Abstract

We report the formulation of an ABCD matrix for reflection and refraction of Gaussian light beams at the surfaces of the hyperboloid of revolution that separate media of different refractive indices. The analysis includes an arbitrary angle of incidence and is based on matching the optical phase at the interface. Finally, we deduce expressions for spot sizes and wave-front radii and use them to obtain the ABCD matrix. Based on the formulated ABCD matrix for refraction under paraxial approximation, we also report a simple theoretical investigation of the coupling efficiency of a laser diode to a single-mode fiber with a hyperbolic lens formed on its tip.

© 1997 Optical Society of America

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References

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  1. H. M. Presby, C. A. Edwards, “Near 100% efficient fibre microlenses,” Electron. Lett. 28, 582–584 (1992).
    [CrossRef]
  2. C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser to fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
    [CrossRef]
  3. J. John, T. S. M. Maclean, H. Ghafouri-Shiraz, J. Niolett, “Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength,” IEE Proc. Optoelectron. 141, 178–184 (1994).
    [CrossRef]
  4. G. A. Massey, A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt. 8, 975–978 (1969).
    [CrossRef] [PubMed]
  5. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]
  6. S Solimeno, B Crosignani, P Diporto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, New York, 1986), Chap. 2, pp. 81–89.
  7. S. Sarkar, K. Thyagarajan, A. Kumar, “Gaussian approximation of the fundamental mode in single-mode elliptical core fibers,” Opt. Commun. 49, 178–183 (1984).
    [CrossRef]
  8. D. Marcuse, “Loss analysis of single-mode fibre splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  9. S. N. Sarkar, B. P. Pal, K. Thyagarajan, “Lens coupling of laser diodes to monomode elliptic core fibers,” J. Opt. Commun. 7, 92–96 (1986).
    [CrossRef]
  10. H. M. Presby, C. A. Edwards, “Efficient coupling of polarisation maintaining fibre to laser diodes,” IEEE Photon. Technol. Lett. 4, 897–899 (1992).
    [CrossRef]

1994 (1)

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz, J. Niolett, “Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength,” IEE Proc. Optoelectron. 141, 178–184 (1994).
[CrossRef]

1993 (1)

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser to fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

1992 (2)

H. M. Presby, C. A. Edwards, “Near 100% efficient fibre microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

H. M. Presby, C. A. Edwards, “Efficient coupling of polarisation maintaining fibre to laser diodes,” IEEE Photon. Technol. Lett. 4, 897–899 (1992).
[CrossRef]

1986 (1)

S. N. Sarkar, B. P. Pal, K. Thyagarajan, “Lens coupling of laser diodes to monomode elliptic core fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

1984 (1)

S. Sarkar, K. Thyagarajan, A. Kumar, “Gaussian approximation of the fundamental mode in single-mode elliptical core fibers,” Opt. Commun. 49, 178–183 (1984).
[CrossRef]

1977 (1)

D. Marcuse, “Loss analysis of single-mode fibre splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1969 (1)

Crosignani, B

S Solimeno, B Crosignani, P Diporto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, New York, 1986), Chap. 2, pp. 81–89.

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Diporto, P

S Solimeno, B Crosignani, P Diporto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, New York, 1986), Chap. 2, pp. 81–89.

Dragone, C.

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser to fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

Edwards, C. A.

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser to fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

H. M. Presby, C. A. Edwards, “Efficient coupling of polarisation maintaining fibre to laser diodes,” IEEE Photon. Technol. Lett. 4, 897–899 (1992).
[CrossRef]

H. M. Presby, C. A. Edwards, “Near 100% efficient fibre microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

Ghafouri-Shiraz, H.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz, J. Niolett, “Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength,” IEE Proc. Optoelectron. 141, 178–184 (1994).
[CrossRef]

John, J.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz, J. Niolett, “Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength,” IEE Proc. Optoelectron. 141, 178–184 (1994).
[CrossRef]

Kumar, A.

S. Sarkar, K. Thyagarajan, A. Kumar, “Gaussian approximation of the fundamental mode in single-mode elliptical core fibers,” Opt. Commun. 49, 178–183 (1984).
[CrossRef]

Maclean, T. S. M.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz, J. Niolett, “Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength,” IEE Proc. Optoelectron. 141, 178–184 (1994).
[CrossRef]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fibre splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Massey, G. A.

Niolett, J.

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz, J. Niolett, “Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength,” IEE Proc. Optoelectron. 141, 178–184 (1994).
[CrossRef]

Pal, B. P.

S. N. Sarkar, B. P. Pal, K. Thyagarajan, “Lens coupling of laser diodes to monomode elliptic core fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

Presby, H. M.

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser to fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

H. M. Presby, C. A. Edwards, “Near 100% efficient fibre microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

H. M. Presby, C. A. Edwards, “Efficient coupling of polarisation maintaining fibre to laser diodes,” IEEE Photon. Technol. Lett. 4, 897–899 (1992).
[CrossRef]

Sarkar, S.

S. Sarkar, K. Thyagarajan, A. Kumar, “Gaussian approximation of the fundamental mode in single-mode elliptical core fibers,” Opt. Commun. 49, 178–183 (1984).
[CrossRef]

Sarkar, S. N.

S. N. Sarkar, B. P. Pal, K. Thyagarajan, “Lens coupling of laser diodes to monomode elliptic core fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

Siegman, A. E.

Solimeno, S

S Solimeno, B Crosignani, P Diporto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, New York, 1986), Chap. 2, pp. 81–89.

Thyagarajan, K.

S. N. Sarkar, B. P. Pal, K. Thyagarajan, “Lens coupling of laser diodes to monomode elliptic core fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

S. Sarkar, K. Thyagarajan, A. Kumar, “Gaussian approximation of the fundamental mode in single-mode elliptical core fibers,” Opt. Commun. 49, 178–183 (1984).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of single-mode fibre splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Electron. Lett. (1)

H. M. Presby, C. A. Edwards, “Near 100% efficient fibre microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

IEE Proc. Optoelectron. (1)

J. John, T. S. M. Maclean, H. Ghafouri-Shiraz, J. Niolett, “Matching of single-mode fibre to laser diode by microlenses at 1.5 μm wavelength,” IEE Proc. Optoelectron. 141, 178–184 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. M. Presby, C. A. Edwards, “Efficient coupling of polarisation maintaining fibre to laser diodes,” IEEE Photon. Technol. Lett. 4, 897–899 (1992).
[CrossRef]

J. Lightwave Technol. (1)

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser to fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

J. Opt. Commun. (1)

S. N. Sarkar, B. P. Pal, K. Thyagarajan, “Lens coupling of laser diodes to monomode elliptic core fibers,” J. Opt. Commun. 7, 92–96 (1986).
[CrossRef]

Opt. Commun. (1)

S. Sarkar, K. Thyagarajan, A. Kumar, “Gaussian approximation of the fundamental mode in single-mode elliptical core fibers,” Opt. Commun. 49, 178–183 (1984).
[CrossRef]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Other (1)

S Solimeno, B Crosignani, P Diporto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, New York, 1986), Chap. 2, pp. 81–89.

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Figures (3)

Fig. 1
Fig. 1

Coordinate systems for a hyperboloid interface and for incident, reflected, and refracted beams.

Fig. 2
Fig. 2

Geometry of an optical beam emitted from input plane 1 of a laser diode (L.D.) and refracted through a hyperbolic lens onto plane 2, which is the end face of a single-mode fiber (S.M.F.).

Fig. 3
Fig. 3

Variation of maximum coupling efficiency with the effective focal length of a hyperbolic lens on the tip of a fiber. Comparison of our theoretical results with the results in Refs. 1, 3, and 10: ×, published experimental point1,10; solid curve, our theory based on an ABCD matrix; dotted curve, theory3 derived with the planar wave model; dashed curve, theory3 derived with the spherical wave model.

Equations (26)

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Z 2 a 2 - X 2 + Y 2 b 2 = 1 ,
X = x 1 + A tan θ 1 ,             Y = y 1 ,             Z = z 1 + a ,
( z 1 + a ) 2 a 2 - ( x 1 + A tan θ 1 ) 2 + y 1 2 b 2 = 1.
z 1 = x 1 tan θ 1 + ( x 1 2 + y 1 2 ) / 2 A .
U i ( x i , y i , z i ) = A i exp [ - j ϕ i ( x i , y i , z i ) ] ,             i = 1 , 2 , 3 ,
ϕ i ( x i , y i , z i ) = k i z i + k i 2 ( x i 2 q T i + y i 2 q S i ) .
k 1 = 2 π n 1 / λ 0 = k 2 ,             k 3 = 2 π n 2 / λ 0 , 1 / q i = 1 / R i - j λ 0 / π n i w i 2 .
ϕ 1 ( x 1 , y 1 ) = k 1 ( x 1 tan θ 1 ) + k 1 x 1 2 2 ( 1 q T 1 + 1 A ) + k 1 y 1 2 2 ( 1 q S 1 + 1 A ) .
ϕ 1 ( x 1 , y 1 ) = ϕ 2 ( x 1 , y 1 ) = ϕ 3 ( x 1 , y 1 ) ,
x 3 = x 1 cos ( θ 1 - θ 2 ) + z 1 sin ( θ 1 - θ 2 ) , y 3 = y 1 , z 3 = z 1 cos ( θ 1 - θ 2 ) - x 1 sin ( θ 1 - θ 2 ) .
ϕ 3 ( x 1 , y 1 ) = k 3 { x 1 [ tan θ 1 cos ( θ 1 - θ 2 ) - sin ( θ 1 - θ 2 ) ] } + k 3 x 1 2 2 [ 1 q T 3 + cos ( θ 1 - θ 2 ) A + sin 2 ( θ 1 - θ 2 ) tan θ 1 q T 3 ] + k 3 y 1 2 2 [ 1 q S 3 + cos ( θ 1 - θ 2 ) A ] .
1 q T 3 , S 3 = 1 n q T 1 , S 1 + 1 - n cos ( θ 1 - θ 2 ) n A ,
1 q T 3 , S 3 = 1 n q T 1 , S 1 + 1 - n cos θ 1 n A .
( w T 3 , S 3 w T 3 , S 3 R T 3 , S 3 ) = ( 1 0 1 - n cos θ 1 n b 2 / a 1 n ) ( w T 1 , S 1 w T 1 , S 1 R T 1 , S 1 ) .
x 2 = - x 1 cos 2 θ 1 - z 1 sin 2 θ 1 , y 2 = y 1 , z 2 = x 1 sin 2 θ 1 - z 1 cos 2 θ 1 .
ϕ 2 ( x 1 , y 1 ) = k 1 x 1 tan θ 1 + k 1 x 1 2 2 ( 1 q T 2 - cos 2 θ 1 A ) + k 1 y 1 2 2 ( 1 q S 2 - cos 2 θ 1 A ) .
1 q T 2 , S 2 = 1 q T 1 , S 1 + 2 cos 2 θ 1 A .
( w T 2 , S 2 w T 2 , S 2 R T 2 , S 2 ) = ( 1 0 2 cos 2 θ 1 b 2 / a 1 ) ( w T 1 , S 1 w T 1 , S 1 R T 1 , S 1 ) .
ψ u = exp [ - ( x 2 / w 1 x 2 + y 2 / w 1 y 2 ) ] × exp [ - ( j k 1 ) ( x 2 + y 2 ) / 2 R 1 ] .
ψ f = exp [ - ( x 2 + y 2 ) / w f 2 ] ,
w f = ρ ( 0.65 + 1.619 / v 1.5 + 2.879 / v 6 ) ,
ψ v = exp [ - ( x 2 / w 3 x 2 + y 2 / w 3 y 2 ) ] × exp [ ( - j k 3 ) ( x 2 / R 3 x + y 2 / R 3 y ) / 2 ] ,
M = ( A B C D ) = ( 1 d 0 1 ) ( 1 0 ( 1 - n ) / ( n b 2 / a ) 1 / n ) ( 1 f 0 1 ) .
w 3 x , 3 y 2 = A 1 2 w 1 x , 1 y 2 + ( λ 1 2 B 2 ) / w 1 x , 1 y 2 n ( A 1 D - B C 1 ) , 1 R 3 x , 3 Y = A 1 C 1 w 1 x , 1 y 2 + ( λ 1 2 BD ) / w 1 x , 1 y 2 A 1 2 w 1 x , 1 y 2 + ( λ 1 2 B 2 ) / w 1 x , 1 y 2 ,
η = | ψ v ψ f * d x d y | 2 ψ v 2 d x d y ψ f 2 d x d y .
η = 4 w 3 x w 3 y ( w f ) 2 [ ( w f 2 + w 3 x 2 ) 2 + ( k 3 2 w f 2 w 3 x 4 ) / ( 4 R 3 x 2 ) ] 0.5 [ ( w f 2 + w 3 y 2 ) 2 + ( k 3 2 w f 4 w 3 y 4 ) / ( 4 R 3 y 2 ) ] 0.5 .

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