Abstract

An analysis of light transmission through a tapered hyperbolic secant waveguide is presented. Focusing and collimation conditions by this waveguide are obtained. The design of a beam-size controller device is proposed and compared with a similar device designed with nontapered hyperbolic secant waveguides.

© 1997 Optical Society of America

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References

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  1. A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London, Ser. A 233, 216–225 (1954).
    [CrossRef]
  2. 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] [PubMed]
  3. M. Murty, “Gradient-index optics: some comments,” Appl. Opt. 20, 2180–2181 (1981).
    [CrossRef] [PubMed]
  4. C. Bao, C. Gómez-Reino, “Collimation and focusing properties of a planar waveguide with hyperbolic secant refractive index profile,” Opt. Quantum Electron. 27, 897–907 (1995).
    [CrossRef]
  5. D. W. Hewak, J. W. Y. Lit, “Solution deposited optical waveguide lens,” Appl. Opt. 28, 4190–4198 (1989).
    [CrossRef] [PubMed]
  6. D. Bertilone, A. Ankiewicz, C. Pask, “Wave propagation in a graded-index taper,” Appl. Opt. 26, 2213–2221 (1987).
    [CrossRef] [PubMed]
  7. J. R. Flores, C. Gómez-Reino, E. Acosta, J. Liñares, “Geometrical optics of gradient index lenses,” Opt. Eng. 28, 1173–1179 (1989).
    [CrossRef]
  8. J. N. McMullin, “The ABCD matrix in arbitrarily tapered quadratic-index waveguides,” Appl. Opt. 25, 2184–2187 (1986).
    [CrossRef] [PubMed]
  9. C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Image and transform transmission through divergent conical GRIN rods,” Opt. Commun. 55, 5–7 (1985).
    [CrossRef]
  10. C. Gómez-Reino, J. Sochaki, “Imaging and transforming capabilities of GRIN rods with noncylindrical surfaces of constant index: a family of exact solutions,” Appl. Opt. 24, 4375–4378 (1985).
    [CrossRef]
  11. C. Gómez-Reino, E. Larrea, “Paraxial imaging and transforming in a medium with gradient-index: transmittance function,” Appl. Opt. 21, 4271–4275 (1982).
    [CrossRef] [PubMed]
  12. A. A. Tovar, L. W. Casperson, “Beam propagation in parabolically tapered graded-index waveguides,” Appl. Opt. 33, 7733–7739 (1994).
    [CrossRef] [PubMed]
  13. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 1.
  14. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 9.3.
  15. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), Chap. 8.
  16. C. Gómez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).
  17. K. Iizuka, Engineering Optics, 2nd ed. (Springer-Verlag, Berlin, 1987).
  18. 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]

1996 (1)

1995 (1)

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

1994 (2)

1992 (1)

C. Gómez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

1989 (2)

D. W. Hewak, J. W. Y. Lit, “Solution deposited optical waveguide lens,” Appl. Opt. 28, 4190–4198 (1989).
[CrossRef] [PubMed]

J. R. Flores, C. Gómez-Reino, E. Acosta, J. Liñares, “Geometrical optics of gradient index lenses,” Opt. Eng. 28, 1173–1179 (1989).
[CrossRef]

1987 (1)

1986 (1)

1985 (2)

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Image and transform transmission through divergent conical GRIN rods,” Opt. Commun. 55, 5–7 (1985).
[CrossRef]

C. Gómez-Reino, J. Sochaki, “Imaging and transforming capabilities of GRIN rods with noncylindrical surfaces of constant index: a family of exact solutions,” Appl. Opt. 24, 4375–4378 (1985).
[CrossRef]

1982 (1)

1981 (1)

1954 (1)

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London, Ser. A 233, 216–225 (1954).
[CrossRef]

Acosta, E.

J. R. Flores, C. Gómez-Reino, E. Acosta, J. Liñares, “Geometrical optics of gradient index lenses,” Opt. Eng. 28, 1173–1179 (1989).
[CrossRef]

Ankiewicz, A.

Bao, C.

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,” Opt. Quantum Electron. 27, 897–907 (1995).
[CrossRef]

Bertilone, D.

Casperson, L. W.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 9.3.

Fletcher, A.

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London, Ser. A 233, 216–225 (1954).
[CrossRef]

Flores, J. R.

J. R. Flores, C. Gómez-Reino, E. Acosta, J. Liñares, “Geometrical optics of gradient index lenses,” Opt. Eng. 28, 1173–1179 (1989).
[CrossRef]

Gómez-Reino, C.

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,” 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] [PubMed]

C. Gómez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

J. R. Flores, C. Gómez-Reino, E. Acosta, J. Liñares, “Geometrical optics of gradient index lenses,” Opt. Eng. 28, 1173–1179 (1989).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Image and transform transmission through divergent conical GRIN rods,” Opt. Commun. 55, 5–7 (1985).
[CrossRef]

C. Gómez-Reino, J. Sochaki, “Imaging and transforming capabilities of GRIN rods with noncylindrical surfaces of constant index: a family of exact solutions,” Appl. Opt. 24, 4375–4378 (1985).
[CrossRef]

C. Gómez-Reino, E. Larrea, “Paraxial imaging and transforming in a medium with gradient-index: transmittance function,” Appl. Opt. 21, 4271–4275 (1982).
[CrossRef] [PubMed]

Hewak, D. W.

Iizuka, K.

K. Iizuka, Engineering Optics, 2nd ed. (Springer-Verlag, Berlin, 1987).

Larrea, E.

Liñares, J.

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] [PubMed]

J. R. Flores, C. Gómez-Reino, E. Acosta, J. Liñares, “Geometrical optics of gradient index lenses,” Opt. Eng. 28, 1173–1179 (1989).
[CrossRef]

Lit, J. W. Y.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 1.

McMullin, J. N.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 9.3.

Murphy, T.

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London, Ser. A 233, 216–225 (1954).
[CrossRef]

Murty, M.

Pask, C.

Pérez, M. V.

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. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Image and transform transmission through divergent conical GRIN rods,” Opt. Commun. 55, 5–7 (1985).
[CrossRef]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), Chap. 8.

Sochacka, M.

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Image and transform transmission through divergent conical GRIN rods,” Opt. Commun. 55, 5–7 (1985).
[CrossRef]

Sochacki, J.

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Image and transform transmission through divergent conical GRIN rods,” Opt. Commun. 55, 5–7 (1985).
[CrossRef]

Sochaki, J.

Tovar, A. A.

Young, A.

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London, Ser. A 233, 216–225 (1954).
[CrossRef]

Appl. Opt. (8)

Int. J. Optoelectron. (1)

C. Gómez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

Opt. Commun. (1)

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Image and transform transmission through divergent conical GRIN rods,” Opt. Commun. 55, 5–7 (1985).
[CrossRef]

Opt. Eng. (1)

J. R. Flores, C. Gómez-Reino, E. Acosta, J. Liñares, “Geometrical optics of gradient index lenses,” Opt. Eng. 28, 1173–1179 (1989).
[CrossRef]

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,” Opt. Quantum Electron. 27, 897–907 (1995).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London, Ser. A 233, 216–225 (1954).
[CrossRef]

Other (4)

K. Iizuka, Engineering Optics, 2nd ed. (Springer-Verlag, Berlin, 1987).

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 1.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 9.3.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), Chap. 8.

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Figures (6)

Fig. 1
Fig. 1

Equi-index curves of a planar waveguide with a HS transverse index profile modulated by a divergent conical axial index.

Fig. 2
Fig. 2

Ray propagation for an on-axis point source at the input face of a HS waveguide with a divergent conical taper function. Calculations have been made for n0=1.5, α0=0.01 μm-1, and L=5 mm.

Fig. 3
Fig. 3

Equi-index curves of a planar waveguide with a HS transverse index profile modulated by a convergent parabolical axial index.

Fig. 4
Fig. 4

Ray propagation for an on-axis point source at the input face of a HS waveguide with a convergent parabolical taper function. Calculations have been made for n0=1.5, α0 =0.01 μm-1, and L=1 mm.

Fig. 5
Fig. 5

Beam-size-controller device by a HS tapered waveguide. Note that the output beam size depends on the waveguide length.

Fig. 6
Fig. 6

Relationship between the input and the output collimated beam sizes versus mth order of collimation for divergent conical and convergent parabolical taper functions. Calculations have been made in both cases for n0=1.5, α0 =0.01 μm-1, and L=10 mm.

Equations (50)

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n(x, z)=n0 sech[α(z)x],
n(x, z)x¨=(1+x˙2)n(x, z)x-x˙ n(x, z)z,
x¨α2=(1+x˙2)xx˙ α˙α2-αα2tanh[α(z)x].
|α˙|α21,
x¨=-(1+x˙2)α(z)tanh[α(z)x].
u(z, x)=sinh[α(z)x].
u˙(z, x)α(z)x˙ cosh[α(z)x].
uu˙=H2(z)H1(z)H˙2(z)H˙1(z) u0u˙0.
H1(z)=[α0α(z)]-1/2 sin0z α(z)dz,
H2(z)=[α0/α(z)]1/2 cos0z α(z)dz;
H˙1(z)=[α(z)/α0]1/2 cos0z α(z)dz=α(z)α0 H2(z),
H˙2(z)=-[α0α(z)]1/2 sin0z α(z)dz=-α0α(z)H1(z),
x(z)=1α(z) sinh-1[u(z)],
x˙(z)=u˙(z)α(z)cosh[α(z)x(z)].
H1(zm)=H˙2(zm)=0,
0zm α(z)dz=mπ(mnatural).
H2(zm)=(-1)mα0α(zm)1/2.
x(zm)=(-1)mα(zm) sinh-1α0α(zm)1/2sinh(α0x0),
x˙(zm)=(-1)mα0α(zm)1/2cosh(α0x0)cosh[α(zm)x(zm)] x˙0,
Mt=dx(zm)dx0=(-1)m α03/2 cosh(α0x0)α(zm)[α(zm)+α0 sinh2(α0x0)]1/2,
Ma=dx˙(zm)dx˙0=(-1)mα0α(zm)1/2 cosh(α0x0)cosh[α(zm)x(zm)].
x(zm)=0,
x˙(zm)=Max˙0,
Ma=(-1)mα0α(zm)1/2
H2(zp)=H˙1(zp)=0,
0zp α(z)dz=(2p+1)(π/2)(pnatural).
H1(zp)=(-1)p[α0α(zp)]-1/2
x(zp)=(-1)pα(zp) sinh-1{u˙0[α0α(zp)]-1/2},
x˙(zp)=-(-1)p+1[α0α(zp)]1/2u0α(zp)cosh(sinh-1{[α0α(zp)]-1/2u˙0}).
α(z)=α01+z/L,
H1(z)=1α0 1+zL1/2 sin[α0L ln(1+z/L)],
H2(z)=1+zL1/2 cos[α0L ln(1+z/L)],
zm=Lexpmπα0L-1.
x(z)=(L+z)α0L sinh-1[H2(z)u0+H1(z)u˙0],
x˙(z)=(L+z)[H˙2(z)u0+H˙1(z)u˙0]α0L cosh{sinh-1[H2(z)u0+H1(z)u˙0]}.
zp=Lexp(2p+1)π2α0L-1.
α(z)=α01-(z/L)2.
H1(z)=(L2-z2)1/2α0L sin[α0L tanh-1(z/L)],
H2(z)=1-z2L21/2 cos[α0L tanh-1(z/L)],
zm=L tanhmπα0L.
x(z)=(L2-z2)α0L2 sinh-1[H2(z)u0+H1(z)u˙0],
x˙(z)=(L2-z2)[H˙2(z)u0+H˙1(z)u˙0]α0L2 cosh{sinh-1[H2(z)u0+H1(z)u˙0]}.
zp=L tanh(2p+1)π2α0L.
u0i=sinh(α0x0i),
u˙0i=0.
x(z)=1α(z) sinh-1[u0iH2(z)],
x˙(z)=u0iH˙2(z)α(z)cosh{sinh-1[u0iH2(z)]}.
x(zm)=(-1)mα(zm) sinh-1u0iα0α(zm)1/2,
x˙(zm)=0.
AoAi=α0α(zm) sinh-1u0Mα0α(zm)1/2sinh-1(u0M),

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