c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[CrossRef]

1973

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Bandres, M.

Bernardini, C.

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The general wavefunction for a particle under uniform force,” Eur. J. Phys. 20, 477–482 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968), Chap. 1.

Casperson, L. W.

Chen, Z.

Efremidis, N. K.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The general wavefunction for a particle under uniform force,” Eur. J. Phys. 20, 477–482 (1999).
[CrossRef]

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.

Guattari, G.

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The general wavefunction for a particle under uniform force,” Eur. J. Phys. 20, 477–482 (1999).
[CrossRef]

Gutiérrez-Vega, J.

Hesselink, L.

Hu, Y.

Kalnins, E. G.

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt +Uxx−c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[CrossRef]

Khonina, S. N.

S. N. Khonina, A. S. Striletz, A. A. Kovalev, and V. V. Kotlyar, “Propagation of laser vortex beams in a parabolic optical fiber,” Proc. SPIE 7523, 7523B (2009).

Koç, A.

Kotlyar, V. V.

S. N. Khonina, A. S. Striletz, A. A. Kovalev, and V. V. Kotlyar, “Propagation of laser vortex beams in a parabolic optical fiber,” Proc. SPIE 7523, 7523B (2009).

Kovalev, A. A.

S. N. Khonina, A. S. Striletz, A. A. Kovalev, and V. V. Kotlyar, “Propagation of laser vortex beams in a parabolic optical fiber,” Proc. SPIE 7523, 7523B (2009).

Kutay, M.

Liu, S.

Lou, C.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

Miller, W.

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt +Uxx−c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[CrossRef]

Ozaktas, H.

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The general wavefunction for a particle under uniform force,” Eur. J. Phys. 20, 477–482 (1999).
[CrossRef]

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

Sari, I.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Song, D.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Striletz, A. S.

S. N. Khonina, A. S. Striletz, A. A. Kovalev, and V. V. Kotlyar, “Propagation of laser vortex beams in a parabolic optical fiber,” Proc. SPIE 7523, 7523B (2009).

Wang, L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968), Chap. 1.

Ye, Zh.

Zhang, P.

Zhang, Y.

Zhao, J.

Zheng, C.

Am. J. Phys.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Appl. Opt.

Eur. J. Phys.

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The general wavefunction for a particle under uniform force,” Eur. J. Phys. 20, 477–482 (1999).
[CrossRef]

J. Math. Phys.

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt +Uxx−c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. SPIE

S. N. Khonina, A. S. Striletz, A. A. Kovalev, and V. V. Kotlyar, “Propagation of laser vortex beams in a parabolic optical fiber,” Proc. SPIE 7523, 7523B (2009).

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968), Chap. 1.

A. E. Siegman, Lasers (University Science, 1986).

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

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