Abstract

Propagation and focusing of nonparaxial Gaussian beams with spherical wavefront in a graded-index medium are investigated using quantum-theoretical method of coherent states. Explicit expressions for the trajectory and width of a beam taking into account all correction terms to the paraxial approximation have been obtained. Electromagnetic field distributions in longitudinal and lateral directions are simulated. Diffraction of strongly focused high-aperture wave beams is investigated theoretically. The ratio of intensities of evanescent and propagating fields is calculated for different values of focused spot.

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  1. H. Kogelnik, "On the propagation of Gaussian beams of light through lenslike media including those with a loss and gain variation," Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  2. L. W. Casperson, "Gaussian light beams in inhomogeneous media," Appl. Opt. 12, 2434 (1973).
    [CrossRef] [PubMed]
  3. D. D. Lowental, "Far-field diffraction patterns for Gaussian beams in the presence of small spherical aberrations," J. Opt. Soc. Am. 65, 853 (1975).
    [CrossRef]
  4. L. W. Casperson, "Synthesis of Gaussian beam optical systems," Appl. Opt. 20, 2243 (1981).
    [CrossRef] [PubMed]
  5. S. A. Self, "Focusing of spherical Gaussian beams," Appl. Opt. 22, 658 (1983).
    [CrossRef] [PubMed]
  6. R. M. Herman, J. Pardo and T.A. Wiggins, "Diffraction and focusing of Gaussian beams," Appl. Opt. 24, 1346 (1985).
    [CrossRef] [PubMed]
  7. A. Kujawski, "Focusing of Gaussian beams described in terms of complex rays," Appl. Opt. 28, 2458 (1989).
    [CrossRef] [PubMed]
  8. A. R. Al-Rashed and B. E. A. Saleh, "Decentered Gaussian beams," Appl. Opt. 34, 6819 (1995).
    [CrossRef] [PubMed]
  9. M. Lax, W. H. Luisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. 11, 1365 (1975).
    [CrossRef]
  10. G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575 (1979).
    [CrossRef]
  11. T. Takenaka, M. Yokota and O. Fukumitsu, "Propagation of light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 826 (1985).
    [CrossRef]
  12. G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693 (1983).
    [CrossRef]
  13. S. Nemoto, "Nonparaxial Gaussian beams," Appl. Opt. 29, 1940 (1990).
    [CrossRef] [PubMed]
  14. Q. Cao and X. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144 (1998).
    [CrossRef]
  15. A. Wunsche, "Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams," J. Opt. Soc. Am. A 9, 765 (1992).
    [CrossRef]
  16. X. D. Zeng, C. H. Liang and Y. Y. An, "Far-firld radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation," Appl. Opt. 36, 2042 (1997).
    [CrossRef] [PubMed]
  17. H. Laabs, "Propagation of Hermite-Gaussian-beams beyond the paraxial approximation," Opt. Commun. 147, 1 (1998).
    [CrossRef]
  18. C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381 (1999).
    [CrossRef]
  19. C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579 (2001).
    [CrossRef]
  20. Z. Ulanowski and I. K. Ludlow, "Scalar field of nonparaxial Gaussian beams," Opt. Lett. 25, 1792 (2000).
    [CrossRef]
  21. R. Borghi, M. Santarsiero and M. A. Porras, "Nonparaxial Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 1618 (2001).
    [CrossRef]
  22. L. W. Casperson, "Beam propagation in periodic quadratic-index waveguides," Appl. Opt. 24, 4395 (1985).
    [CrossRef] [PubMed]
  23. J. N. McMullin, "The ABCD matrix in arbitrarily quadratic-index waveguides," Appl. Opt. 25, 2184 (1986).
    [CrossRef] [PubMed]
  24. D. Bertilone and C. Pask, "Exact ray paths in a graded-index taper," Appl. Opt. 26, 1189 (1987).
    [CrossRef] [PubMed]
  25. D. Bertilone, A. Ankiewicz and C. Pask, "Wave propagation in a graded-index taper," Appl. Opt. 26, 2213 (1987).
    [CrossRef] [PubMed]
  26. J. N. McMullin, "The ABCD matrix in graded-index tapers used for beam expansion and compression," Appl. Opt. 28, 1298 (1989).
    [CrossRef] [PubMed]
  27. A. A. Tovar and L. W. Casperson, "Beam propagation in parabolically tapered graded-index waveguides," Appl. Opt. 33, 7733 (1994).
    [CrossRef] [PubMed]
  28. J. Linares and C. Gomez-Reino, "Arbitrary single-mode coupling by tapered and nontapered GRIN fiber lenses," Appl. Opt. 29, 4003 (1990).
    [CrossRef] [PubMed]
  29. S. G. Krivoshlykov and E. G. Sauter, "Propagation and focusing of nonparaxial Gaussian beams with spherical wave fronts in graded-index waveguides with polynomial profiles," J. Opt. Soc. Am. A 10,262 (1993).
    [CrossRef]
  30. N. I. Petrov, "Candidate's Thesis in Physico-Mathematical Sciences" (Moscow: General Physics Institute, Russian Academy of Sciences, 1985).
  31. N. I. Petrov, "Nonparaxial focusing of wave beams in a graded-index medium," Rus. J. Quantum Electronics 29, 249 (1999).
    [CrossRef]
  32. D. Marcuse, Light transmission optics (Van Nostrand, New York, 1972).
  33. J. A. Arnaud, Beam and fiber optics (Academic Press, New York, 1976).
  34. R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766 (1963).
    [CrossRef]
  35. E. Schrodinger, Naturwissenschaften, 14, 664 (1926).
    [CrossRef]
  36. S. G. Krivoshlykov, N. I. Petrov and I. N. Sisakyan, "Correlated coherent states and propagation of arbitrary gaussian beams in graded-index media with loss and gain," Sov. J. Quantum Electronics 7, 1424 (1986).
  37. H. A. Eide and J. J. Stamnes, "Exact and approximate solutions for focusing of two-dimensional waves," J. Opt. Soc. Am. A 15, 1292 (1998).
    [CrossRef]
  38. Q. Wu, R. D. Grober, D. Gammon, D. S. Katzer, "Imaging Spectroscopy of Two-Dimensional Excitons in a Narrow GaAs/AlGaAs Quantum Well," Phys. Rev. Letts. 83, 2652 (1999).
    [CrossRef]
  39. G. von Freymann, et.al., "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probes?," Appl. Phys. Letts. 73, 1170 (1998).
    [CrossRef]
  40. E. A. J. Marcatili, "Dielectric tapers with curved axes and no loss," IEEE J. Quant. Electr. QE-21,307 (1985).
    [CrossRef]
  41. D. Bertilone, "Ray propagation and compression in a strictly adiabatic taper," Opt. Quant. Electr. 19, 361 (1987).
    [CrossRef]
  42. D. Bertilone, J. Love and C. Pask, "Splicing of optical waveguides with lossless graded-index tapers," Opt. Quant. Electr. 20, 501 (1988).
    [CrossRef]
  43. L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon, Oxford, 1965).
  44. N. I. Petrov, "Depolarization of light in a graded-index isotropic medium," J. Mod. Opt. 43, 2239 (1996).
    [CrossRef]
  45. N. I. Petrov, "Evolution of Berry's phase in a graded-index medium," Phys. Lett. A 234, 239 (1997).
    [CrossRef]
  46. S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, "Focusing light to a tighter spot," Opt. Comms. 179, 1 (2000).
    [CrossRef]

Other (46)

H. Kogelnik, "On the propagation of Gaussian beams of light through lenslike media including those with a loss and gain variation," Appl. Opt. 4, 1562 (1965).
[CrossRef]

L. W. Casperson, "Gaussian light beams in inhomogeneous media," Appl. Opt. 12, 2434 (1973).
[CrossRef] [PubMed]

D. D. Lowental, "Far-field diffraction patterns for Gaussian beams in the presence of small spherical aberrations," J. Opt. Soc. Am. 65, 853 (1975).
[CrossRef]

L. W. Casperson, "Synthesis of Gaussian beam optical systems," Appl. Opt. 20, 2243 (1981).
[CrossRef] [PubMed]

S. A. Self, "Focusing of spherical Gaussian beams," Appl. Opt. 22, 658 (1983).
[CrossRef] [PubMed]

R. M. Herman, J. Pardo and T.A. Wiggins, "Diffraction and focusing of Gaussian beams," Appl. Opt. 24, 1346 (1985).
[CrossRef] [PubMed]

A. Kujawski, "Focusing of Gaussian beams described in terms of complex rays," Appl. Opt. 28, 2458 (1989).
[CrossRef] [PubMed]

A. R. Al-Rashed and B. E. A. Saleh, "Decentered Gaussian beams," Appl. Opt. 34, 6819 (1995).
[CrossRef] [PubMed]

M. Lax, W. H. Luisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. 11, 1365 (1975).
[CrossRef]

G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575 (1979).
[CrossRef]

T. Takenaka, M. Yokota and O. Fukumitsu, "Propagation of light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 826 (1985).
[CrossRef]

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693 (1983).
[CrossRef]

S. Nemoto, "Nonparaxial Gaussian beams," Appl. Opt. 29, 1940 (1990).
[CrossRef] [PubMed]

Q. Cao and X. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144 (1998).
[CrossRef]

A. Wunsche, "Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams," J. Opt. Soc. Am. A 9, 765 (1992).
[CrossRef]

X. D. Zeng, C. H. Liang and Y. Y. An, "Far-firld radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation," Appl. Opt. 36, 2042 (1997).
[CrossRef] [PubMed]

H. Laabs, "Propagation of Hermite-Gaussian-beams beyond the paraxial approximation," Opt. Commun. 147, 1 (1998).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381 (1999).
[CrossRef]

C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579 (2001).
[CrossRef]

Z. Ulanowski and I. K. Ludlow, "Scalar field of nonparaxial Gaussian beams," Opt. Lett. 25, 1792 (2000).
[CrossRef]

R. Borghi, M. Santarsiero and M. A. Porras, "Nonparaxial Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 1618 (2001).
[CrossRef]

L. W. Casperson, "Beam propagation in periodic quadratic-index waveguides," Appl. Opt. 24, 4395 (1985).
[CrossRef] [PubMed]

J. N. McMullin, "The ABCD matrix in arbitrarily quadratic-index waveguides," Appl. Opt. 25, 2184 (1986).
[CrossRef] [PubMed]

D. Bertilone and C. Pask, "Exact ray paths in a graded-index taper," Appl. Opt. 26, 1189 (1987).
[CrossRef] [PubMed]

D. Bertilone, A. Ankiewicz and C. Pask, "Wave propagation in a graded-index taper," Appl. Opt. 26, 2213 (1987).
[CrossRef] [PubMed]

J. N. McMullin, "The ABCD matrix in graded-index tapers used for beam expansion and compression," Appl. Opt. 28, 1298 (1989).
[CrossRef] [PubMed]

A. A. Tovar and L. W. Casperson, "Beam propagation in parabolically tapered graded-index waveguides," Appl. Opt. 33, 7733 (1994).
[CrossRef] [PubMed]

J. Linares and C. Gomez-Reino, "Arbitrary single-mode coupling by tapered and nontapered GRIN fiber lenses," Appl. Opt. 29, 4003 (1990).
[CrossRef] [PubMed]

S. G. Krivoshlykov and E. G. Sauter, "Propagation and focusing of nonparaxial Gaussian beams with spherical wave fronts in graded-index waveguides with polynomial profiles," J. Opt. Soc. Am. A 10,262 (1993).
[CrossRef]

N. I. Petrov, "Candidate's Thesis in Physico-Mathematical Sciences" (Moscow: General Physics Institute, Russian Academy of Sciences, 1985).

N. I. Petrov, "Nonparaxial focusing of wave beams in a graded-index medium," Rus. J. Quantum Electronics 29, 249 (1999).
[CrossRef]

D. Marcuse, Light transmission optics (Van Nostrand, New York, 1972).

J. A. Arnaud, Beam and fiber optics (Academic Press, New York, 1976).

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766 (1963).
[CrossRef]

E. Schrodinger, Naturwissenschaften, 14, 664 (1926).
[CrossRef]

S. G. Krivoshlykov, N. I. Petrov and I. N. Sisakyan, "Correlated coherent states and propagation of arbitrary gaussian beams in graded-index media with loss and gain," Sov. J. Quantum Electronics 7, 1424 (1986).

H. A. Eide and J. J. Stamnes, "Exact and approximate solutions for focusing of two-dimensional waves," J. Opt. Soc. Am. A 15, 1292 (1998).
[CrossRef]

Q. Wu, R. D. Grober, D. Gammon, D. S. Katzer, "Imaging Spectroscopy of Two-Dimensional Excitons in a Narrow GaAs/AlGaAs Quantum Well," Phys. Rev. Letts. 83, 2652 (1999).
[CrossRef]

G. von Freymann, et.al., "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probes?," Appl. Phys. Letts. 73, 1170 (1998).
[CrossRef]

E. A. J. Marcatili, "Dielectric tapers with curved axes and no loss," IEEE J. Quant. Electr. QE-21,307 (1985).
[CrossRef]

D. Bertilone, "Ray propagation and compression in a strictly adiabatic taper," Opt. Quant. Electr. 19, 361 (1987).
[CrossRef]

D. Bertilone, J. Love and C. Pask, "Splicing of optical waveguides with lossless graded-index tapers," Opt. Quant. Electr. 20, 501 (1988).
[CrossRef]

L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon, Oxford, 1965).

N. I. Petrov, "Depolarization of light in a graded-index isotropic medium," J. Mod. Opt. 43, 2239 (1996).
[CrossRef]

N. I. Petrov, "Evolution of Berry's phase in a graded-index medium," Phys. Lett. A 234, 239 (1997).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, "Focusing light to a tighter spot," Opt. Comms. 179, 1 (2000).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Trajectories of wave beams (dashed curves) of wavelength λ=0.65 µm accompanied by its width (solid curves) for gradient parameter ω=5·10-2 µm-1 with initial inclination angles 15° and 60° (a) and initial axis displacements 7.76 and 26 µm (b).

Fig. 2.
Fig. 2.

Variation with distance of the field intensity along the axial line of a graded-index medium withω=5·10-2 µm-1 and n 0=1.5; half-width of the incident beam a 0=15 µm.

Fig. 3.
Fig. 3.

Distribution of the field intensity in a transverse plane of a waveguide at various distances from the initial plane z=0 (a), z=40 (b) and z=44.9 µm (c) (the parameters of the beam and the medium correspond to those in Fig.2).

Fig. 4.
Fig. 4.

Distribution of the evanescent (red curve) and propagating (blue curve) fields in a transverse direction at the initial plane.

Fig. 5.
Fig. 5.

Distribution of the total field intensity at the initial plane.

Fig. 6.
Fig. 6.

Distribution of the propagating field intensity at the initial plane.

Fig. 7.
Fig. 7.

The ratio of energy guided by propagating modes P g (solid curves) and evanescent modes P ev (dashed curves) to the total input energy P 0 as function of spot size a 0 for different values of refractive index: 1-n 0=1.5; 2-n 0=2; 3-n 0=3.87.

Fig. 8.
Fig. 8.

Variation in the angle θ 0 at which the far-field intensity drops to one half and 1/e of the intensity on the axis for different values of the initial spot size.

Equations (24)

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

Δ E + k 2 n 2 E + ( E · In n 2 ) = 0 ,
2 E x 2 + 2 E y 2 + 2 E z 2 + k 2 n 2 ( x , y , z ) E = 0
H ̂ ψ ( x , y ) = ε ψ ( x , y ) ,
H ̂ = 1 2 k 2 [ 2 x 2 + 2 y 2 ] + 1 2 ( n 0 2 n 2 ( x , y ) ) .
A ̂ = i [ A ̂ , β ̂ ] ,
n 2 = n 0 2 ω 2 ( x 2 + y 2 ) ,
β ̂ = k n 0 ( 1 H ̂ n 0 2 H ̂ 2 2 n 0 4 H ̂ 3 2 n 0 6 ) ,
x ̅ ( z ) = ψ α ( z ) x ̂ ψ α ( z ) = ψ ( 0 ) U ̂ + x ̂ U ̂ ψ ( 0 ) = ψ α ( 0 ) x ̂ ( z ) ψ α ( 0 ) ,
Δ x α 2 = x ̂ 2 α x ̂ α 2 .
x α = α k ω 2 exp { α 2 ( cos ω k n 0 3 z 1 ) } cos [ ( ω n 0 + ω 2 k n 0 3 ) z + α 2 sin ω 2 k n 0 3 z θ ]
x α = α k ω 2 e α 2 m = 0 α 2 m m ! cos [ k n 0 ( 1 2 ω k n 0 2 ( m + 1 2 ) 1 2 ω k n 0 2 ( m + 3 2 ) ) z θ ]
Δ x α 2 = 1 2 k ω { 1 + 2 α 2 + 2 α 2 e α 2 m = 0 α 2 m m ! cos [ k n 0 ( 1 2 ω k n 0 2 ( m + 1 2 ) 1 2 ω k n 0 2 ( m + 5 2 ) ) z 2 θ ] }
x α 2
ψ ( x , 0 ) = ( 2 π ) 1 4 1 a 0 exp ( x 2 a 0 2 i k 2 R x 2 ) ,
Ψ ( x , z ) = ( k ω π ) 1 4 exp ( k ω 2 x 2 ) ( u v u * v * ) 1 4 1 u m = 0 N H m ( 0 ) m ! 2 m 2 ( v 2 u ) m 2 H m ( x k ω ) e i β m z ,
where v = ( v r 2 + v i 2 ) 1 2 e i φ , v r = 1 2 ( 1 ω μ ω cos χ ) ( μ cos χ ) 1 2 ,
v i = 1 2 μ ω sin χ ( μ cos χ ) 1 2 , u = ( u r 2 + u i 2 ) 1 2 e i ,
u r = 1 2 ( 1 ω + μ ω cos χ ) ( μ cos χ ) 1 2 , u i = 1 2 μ ω sin χ ( μ cos χ ) 1 2
φ = arctan [ v i v r ] , ϑ = arctan [ u i u r ] , χ = arctan ( k a 0 2 2 R ) , μ = R sin χ ,
β m = k n 0 [ 1 2 ω k n 0 2 ( m + 1 2 ) ] 1 2 .
Δ x 0 2 ( z ) = 1 2 k ω { 1 + 2 v 2 + v r 2 + v i 2 u 2 m = 0 H 2 m ( 0 ) H 2 m + 2 ( 0 ) ( 2 m ) ! ( v 2 u ) 2 m cos [ ( β 2 m β 2 m + 2 ) z φ + ϑ ] }
L f π n 0 / 2 ω [ 1 + 3 8 ( 4 + k 2 a 0 4 ω 2 k 2 n 0 2 a 0 2 ) ] ,
Δ x 2 1 4 k 2 Δ p 2 λ 2 16 π 2 n 0 2 ,
R 2 Δ x 4 Δ x 2 Δ p 2 1 / 4 k 2 ,

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