Abstract

We report the formulation of an ABCD matrix for reflection and refraction of Gaussian light beams at the surface of a parabola of revolution that separate media of different refractive indices based on optical phase matching. The equations for the spot sizes and wave-front radii of the beams are also obtained by using the ABCD matrix. With these matrices, we can more conveniently design and evaluate some special optical systems, including these kinds of elements.

© 2005 Optical Society of America

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References

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  1. 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]
  2. S. Gangopadhyay, S. N. Sarkar, “Laser diode to single-mode fibre excitation via hyperbolic lens on the fibre tip: formulation of ABCD matrix and efficiency computation,” Opt. Commun. 132, 55–60 (1996).
    [CrossRef]
  3. S. Gangopadhyay, S. Sarkar, “ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for laser diode to single-mode fiber coupling by way of a hyperbolic lens on the fiber tip,” Appl. Opt. 36, 8582–8586 (1997).
    [CrossRef]
  4. F. A. Rahman, K. Takahashi, C. H. Teik, “Theoretical analysis of coupling between laser diodes and conically lensed single-mode fibers utilising ABCD matrix method,” Opt. Commun. 215, 61–67 (2003).
    [CrossRef]
  5. A. E. Siegman, “ABCD-matrix elements for a curved diffraction grating,” J. Opt. Soc. Am. A 2, 1793 (1985).
    [CrossRef]
  6. X. Zeng, Y. An, “Coupling light from a laser diode into a multimode fiber,” Appl. Opt. 42, 4427–4430 (2003).
    [CrossRef] [PubMed]
  7. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]

2003 (2)

F. A. Rahman, K. Takahashi, C. H. Teik, “Theoretical analysis of coupling between laser diodes and conically lensed single-mode fibers utilising ABCD matrix method,” Opt. Commun. 215, 61–67 (2003).
[CrossRef]

X. Zeng, Y. An, “Coupling light from a laser diode into a multimode fiber,” Appl. Opt. 42, 4427–4430 (2003).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

S. Gangopadhyay, S. N. Sarkar, “Laser diode to single-mode fibre excitation via hyperbolic lens on the fibre tip: formulation of ABCD matrix and efficiency computation,” Opt. Commun. 132, 55–60 (1996).
[CrossRef]

1985 (1)

1972 (1)

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

1969 (1)

An, Y.

Deschamps, G. A.

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

Gangopadhyay, S.

Massey, G. A.

Rahman, F. A.

F. A. Rahman, K. Takahashi, C. H. Teik, “Theoretical analysis of coupling between laser diodes and conically lensed single-mode fibers utilising ABCD matrix method,” Opt. Commun. 215, 61–67 (2003).
[CrossRef]

Sarkar, S.

Sarkar, S. N.

S. Gangopadhyay, S. N. Sarkar, “Laser diode to single-mode fibre excitation via hyperbolic lens on the fibre tip: formulation of ABCD matrix and efficiency computation,” Opt. Commun. 132, 55–60 (1996).
[CrossRef]

Siegman, A. E.

Takahashi, K.

F. A. Rahman, K. Takahashi, C. H. Teik, “Theoretical analysis of coupling between laser diodes and conically lensed single-mode fibers utilising ABCD matrix method,” Opt. Commun. 215, 61–67 (2003).
[CrossRef]

Teik, C. H.

F. A. Rahman, K. Takahashi, C. H. Teik, “Theoretical analysis of coupling between laser diodes and conically lensed single-mode fibers utilising ABCD matrix method,” Opt. Commun. 215, 61–67 (2003).
[CrossRef]

Zeng, X.

Appl. Opt. (3)

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

Opt. Commun. (2)

F. A. Rahman, K. Takahashi, C. H. Teik, “Theoretical analysis of coupling between laser diodes and conically lensed single-mode fibers utilising ABCD matrix method,” Opt. Commun. 215, 61–67 (2003).
[CrossRef]

S. Gangopadhyay, S. N. Sarkar, “Laser diode to single-mode fibre excitation via hyperbolic lens on the fibre tip: formulation of ABCD matrix and efficiency computation,” Opt. Commun. 132, 55–60 (1996).
[CrossRef]

Proc. IEEE (1)

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

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Figures (2)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Illustration for the deduction of Appendix A.

Equations (30)

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x 2 + y 2 = 2 p z ,
x x 1 + p tan θ 1 ,             y = y 1 ,             z z 1 + 0.5 p tan 2 θ 1 ,
( x 1 + p tan θ 1 ) 2 + y 1 2 = 2 p ( z 1 + 0.5 p tan 2 θ 1 ) .
z 1 = x 1 tan θ 1 + ( x 1 2 + y 1 2 ) / 2 p .
U i ( x i , y i , z i ) = A i exp [ - j Φ ( 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 ω 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 p ) + k 1 y 1 2 2 ( 1 q S 1 + 1 p ) .
Φ 1 ( x 1 , y 1 ) = Φ 2 ( x 1 , y 1 ) = Φ 3 ( x 1 , y 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 2 , y 2 , z 2 ) = k 2 z 2 + k 2 2 ( x 2 2 q T 2 + y 2 2 q S 2 ) .
Φ 2 ( x 1 , y 1 ) = k 1 ( x 1 sin 2 θ 1 - z 1 cos 2 θ 1 ) + k 1 2 [ ( - x 1 cos 2 θ 1 - z 1 sin 2 θ 1 ) 2 q T 2 + y 1 2 q S 2 ] = k 1 tan θ 1 - k 1 cos 2 θ 1 x 1 2 2 p + k 1 2 [ x 1 + sin 2 θ 1 ( x 1 2 + y 1 2 ) / 2 p ] 2 q T 2 + k 1 y 1 2 2 ( 1 q S 2 - cos 2 θ 1 p ) .
Φ 2 ( x 1 , y 1 ) = k 1 tan θ 1 + k 1 x 1 2 2 ( 1 q T 2 - cos 2 θ 1 p ) + k 1 y 1 2 2 ( 1 q S 2 - cos 2 θ 1 p ) .
1 q T 2 , S 2 = 1 q T 1 , S 1 + 2 cos 2 θ 1 p .
1 q T 1 = 1 R T 1 - j λ 0 π n 1 ω T 1 2 , 1 q T 2 = 1 R T 2 - j λ 0 π n 1 ω T 2 2 ,
1 q T 2 = 1 q T 1 + 2 cos 2 θ 1 p = 1 R T 1 + 2 cos 2 θ 1 p - j λ 0 π n 1 ω T 1 2 .
1 R T 2 = 2 cos 2 θ 1 p + 1 R T 1 , ω T 1 = ω T 2 .
1 R S 2 = 2 cos 2 θ 1 p + 1 R S 1 , ω S 1 = ω S 2 .
[ ω T 2 , S 2 ω T 2 , S 2 / R T 2 , S 2 ] = [ 1 0 2 cos 2 θ 1 p 1 ] [ ω T 1 , S 1 ω T 1 , S 1 / R T 1 , S 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 [ z 1 cos ( θ 1 - θ 2 ) - x 1 sin ( θ 1 - θ 2 ) ] + k 3 y 1 2 2 q S 3 + k 3 2 q T 3 [ x 1 cos ( θ 1 - θ 2 ) + z 1 sin ( θ 1 - θ 2 ) ] 2 = k 3 x 1 [ tan θ 1 cos ( θ 1 - θ 2 ) - sin ( θ 1 - θ 2 ) ] + k 3 y 1 2 2 [ 1 q S 3 + cos ( θ 1 - θ 2 ) p ] + k 3 cos ( θ 1 - θ 2 ) 2 p x 1 2 + k 3 2 q T 3 × [ x 1 cos θ 2 cos θ 1 + ( x 1 2 + y 1 2 ) sin ( θ 1 - θ 2 ) 2 p ] 2 .
Φ 3 ( x 1 , y 1 ) = k 3 x 1 sin θ 2 cos θ 2 + k 3 x 1 2 2 × [ cos 2 θ 2 cos 2 θ 1 1 q T 3 + cos ( θ 1 - θ 2 ) p ] + k 3 y 1 2 2 [ 1 q S 3 + cos ( θ 1 - θ 2 ) p ] .
1 q T 1 = cos 2 θ 2 cos 2 θ 1 n q T 3 + n cos ( θ 1 - θ 2 ) - 1 p , 1 q S 1 = n q S 3 + n cos ( θ 1 - θ 2 ) - 1 p ,
1 q T 3 = ( n cos 2 θ 1 n 2 - sin 2 θ 1 ) 1 q T 1 + n cos 3 θ 1 ( cos θ 1 - n 2 - sin 2 θ 1 ) p ( n 2 - sin 2 θ 1 ) , 1 q S 3 = 1 n 1 q S 1 + cos θ 1 ( cos θ 1 - n 2 - sin 2 θ 1 ) n p .
ω S 3 = ω S 1 , 1 R S 3 = 1 n 1 R S 1 + cos θ 1 ( cos θ 1 - n 2 - sin 2 θ 1 ) n p , ω T 3 = n 2 - sin 2 θ 1 n cos θ 1 ω T 1 , 1 R T 3 = ( n cos 2 θ 1 n 2 - sin 2 θ 1 ) 1 R T 1 + n cos 3 θ 1 ( cos θ 1 - n 2 - sin 2 θ 1 ) p ( n 2 - sin 2 θ 1 ) .
[ ω S 3 ω S 3 / R S 3 ] = [ 1 0 cos θ 1 ( cos θ 1 - n 2 - sin 2 θ 1 ) n p 1 / n ] [ ω S 1 ω S 1 / R S 1 ] , [ ω T 3 ω T 3 / R T 3 ] = [ n 2 - sin 2 θ 1 n cos θ 1 0 cos 2 θ 1 ( cos θ 1 - n 2 - sin 2 θ 1 ) p n 2 - sin 2 θ 1 cos θ 1 n 2 - sin 2 θ 1 ] × [ ω T 1 ω T 1 / R T 1 ] .
x = ( p / x Q ) ( z + z Q ) ,
tan [ α + ( 90 ° - θ 1 ) ] = p / x Q , x Q 2 = 2 p z Q .
x Q = p tan ( θ 1 - α ) , z Q = 0.5 p tan 2 ( θ 1 - α ) .
x Q p tan θ 1 , z Q 0.5 p tan 2 θ 1 .

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