Abstract

Using the properties of Hermite polynomials, a simple first-order matrix differential equation is developed which describes the propagation of an arbitrary field through an inhomogeneous medium and which can be solved exactly. This method handles both spatially varying refractive index and linear absorption and diffraction. As examples, it is applied to an etalon and a graded index optical fiber.

© 1990 Optical Society of America

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References

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  1. H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  2. M. D. Feit, J. A. Fleck, “Light Propagation in Graded-Index Optical Fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  3. A. Sharma, S. Banerjee, “Method for Propagation of Total Fields for Beams through Optical Waveguides,” Opt. Lett. 14, 96–98 (1989).
    [CrossRef] [PubMed]
  4. W. J. Firth, E. M. Wright, “Theory of Gaussian Beam Optical Bistability,” Opt. Commun. 40, 233–238 (1982).
    [CrossRef]
  5. A. E. Siegman, Lasers (Oxford U.P., London, 1986).
  6. I. W. Busbridge, “Some Integrals Involving Hermite Polynomials,” J. London Math. Soc. 23, 133–141 (1948).
    [CrossRef]
  7. J. H. Marburger, F. S. Felber, “Theory of a Lossless Nonlinear Fabry-Perot Interferometer,” Phys. Rev. A, 177, 335–342 (1978).
    [CrossRef]
  8. R. P. Riesz, R. Simon, “Reflection of a Gaussian Beam from a Dielectric Slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985).
    [CrossRef]
  9. The far-field irradiance is derived from the Fourier transform of the electric field at the surface. Since the F.T. of a GH polynomial is also a GH polynomial of the same order it follows that the F.T. of E(y)=∑i=0∞fiψi(y) exp(iβy) isEFT(k)=πr0∑i=0∞Cifi exp[-r02(β-k)2/4]Hi(r0(β-k)2)ii,where k = [(2π)/λ] sin θ.
  10. W. Nasalski, T. Tamir, L. Lin, “Displacement of the Intensity Peak in Narrow Beams Reflected at a Dielectric Interface,” J. Opt. Soc. Am. A 5, 132–140 (1988).
    [CrossRef]
  11. There seems to be an inconsistency in the work of Riesz and Simon,6 since there are two graphs of reflection amplitude vs sin θ for the case nd = 2.7 with each having the peak of the spatial frequency distribution on different sides of the etalon reflectivity minima [Figs.2(b) and 3(b)]. This by Eq. (14) gives different signs to the angular shift, either towards or away from the normal. Since the etalon reflectivity minima occurs at sinθ ~0.57 and the spectral peak occurs at sinθ = 0.5 (since the angle of incidence is 30°) Δα the angular shift should be −2.8°, i.e., toward the normal instead of away as their work suggests.
  12. R. McDuff, unpublished.
  13. W. N. Bailey, “Some Integrals Involving Hermite Polynomial,” J. London Math. Soc. 23, 291–297 (1948).
    [CrossRef]

1989 (1)

1988 (1)

1985 (1)

1982 (1)

W. J. Firth, E. M. Wright, “Theory of Gaussian Beam Optical Bistability,” Opt. Commun. 40, 233–238 (1982).
[CrossRef]

1978 (2)

M. D. Feit, J. A. Fleck, “Light Propagation in Graded-Index Optical Fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

J. H. Marburger, F. S. Felber, “Theory of a Lossless Nonlinear Fabry-Perot Interferometer,” Phys. Rev. A, 177, 335–342 (1978).
[CrossRef]

1965 (1)

1948 (2)

I. W. Busbridge, “Some Integrals Involving Hermite Polynomials,” J. London Math. Soc. 23, 133–141 (1948).
[CrossRef]

W. N. Bailey, “Some Integrals Involving Hermite Polynomial,” J. London Math. Soc. 23, 291–297 (1948).
[CrossRef]

Bailey, W. N.

W. N. Bailey, “Some Integrals Involving Hermite Polynomial,” J. London Math. Soc. 23, 291–297 (1948).
[CrossRef]

Banerjee, S.

Busbridge, I. W.

I. W. Busbridge, “Some Integrals Involving Hermite Polynomials,” J. London Math. Soc. 23, 133–141 (1948).
[CrossRef]

Feit, M. D.

Felber, F. S.

J. H. Marburger, F. S. Felber, “Theory of a Lossless Nonlinear Fabry-Perot Interferometer,” Phys. Rev. A, 177, 335–342 (1978).
[CrossRef]

Firth, W. J.

W. J. Firth, E. M. Wright, “Theory of Gaussian Beam Optical Bistability,” Opt. Commun. 40, 233–238 (1982).
[CrossRef]

Fleck, J. A.

Kogelnik, H.

Lin, L.

Marburger, J. H.

J. H. Marburger, F. S. Felber, “Theory of a Lossless Nonlinear Fabry-Perot Interferometer,” Phys. Rev. A, 177, 335–342 (1978).
[CrossRef]

McDuff, R.

R. McDuff, unpublished.

Nasalski, W.

Riesz, R. P.

Sharma, A.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U.P., London, 1986).

Simon, R.

Tamir, T.

Wright, E. M.

W. J. Firth, E. M. Wright, “Theory of Gaussian Beam Optical Bistability,” Opt. Commun. 40, 233–238 (1982).
[CrossRef]

Appl. Opt. (2)

J. London Math. Soc. (2)

I. W. Busbridge, “Some Integrals Involving Hermite Polynomials,” J. London Math. Soc. 23, 133–141 (1948).
[CrossRef]

W. N. Bailey, “Some Integrals Involving Hermite Polynomial,” J. London Math. Soc. 23, 291–297 (1948).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

W. J. Firth, E. M. Wright, “Theory of Gaussian Beam Optical Bistability,” Opt. Commun. 40, 233–238 (1982).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

J. H. Marburger, F. S. Felber, “Theory of a Lossless Nonlinear Fabry-Perot Interferometer,” Phys. Rev. A, 177, 335–342 (1978).
[CrossRef]

Other (4)

The far-field irradiance is derived from the Fourier transform of the electric field at the surface. Since the F.T. of a GH polynomial is also a GH polynomial of the same order it follows that the F.T. of E(y)=∑i=0∞fiψi(y) exp(iβy) isEFT(k)=πr0∑i=0∞Cifi exp[-r02(β-k)2/4]Hi(r0(β-k)2)ii,where k = [(2π)/λ] sin θ.

A. E. Siegman, Lasers (Oxford U.P., London, 1986).

There seems to be an inconsistency in the work of Riesz and Simon,6 since there are two graphs of reflection amplitude vs sin θ for the case nd = 2.7 with each having the peak of the spatial frequency distribution on different sides of the etalon reflectivity minima [Figs.2(b) and 3(b)]. This by Eq. (14) gives different signs to the angular shift, either towards or away from the normal. Since the etalon reflectivity minima occurs at sinθ ~0.57 and the spectral peak occurs at sinθ = 0.5 (since the angle of incidence is 30°) Δα the angular shift should be −2.8°, i.e., toward the normal instead of away as their work suggests.

R. McDuff, unpublished.

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Figures (4)

Fig. 1
Fig. 1

Geometry of incident beam. θi is the angle of incidence, w0 the beam waist, and z0 the distance from the waist to the etalon along the direction of propagation.

Fig. 2
Fig. 2

Weakly divergent Gaussian beam incident on a etalon with uniform refractive index equal to n0 + Δn0 in the vicinity of the beam and falling off to n0 away from the beam. (a) Irradiance and phase of the incident beam at the surface. The irradiance is normalized to one. (b) Histogram of the spectrum of GH modes needed to approximate the incident field. (c) Refractive index profile. (d) GH mode spectrum needed to approximate σ(y). (e) Reflectivity (solid) and transmissivity (dashed) vs etalon thickness. Vertical dashed lines are predicted etalon resonances for n = n0 + Δn0.

Fig. 3
Fig. 3

Strongly divergent beam incident upon an uniform etalon. (a) Input irradiance and phase at surface for a tightly focused beam, and (b) a histogram of the spectrum of GH modes needed to approximate the incident field. (c) Reflected irradiance at the surface, and (d) its GH mode spectrum. (e) Far-field reflected irradiance. Note the nonspecular lateral shift in (c) and angular shift in (e).

Fig. 4
Fig. 4

Propagation of a Gaussian beam of λ = 10 μm and w0 = 100 μm which is injected in to a truncated quadratic graded index optical fiber at an angle of 2.25°. The fiber is 400 μm in diameter and has a peak refractive index of 1.01.

Equations (41)

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k ( x , y , z ) = ω c n ( x , y , z ) + i α ( x , y , z ) 2 .
2 E + [ k 0 2 + i k 0 α 0 + k 0 σ ] E = 0 ,
σ ( x , y , z ) = 2 ω c Δ n ( x , y , z ) + i ( Δ α ( x , y , z ) + α 0 n 0 Δ n ( x , y , z ) ) .
t 2 E 0 + 2 i γ [ E 0 z ] + 2 i β [ E 0 y ] + i k 0 α 0 E 0 + k 0 σ E 0 = 0.
E 0 = i j f i j ( z ) ψ i j ( x , y ) ,
σ = i j N i j ( z ) ψ i j ( x , y ) ,
ψ i j = C i j H i [ 2 x r 0 ] H j [ 2 y r 0 ] exp [ - ( r / r 0 ) 2 ] ,
+ - ψ m n ψ i j d x d y = δ m i δ n j .
C i j = 2 r 0 1 π 2 i + j i ! j ! ,
- H m ( x ) H n ( x ) exp ( - x 2 ) d x = π 2 n n ! δ m n .
f m n = + - E 0 ψ m n d x d y .
d f m n d z = α 0 k 0 2 γ f m n 2 β r 0 γ [ n + 1 2 f m , n + 1 - n 2 f m , n - 1 ] + i γ r 0 2 f i j M m n , i j + i 4 k 0 r 0 2 γ f i j N k l A m n , i j , k l ,
M m n , i j = r 0 2 2 + - ψ m n t 2 ψ i j d x d y = - ( i + j + l ) δ m i δ n j + ( i + 1 ) ( i + 2 ) 2 δ m , i + 2 δ n j + ( j + 1 ) ( j + 2 ) 2 δ m i δ n , j + 2 + i ( i - 1 ) 2 δ m , i - 2 δ n j + j ( j - 1 ) 2 δ m i δ n , j - 2
H m + 1 = 2 x H m - 2 m H m - 1 d H m d x = 2 m H m - 1
A m n , i j , k l = 2 r 0 2 + - ψ m n ψ i j ψ k l d x d y = C m n C i j C k l H m i k ( 3 ) H n j l ( 3 ) ,
H m 1 m n ( n ) = - H m 1 ( t ) H m n ( t ) exp ( - n t 2 / 2 ) d t = 0 if i m i is odd .
I m n p a = - H m ( x ) H n ( x ) H p ( x ) exp ( - x 2 / a ) d x = a 1 / 2 ( z 2 ) - k Γ ( k + 1 2 ) r s t ( - m ) s + t ( - n ) t + r ( - p ) r + s r ! s ! t ! ( ½ - k ) r + s + t z r + s + t ,
d f d z = α 0 k 0 2 γ f - 2 β r 0 γ Θ f + i γ r 0 2 M f + i 4 k 0 σ r 0 2 γ R f = - E f
ν ( m , n ) , μ ( i , j ) , f ν f m n Θ ν μ Θ m n , i j = δ m i [ j 2 δ n j - 1 - j + 1 2 δ n , j + 1 ] M ν μ M m n , i j R ν μ R m n , i j = k l N k l A m n , i j , k l = C m n C i j k l N k l C k l H m i k ( 3 ) H n j l ( 3 ) E = + α 0 k 0 2 γ J + 2 β r 0 γ Θ - i γ r 0 2 M - i 2 k 0 σ γ R ,
f = exp [ - 0 z E ( z ) d z ] f 0
f = exp ( - E z ) f 0 .
E ( y , z ) = E f exp ( i γ z ) exp ( i β y ) + E b exp ( - i γ z ) exp ( i β y ) .
2 E f y 2 + 2 i γ [ E f z ] + 2 i β [ E f y ] + i k 0 α 0 E f + k 0 σ E f = 0
2 E b y 2 - 2 i γ [ E b z ] + 2 i β [ E b y ] + i k 0 α 0 E b + k 0 σ E b = 0.
E f = i = 0 f i ( z ) ψ i ( y ) E b = i = 0 b i ( z ) ψ i ( y )
ψ i = C i H i ( 2 y r 0 ) exp [ - ( y r 0 ) 2 ]
C i = [ π 1 / 2 r 0 2 ( i - 1 / 2 ) i ! ] - 1 / 2 .
d f d z = - E f d b d z = + E b ,
E = α 0 k 0 2 γ J + 2 β r 0 γ Θ - i γ r 0 2 M - i k 0 σ r 0 2 3 / 2 γ R Θ m j = j 2 δ m , j - 1 - j + 1 2 δ m , j + 1 M m j = - ( j + ½ ) δ m j + ( j + 1 ) ( j + 2 ) 2 δ m , j + 2 + j ( j - 1 ) 2 δ m , j - 2 R m j = C m C j i C i N i H m i j ( 3 ) .
E i = j I j ψ j exp [ i ( β 0 y + γ 0 z ) ] E r = j R j ψ j exp [ i ( β 0 y - γ 0 z ) ] E t = j T j ψ j exp [ i ( β 0 y + γ 0 z ) ]
( 1 + η ) f x ( 0 ) + ( 1 - η ) b x ( 0 ) = 2 I x ( 0 ) ( 1 - η ) f x ( L ) exp ( i γ L ) + ( 1 + η ) b x ( L ) exp ( - i γ L ) = 0 ,
η = n 0 cos ( θ t ) cos ( θ i ) .
f ( z ) = exp ( - E z ) f 0 b ( z ) = exp ( - E z ) b 0 ,
f o x = - 2 ( 1 + η ) Δ - 1 exp ( E L ) exp ( - i γ L ) I x b o x = 2 I o x - ( 1 + η ) f o x ( 1 - η ) Δ = ( 1 - η ) 2 exp ( - E L ) exp ( i γ L ) - ( 1 + η ) 2 exp ( E L ) exp ( - i γ L )
n ( y ) = { 1.0 y > 200 μ m - 2.5 × 10 5 y 2 + 1.01 y 200 μ m .
g m n = i j a f i j C m n C i j D m i ( a ) ,
D m i ( a ) = r 0 a 2 + 1 I m i α β α = 2 a 2 + 1             β = 2 a a 2 + 1 .
I m n α β = - H m ( α x ) H n ( β x ) exp ( - x 2 ) d x = 0 if m + n is odd , = π m ! n ! s = 0 n / 2 ( α β ) n - 2 s ( α 2 - 1 ) s + ( m - n ) / 2 ( β 2 - 1 ) s { s + ( m - n ) / 2 } ! ( n - 2 s ) ! s ! × 2 n - s             for m n .
g m n = i j f i j C m n C i j T m i ( - a ) T n j ( - b ) ,
T m i ( a ) = π 1 / 2 2 i m ! r 0 2 exp ( - [ a 2 r 0 ] 2 ) [ a ( 2 - 1 ) 2 r 0 ] i - m × L m i - m [ a 2 ( 2 - 1 ) r 0 2 ]             if m i = π 1 / 2 2 m i ! r 0 2 exp ( - [ a 2 r 0 ] 2 ) [ a 2 r 0 ] m - i × L i m - i [ a 2 ( 2 - 1 ) r 0 2 ]             if i < m .
EFT(k)=πr0i=0Cifiexp[-r02(β-k)2/4]Hi(r0(β-k)2)ii,

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