Abstract

A theoretical analysis is made of Gaussian beam propagation in media where the refractive index and/or the gain constant varies quadratically with the distance from the optic axis. Use of a complex beam parameter simplifies the analysis. A differential equation for the complex beam parameter is deduced from the wave equation for the complex beam parameter is deduced from the wave equation, and various of its solutions are discussed. This includes a discussion of light propagation in media with a gain profile, which are found capable of supporting stationary beams of constant diameter.

© 1965 Optical Society of America

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References

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  1. S. A. Collins, Appl. Opt. 3,1263 (1964);T. Li, Appl. Opt. 3, 1315 (1964);J. P. Gordon, Bell System Tech. J. 43, 1826 (1964).
    [CrossRef]
  2. G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), pp. 379–395.
  3. H. Kogelnik, Bell System Tech. J. 44, 455 (1965).
  4. D. A. Kleinman, A. Ashkin, G. D. Boyd, to be published;H. Kogelnik, Proc. Symp. Quasi-Optics (Polytechnic Inst.Brooklyn, 1964), pp. 333–347;A. G. van Nie, Philips Res. Rept. 19, 378 (1964).
  5. M. A. Leontovich, V. A. Fok, Zh. Eksperim. i Teor. Fiz. 16, 557 (1946);L. A. Vainshtein, Zh. Techn. Fiz. 34, 193 (1964);Soviet Phys.—Tech. Phys. 9, 157 (1964).
  6. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [CrossRef]
  7. D. W. Berreman, J. Opt. Soc. Am. 55, 239 (1965);D. Marcuse, S. E. Miller, Bell System Tech. J. 43, 1759 (1964);E. A. J. Marcatili, Bell System Tech. J. 43, 2887 (1964).
    [CrossRef]
  8. W. Brower, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
  9. G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961);G. Goubau, F. Schwering, Inst. Radio Engrs. Trans. AP-9, 248 (1961);A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963);H. Kogelnik in Advances in Lasers, A. K. Levine, ed. (Dekker, New York, 1965).
    [CrossRef]
  10. J. R. Pierce, Proc. Natl. Acad. Sci. (U.S.) 47, 1808 (1961).
    [CrossRef]
  11. J. W. Kluver, submitted to J. Appl. Phys.

1965 (3)

1964 (1)

1961 (2)

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961);G. Goubau, F. Schwering, Inst. Radio Engrs. Trans. AP-9, 248 (1961);A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963);H. Kogelnik in Advances in Lasers, A. K. Levine, ed. (Dekker, New York, 1965).
[CrossRef]

J. R. Pierce, Proc. Natl. Acad. Sci. (U.S.) 47, 1808 (1961).
[CrossRef]

1946 (1)

M. A. Leontovich, V. A. Fok, Zh. Eksperim. i Teor. Fiz. 16, 557 (1946);L. A. Vainshtein, Zh. Techn. Fiz. 34, 193 (1964);Soviet Phys.—Tech. Phys. 9, 157 (1964).

Ashkin, A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, to be published;H. Kogelnik, Proc. Symp. Quasi-Optics (Polytechnic Inst.Brooklyn, 1964), pp. 333–347;A. G. van Nie, Philips Res. Rept. 19, 378 (1964).

Berreman, D. W.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961);G. Goubau, F. Schwering, Inst. Radio Engrs. Trans. AP-9, 248 (1961);A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963);H. Kogelnik in Advances in Lasers, A. K. Levine, ed. (Dekker, New York, 1965).
[CrossRef]

D. A. Kleinman, A. Ashkin, G. D. Boyd, to be published;H. Kogelnik, Proc. Symp. Quasi-Optics (Polytechnic Inst.Brooklyn, 1964), pp. 333–347;A. G. van Nie, Philips Res. Rept. 19, 378 (1964).

Brower, W.

W. Brower, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

Collins, S. A.

Deschamps, G. A.

G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), pp. 379–395.

Fok, V. A.

M. A. Leontovich, V. A. Fok, Zh. Eksperim. i Teor. Fiz. 16, 557 (1946);L. A. Vainshtein, Zh. Techn. Fiz. 34, 193 (1964);Soviet Phys.—Tech. Phys. 9, 157 (1964).

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961);G. Goubau, F. Schwering, Inst. Radio Engrs. Trans. AP-9, 248 (1961);A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963);H. Kogelnik in Advances in Lasers, A. K. Levine, ed. (Dekker, New York, 1965).
[CrossRef]

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, to be published;H. Kogelnik, Proc. Symp. Quasi-Optics (Polytechnic Inst.Brooklyn, 1964), pp. 333–347;A. G. van Nie, Philips Res. Rept. 19, 378 (1964).

Kluver, J. W.

J. W. Kluver, submitted to J. Appl. Phys.

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

Leontovich, M. A.

M. A. Leontovich, V. A. Fok, Zh. Eksperim. i Teor. Fiz. 16, 557 (1946);L. A. Vainshtein, Zh. Techn. Fiz. 34, 193 (1964);Soviet Phys.—Tech. Phys. 9, 157 (1964).

Mast, P. E.

G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), pp. 379–395.

Pierce, J. R.

J. R. Pierce, Proc. Natl. Acad. Sci. (U.S.) 47, 1808 (1961).
[CrossRef]

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Appl. Opt. (1)

Bell System Tech. J. (2)

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961);G. Goubau, F. Schwering, Inst. Radio Engrs. Trans. AP-9, 248 (1961);A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963);H. Kogelnik in Advances in Lasers, A. K. Levine, ed. (Dekker, New York, 1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Proc. Natl. Acad. Sci. (U.S.) (1)

J. R. Pierce, Proc. Natl. Acad. Sci. (U.S.) 47, 1808 (1961).
[CrossRef]

Zh. Eksperim. i Teor. Fiz. (1)

M. A. Leontovich, V. A. Fok, Zh. Eksperim. i Teor. Fiz. 16, 557 (1946);L. A. Vainshtein, Zh. Techn. Fiz. 34, 193 (1964);Soviet Phys.—Tech. Phys. 9, 157 (1964).

Other (4)

W. Brower, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

D. A. Kleinman, A. Ashkin, G. D. Boyd, to be published;H. Kogelnik, Proc. Symp. Quasi-Optics (Polytechnic Inst.Brooklyn, 1964), pp. 333–347;A. G. van Nie, Philips Res. Rept. 19, 378 (1964).

G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), pp. 379–395.

J. W. Kluver, submitted to J. Appl. Phys.

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Figures (3)

Fig. 1
Fig. 1

Wavefronts of spherical wave.

Fig. 2
Fig. 2

Ray path in lenslike medium and reference planes.

Fig. 3
Fig. 3

Gaussian beam propagating through lens system and corresponding circuit analogs.

Equations (80)

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R 2 = R 1 + z .
1 R 2 = 1 R 1 1 f .
n ( r , z ) = n 0 ( z ) 1 2 n 2 ( z ) · r 2 .
d d z ( n 0 d x d z ) = n 2 x ,
( x 2 x 2 ) = ( A B C D ) ( x 1 x 1 ) .
R = x x ·
R 2 = A R 1 + B C R 1 + D ·
E ( r , z ) = w 0 w exp { j ( k z + φ ) r 2 ( 1 w 2 + j k 2 R ) } ,
w 2 = w 0 2 [ 1 + ( λ z π w 0 2 ) 2 ] ,
R = z [ 1 + ( π w 0 2 λ z ) 2 ] ,
φ = tan 1 ( λ z π w 0 2 ) .
1 q = 1 R j λ π w 2 .
q 2 = q 0 + z ,
q 0 = j π w 0 2 λ .
q 2 = q 1 + z .
1 q 2 = 1 q 1 1 f .
q 2 = A q 1 + B C q 1 + D .
Δ E + k 2 E = 0 ,
k 2 = ω 2 μ ( 1 j σ ω ) .
k 2 ( r , z ) = k 0 2 k 0 k 2 r 2 .
E = ψ ( x , y , z ) · exp ( j Φ ) ,
Φ ( z ) = k 0 ( z ) ,
Δ t ψ 2 j k 0 ψ j k 0 ψ k 0 k 2 r 2 ψ = 0 ,
Δ t = Δ 2 z 2 = 2 x 2 + 2 y 2 .
ψ = exp j ( p + 1 2 Q r 2 ) .
Q 2 r 2 + 2 j Q + k 0 ( 2 P + Q r 2 ) + j k 0 + k 0 k 2 r 2 = 0 .
P = j ( Q k 0 + k 0 2 k 0 ) ,
Q 2 + k 0 Q + k 0 k 2 = 0 .
Q = k 0 x x .
( k 0 x ) + k 2 x = 0 .
Q m = j k 0 k 2 .
x = a e j γ z + b e j γ z ,
γ = k 2 k 0 .
Q 2 = Q m Q 1 + Q m + ( Q 1 Q m ) e 2 j γ z Q 1 + Q m ( Q 1 Q m ) e 2 j γ z ,
k = β + j α ,
k 2 = β 2 + 2 j α β .
α = σ 2 μ .
P 2 = P 1 : e 2 α z .
α = dB gain 8.686 ( z )
β = β 0 1 2 β 2 r 2 .
n = λ 2 π β = n 0 1 2 n 2 r 2 ,
α = α 0 1 2 α 2 r 2 .
k = k 0 1 2 k 2 r 2 .
k 0 = β 0 + j α 0 ,
k 2 = β 2 + j α 2 .
k 0 k 2 = β 0 β 2 + j ( α 0 β 2 + α 2 β 0 ) .
1 q = λ 2 π Q .
d q d z = 1 .
q 2 = q 1 + z ,
p 2 p 1 = j 0 z d z q 1 + z = j ln q 1 q 1 + z .
ln ( u + j υ ) = In u 2 + υ 2 + j tan 1 υ u .
1 q 2 + n 0 ( 1 q ) + n 0 n 2 = 0 .
1 q m = j n 0 n 2 = j λ π w m 2 .
1 R = n 0 w w ,
n 0 w ( n 0 w ) = n 0 n 2 + λ 2 π 2 w 4 .
k 0 k 2 = j 2 π λ α 2 .
1 q m = λ 2 π Q m = 1 2 ( 1 j ) α 2 λ π .
R m = π w m 2 λ = 2 π α 2 λ
w m 2 = 2 λ π α 2 .
α = α 0 ( 1 r 2 r 0 2 ) .
α 2 = 2 α 0 / r 0 2 ,
R m = r 0 2 π λ α , w m 2 = r 0 2 λ π α 0 .
α 0 = 11.5 m 1 .
R m = 79 cm , w m = 0.94 mm .
p = j q m = ( 1 + j ) z R m .
p 2 = p 1 ( 1 + j ) 1 R m .
exp ( α 0 1 R m ) z .
γ = k 2 k 0 = j q m = 1 + j R m .
1 q 2 = 1 q m ( q 1 + q m ) ( q 1 q m ) exp 2 ( j 1 ) z R m ( q 1 + q m ) + ( q 1 q m ) exp 2 ( j 1 ) z R m .
q 1 q m q 1 + q m = exp 2 ( δ j φ ) .
σ = 2 ( z R m δ ) , ρ = 2 ( z R m φ ) ,
1 q 2 = 1 q m e σ e j ρ e σ + e j ρ = 1 q m sh σ j sin ρ ch σ + cos ρ .
R 2 R m = c h σ + cos ρ s h σ sin ρ .
w 2 2 w 2 2 = c h σ + cos ρ s h σ + sin ρ .
Y = j q = λ π w 2 + j R .
z = j q ,
d d z ( 1 Z L d V d z ) = Y L V ,
j Z L = k 0 ( z )
Y L = j k 2 ( z ) .
I ( z ) = x Z L .

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