Abstract

Nonparaxial diffraction-limited propagation of light with amplitude distribution in hyperbolic functions through an inhomogeneous planar medium with a hyperbolic secant refractive-index profile is studied by means of the stationary phase method. The irradiance distribution at geometrical shadow, edge of shadow, and a geometrically illuminated region is analyzed for a particular case.

© 2000 Optical Society of America

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References

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  1. C. Bao, C. Gómez-Reino, “Collimation and focusing properties of a planar waveguide with hyperbolic secant refractive index profile: light deflector and collimator design,” Opt. Quantum Electron. 27, 897–907 (1995).
    [CrossRef]
  2. C. Bao, M. V. Pérez, C. Gómez-Reino, “Coupling of planar waveguides with hyperbolic secant refractive index profile by butt-joining: beam size controller design,” Opt. Lett. 21, 1078–1080 (1996).
    [CrossRef] [PubMed]
  3. C. Bao, C. Gómez-Reino, M. V. Pérez, “GRIN tapered hyperbolic secant planar waveguide for focusing, collimation and beam size control,” J. Opt. Soc. Am. A 14, 1754–1759 (1997).
    [CrossRef]
  4. C. Gómez-Reino, M. V. Pérez, C. Bao, M. T. Flores-Arias, S. Vidal, “Diffraction-free and diffraction-limited in graded-index planar waveguides with hyperbolic secant refractive index profile,” J. Mod. Opt. 47, 91–102 (2000).
  5. D. W. Hewak, J. W. Y. Lit, “Solution deposited optical waveguide lens,” Appl. Opt. 28, 4190–4198 (1989).
    [CrossRef] [PubMed]
  6. R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
    [CrossRef]
  7. N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
    [CrossRef]
  8. T. Tamir, ed., Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1990), Chap. 2, pp. 52–54.
  9. M. J. Adams, Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 4, pp 132–135.
  10. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12, pp. 263–267.
  11. T. Kozek, “Design of optical planar waveguides with prescribed propagation constants,” in Optical Fibers and Their Applications IV, R. S. Romaniuk, M. Szustakowski, eds., Proc. SPIE670, 226–233 (1986); “Synthesis of optical planar waveguides,” in Integrated Optical Circuit Engineering IV, M. A. Mentzer, S. Sriram, eds., Proc. SPIE 704, 44–50 (1986).
    [CrossRef]
  12. C. Paré, L. Gagnon, P. A. Bélanger, “Aspherical laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
    [CrossRef] [PubMed]
  13. I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), Chap. 9, pp. 1039–1057.
  14. J. Liñares, C. Gómez-Reino, “Optical propagator in a graded-index medium with a hyperbolic secant refractive index profile,” Appl. Opt. 33, 3427–3431 (1994).
    [CrossRef]
  15. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964), Chap. 4, pp. 282–285.
  16. A. Papoulis, Systems and Transforms with Applications in Optics (Prentice-Hall, Englewood Cliffs, N.J., 1968), Chap. 7.
  17. J. J. Stammes, Waves in Focal Regions (Adam-Hilger, Bristol, UK, 1986), Chap. 8.

2000 (1)

C. Gómez-Reino, M. V. Pérez, C. Bao, M. T. Flores-Arias, S. Vidal, “Diffraction-free and diffraction-limited in graded-index planar waveguides with hyperbolic secant refractive index profile,” J. Mod. Opt. 47, 91–102 (2000).

1997 (1)

1996 (1)

1995 (1)

C. Bao, C. Gómez-Reino, “Collimation and focusing properties of a planar waveguide with hyperbolic secant refractive index profile: light deflector and collimator design,” Opt. Quantum Electron. 27, 897–907 (1995).
[CrossRef]

1994 (1)

1992 (1)

C. Paré, L. Gagnon, P. A. Bélanger, “Aspherical laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef] [PubMed]

1989 (1)

1988 (2)

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
[CrossRef]

Adams, M. J.

M. J. Adams, Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 4, pp 132–135.

Bao, C.

C. Gómez-Reino, M. V. Pérez, C. Bao, M. T. Flores-Arias, S. Vidal, “Diffraction-free and diffraction-limited in graded-index planar waveguides with hyperbolic secant refractive index profile,” J. Mod. Opt. 47, 91–102 (2000).

C. Bao, C. Gómez-Reino, M. V. Pérez, “GRIN tapered hyperbolic secant planar waveguide for focusing, collimation and beam size control,” J. Opt. Soc. Am. A 14, 1754–1759 (1997).
[CrossRef]

C. Bao, M. V. Pérez, C. Gómez-Reino, “Coupling of planar waveguides with hyperbolic secant refractive index profile by butt-joining: beam size controller design,” Opt. Lett. 21, 1078–1080 (1996).
[CrossRef] [PubMed]

C. Bao, C. Gómez-Reino, “Collimation and focusing properties of a planar waveguide with hyperbolic secant refractive index profile: light deflector and collimator design,” Opt. Quantum Electron. 27, 897–907 (1995).
[CrossRef]

Bélanger, P. A.

C. Paré, L. Gagnon, P. A. Bélanger, “Aspherical laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef] [PubMed]

Flores-Arias, M. T.

C. Gómez-Reino, M. V. Pérez, C. Bao, M. T. Flores-Arias, S. Vidal, “Diffraction-free and diffraction-limited in graded-index planar waveguides with hyperbolic secant refractive index profile,” J. Mod. Opt. 47, 91–102 (2000).

Gagnon, L.

C. Paré, L. Gagnon, P. A. Bélanger, “Aspherical laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef] [PubMed]

Gómez-Reino, C.

C. Gómez-Reino, M. V. Pérez, C. Bao, M. T. Flores-Arias, S. Vidal, “Diffraction-free and diffraction-limited in graded-index planar waveguides with hyperbolic secant refractive index profile,” J. Mod. Opt. 47, 91–102 (2000).

C. Bao, C. Gómez-Reino, M. V. Pérez, “GRIN tapered hyperbolic secant planar waveguide for focusing, collimation and beam size control,” J. Opt. Soc. Am. A 14, 1754–1759 (1997).
[CrossRef]

C. Bao, M. V. Pérez, C. Gómez-Reino, “Coupling of planar waveguides with hyperbolic secant refractive index profile by butt-joining: beam size controller design,” Opt. Lett. 21, 1078–1080 (1996).
[CrossRef] [PubMed]

C. Bao, C. Gómez-Reino, “Collimation and focusing properties of a planar waveguide with hyperbolic secant refractive index profile: light deflector and collimator design,” Opt. Quantum Electron. 27, 897–907 (1995).
[CrossRef]

J. Liñares, C. Gómez-Reino, “Optical propagator in a graded-index medium with a hyperbolic secant refractive index profile,” Appl. Opt. 33, 3427–3431 (1994).
[CrossRef]

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), Chap. 9, pp. 1039–1057.

Hewak, D. W.

Jinguji, K.

N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
[CrossRef]

Kawachi, M.

N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
[CrossRef]

Kozek, T.

T. Kozek, “Design of optical planar waveguides with prescribed propagation constants,” in Optical Fibers and Their Applications IV, R. S. Romaniuk, M. Szustakowski, eds., Proc. SPIE670, 226–233 (1986); “Synthesis of optical planar waveguides,” in Integrated Optical Circuit Engineering IV, M. A. Mentzer, S. Sriram, eds., Proc. SPIE 704, 44–50 (1986).
[CrossRef]

Liñares, J.

Lit, J. W. Y.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12, pp. 263–267.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964), Chap. 4, pp. 282–285.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (Prentice-Hall, Englewood Cliffs, N.J., 1968), Chap. 7.

Paré, C.

C. Paré, L. Gagnon, P. A. Bélanger, “Aspherical laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef] [PubMed]

Pérez, M. V.

Ramaswamy, R. V.

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), Chap. 9, pp. 1039–1057.

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12, pp. 263–267.

Srivastava, R.

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

Stammes, J. J.

J. J. Stammes, Waves in Focal Regions (Adam-Hilger, Bristol, UK, 1986), Chap. 8.

Takoto, N.

N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
[CrossRef]

Toba, H.

N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
[CrossRef]

Vidal, S.

C. Gómez-Reino, M. V. Pérez, C. Bao, M. T. Flores-Arias, S. Vidal, “Diffraction-free and diffraction-limited in graded-index planar waveguides with hyperbolic secant refractive index profile,” J. Mod. Opt. 47, 91–102 (2000).

Yasu, M.

N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
[CrossRef]

Appl. Opt. (2)

J. Lightwave Technol. (2)

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

N. Takoto, K. Jinguji, M. Yasu, H. Toba, M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. 6, 1003–1010 (1988).
[CrossRef]

J. Mod. Opt. (1)

C. Gómez-Reino, M. V. Pérez, C. Bao, M. T. Flores-Arias, S. Vidal, “Diffraction-free and diffraction-limited in graded-index planar waveguides with hyperbolic secant refractive index profile,” J. Mod. Opt. 47, 91–102 (2000).

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

C. Bao, C. Gómez-Reino, “Collimation and focusing properties of a planar waveguide with hyperbolic secant refractive index profile: light deflector and collimator design,” Opt. Quantum Electron. 27, 897–907 (1995).
[CrossRef]

Phys. Rev. A (1)

C. Paré, L. Gagnon, P. A. Bélanger, “Aspherical laser resonators: an analogy with quantum mechanics,” Phys. Rev. A 46, 4150–4160 (1992).
[CrossRef] [PubMed]

Other (8)

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), Chap. 9, pp. 1039–1057.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964), Chap. 4, pp. 282–285.

A. Papoulis, Systems and Transforms with Applications in Optics (Prentice-Hall, Englewood Cliffs, N.J., 1968), Chap. 7.

J. J. Stammes, Waves in Focal Regions (Adam-Hilger, Bristol, UK, 1986), Chap. 8.

T. Tamir, ed., Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1990), Chap. 2, pp. 52–54.

M. J. Adams, Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 4, pp 132–135.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12, pp. 263–267.

T. Kozek, “Design of optical planar waveguides with prescribed propagation constants,” in Optical Fibers and Their Applications IV, R. S. Romaniuk, M. Szustakowski, eds., Proc. SPIE670, 226–233 (1986); “Synthesis of optical planar waveguides,” in Integrated Optical Circuit Engineering IV, M. A. Mentzer, S. Sriram, eds., Proc. SPIE 704, 44–50 (1986).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Refractive-index profile of a HS GRIN planar medium.

Fig. 2
Fig. 2

Nonparaxial diffraction-limited propagation of the fundamental mode. (a) Level curves of irradiance on the xz plane along the interval [0, 4π/αγ] for a = 30 µm. (b) Enlarged central region of diffraction pattern at z = π/2αγ for a = 10, 30, 50 µm. In both cases S = 0.5 (α = 38.5 mm-1).

Fig. 3
Fig. 3

Irradiance distribution at z = π/2.1αγ versus x = sinh-1(u)/α for a = 50 µm. Calculations were made for S = 0.25 (α = 60 mm-1), S = 0.5 (α = 38.5 mm-1), and S = 0.75 (α = 29.3 mm-1).

Equations (47)

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n2x=n02-ns2sech2αx|x|a,ns2|x|>a,
ddx2+k2n2xψx=β2ψx,
ψνx=sechS-vαxfv-v,2S-v+1,S-v+1;½1-tanhαx,
βv2=k2ns2+α2S-v,
S=½1+4V2-1,
V=SS+1=kαn02-ns2.
ϕ0x, z=ψ0xexpiβ0z=sechS×αxexpiknsz1+α2S2k2ns21/2,v=0,ϕ1x, z=ψ1xexpiβ1z=sechSαxsinh×αxexpiknsz1+α2S-12k2ns21/2,
v=1.
Vc=v+1v+21/2.
Ψx, z=-aa Ψ0x0Kx, x0; zdx0,
Ψ0x0=sechSαx0,
ux=sinhαx.
Ψu, z=1α-u0au0aKu, u0; zΨ0u01+u02du0,
Ψ0u0=1+u02-S/2.
Ku, u0; z=ki2πγH1zn02-ns21+u021/21/2×expiVtan-1uuH˙1z-u0uH˙1z-u02+α2γ2H12z1+u21/2-tan-1u0u-u0H2zu-u0H2z2+α2γ2H12z1+u021/2,
H1z=sinαγzαγ=-H˙2zα2γ2,
H2z=cosαγz=H˙1z,
γ=l02-ns2/l0,
Ku, u0; z=ki2πγH1zn02-ns21+u021/21/2×expikφu, u0; z,
φu, u0; z=n02-ns2αtan-1×u2+u02-2uu0H2z+α2γ2H12zuu0+H2z1/2.
Ψu, z=Vi2παγH1z1/2-u0au0a Θ0u0×expikφu, u0; zdu0,
Θ0u0=1+u02-3/4+S/2.
φu0u0=ũ0=0,
ũ0=u/H2z.
-u0au0a Θ0u0expikφu, u0; zdu0=i2πkφũ0,ũ01/2Θ˜0×expikφ˜+1ikΘ0u0aexpikφu,u0a;zφu0a-Θ0-u0aexpikφu,-u0a;zφ-u0a,
Θ˜0=H2S+3/2zH22z+u23/4+S/2,
φ˜=n02-ns2αtan-1αγH1zH22z+u21/2,
φũ0,ũ0=H24zn02-ns2α2γH1zu+H22z3/2,
ψu, z=H2S-1/2zH22z+u2S/2expiVtan-1×αγH1zu2+H22z1/2+i2πkγH1zn02-ns21/2×1u2-u02aH22z1+u02aS/2-1/4×(u+u0aH2zu2+u02a-2uu0aH2z+α2γ2H12z1/2 ×expikφu, u0a; z-u-u0aH2z×u2+u02a+2uu0aH2z+α2γ2H12z1/2 expikφu,-u0a; z).
Iu, z=|Ψu, z|2=A2+B+2+B-2-2AB+ coskC--π/4+2AB- coskC+-π/4-2B+B- coskD,
A2=|H2z|2S-1u2+H22z2;
B±2=u2+u02a±2uu0aH2z+α2γ2H12z2πkγ|H1z|n02-ns2u±u0aH2z21+u02aS-1/2;
C±=φ˜-φu,±u0a; z=n02-ns2α×tan-1H2z±uu0aαγH1z-u2+H22zu2+u02a±2uu0aH2z+α2γ2H12z1/2H2z±uu0au2+H22z1/2-αγH1zu2+u02a±2uu0aH2z+α2γ2H12z1/2,
D=φu, u0a; z-φu,-u0a; z=n02-ns2α×tan-1H2z-uu0au2+u02a-2uu0aH2z+α2γ2H12z1/2H22z-u2u02a+u2+u02a+α2γ2H12z2-4u2u02zH22z1/2-H2z+uu0au2+u02a+2uu0aH2z+α2γ2H12z1/2H22z-u2u02a+u2+u02a+α2γ2H12z2-4u2u02zH22z1/2.
-u0au0a Θ0u0expikφu, u0; zdu0=iπ2kφu0a,u0a1/2Θ0u0aexpikφu, u0a; z,
ψu, z=H2S-1/2z2H22z+u2S/2×expiVtan-1αγH1zu2+H22z1/2
Iu, z=|H2z|2S-14u2+H22zS.
-u0au0a Θ0u0expikφu, u0; zdu0=×1ikΘ0u0aexpikφu, u0a; zφu0a-Θ0-u0aexpikφu,-u0a; zφ-u0a.
ψu, z=i2πkγH1zn02-ns21/2×1u2-u02aH22zS/2-1/4u+u0aH2zu2+u02a-2uu0aH2z+α2γ2H12z1/2 expikφu, u0a; z-u-u0aH2zu2+u02a+2uu0aH2z+α2γ2H12z1/2 expikφu,-u0a; z].
Iu, z=1πkn02-ns2α2γ|H1z|u2+u02aH22zu2-u02aH22z21+u02aS-3/2+1γ|H1z|1+u02aS-1/2× 1-{[u2+u02a+α2γ2H12z]2-4u2u02aH22z}1/2u2-u02aH22zcos kD.
Iu, zp=2α1+u2+u02aπku2n02-ns21+u02aS-1/2×sin2Vtan-1uu0a1+u2+u02a1/2,
ψ1u0=u01+u02-S/2,
Iu, z=A12+u02aB+2+B-2+2B+B- coskD+2u0aA1B+ coskC--π/4+B- coskC+-π4
A12=u2H22S-1zu2+H22zS
Iu, z=u2H22S-1z4u2+H22zS
Iu, z=u02aπkn02-nS2×α2γ|H1z|u2+u02aH22zu2-u02aH22z21+u02aS-3/2+1γ|H1z|1+u02aS-1/2×1+u2+u02a+α2γ2H12z2-4u2u02aH22z1/2u2-u02H22zcoskD
Iu, z=2u02aα1+u2+u02aπku2n02-nS21+u02aS-1/2×cos2Vtan-1uu0a1+u2+u02a1/2.

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