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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References

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  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).
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  4. A. E. Siegman, Lasers (University Science, 1986).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. Zh. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of Airy beams with optically induced refractive-index gradient,” Opt. Lett. 36, 3230–3232 (2011).
    [CrossRef]
  15. 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]
  16. 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]
  17. M. Born and E. Wolf, Principles of Optics (Pergamon, 1968), Chap. 1.

2011 (2)

2010 (1)

2009 (1)

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 (1)

2006 (1)

2005 (1)

1999 (1)

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 (1)

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 (1)

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

1974 (1)

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 (1)

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. (1)

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

Appl. Opt. (1)

Eur. J. Phys. (2)

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. (1)

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 (2)

Opt. Express (1)

Opt. Lett. (3)

Proc. SPIE (1)

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 (5)

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).

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

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Figures (6)

Fig. 1.
Fig. 1.

Time-averaged intensity in the Oxz plane of the Gaussian beam (n=0) in medium (3).

Fig. 2.
Fig. 2.

Time-averaged intensity in the Oxz plane of the Hermite–Gaussian beam [(a) n=3 and (b) n=4] in medium (3).

Fig. 3.
Fig. 3.

Binary element to approximate a linear index medium of Eq. (3) (a grating with linearly varied fill factor).

Fig. 4.
Fig. 4.

(a) Subwavelength grating with a linearly varying fill factor and (b) propagation of a Gaussian beam through the grating.

Fig. 5.
Fig. 5.

Propagation of a TM wave in the form of a Gaussian beam through the grating of Fig. 4(a).

Fig. 6.
Fig. 6.

Intensity profiles in the plane z=50λ for the Gaussian beam with (a) TE and (b) TM polarization.

Equations (25)

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E(x,z)=ik2πB×+E(ξ,0)exp[ik2B(Aξ22xξ+Dx2)]dξ.
(ABCD)=(cos(z/a)asin(z/a)sin(z/a)/acos(z/a)),
n2(x)=n02(1αx),
2Ex2+2Ez2+k02n02(1αx)E=0,
E(x,z)=U(x,z)exp(ikz),
2ikUz+2Ux2k2αxU=0
U(x,z)=G0+S(u)×exp[i(Gxxx2+Guuu2+Gxuux+Gxx+Guu)]du,
G0[2kdGxxdz+4Gxx2]x2G0[2kdGuudz+Gxu2]u2G0[2kdGxudz+4GxxGxu]uxG0[2kdGxdz+4GxxGx+k2α]xG0[2kdGudz+2GxuGx]u+[2ikdG0dz+2iG0GxxG0Gx2]=0.
{kdGxxdz+2Gxx2=0,2kdGuudz+Gxu2=0,kdGxudz+2GxxGxu=0,2kdGxdz+4GxxGx+k2α=0,kdGudz+GxuGx=0,2ikdG0dz+2iG0GxxG0Gx2=0.
{G0(z)=A0z+B0exp[iE022k(z+B0)]×exp[ikα296(z+B0)3+iαE0z4],Gxx(z)=k2(z+B0),Guu(z)=D022k(z+B0)+C0,Gxu(z)=D0z+B0,Gx(z)=E0z+B0kα4(z+B0),Gu(z)=αD0(z+B0)4+D0E0k(z+B0)+F0,
U(x,0)=A0exp[i(B0x2+E0x)]×+S(u)exp[i(C0u2+D0ux+F0u)]du,
U(x,z)=D0G02πA0exp[i(Gxxx2+Gxx)]×+U(ξ,0)exp[i(B0ξ2+E0ξ)]×iπGuuC0exp[(GxuxD0ξ+GuF0)24i(GuuC0)]dξ.
U(x,z)=ik2πzexp(ikα2z396)×+U(ξ,0)exp[ik2z(ξx)2ikαz4(x+ξ)]dξ.
2ikUz+2Ux2k2(x2a2x44a4)U=0,
2ikUz+2Ux2k2x2a2U=0.
U(x,z)=ik2πzexp(ikα2z324ikαxz2)×+U(ξ,0)exp[ik2z(ξx+x0)2]dξ.
x0(z)=αz2/4.
U(ξ,0)=exp(ξ2w02)Hn(2ξw0),
U(x,z)=(1)n{w0w(z)exp[iζ(z)]}1/2×exp{[xx0(z)]2w2(z)+ik[xx1(z)]22R(z)}×(z+izRzizR)n/2Hn[2(xx0)w(z)]exp[iΦ(z)],
w(z)=w0(1+z2/zR2)1/2.
x1(z)=[x0(z)αzR2/2].
ζ(z)=arctan(z/zR).
R(z)=z[1+(zR/z)2].
Φ(z)=(kα2z2/24)[2z3R(z)].
εc=ε2+(ε1ε2)(f0+β2),α=β(ε1ε2)dεc.

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