Abstract

This paper presents a series solution for electromagnetic wave propagation in a medium in which the dielectric constant is given by = 02 (z)r2, without recourse to the WKB approximation. The results have been put in a form suitable for numerical computation. Explicit expression has been obtained for the irradiance distribution in the paraxial region, when the incident wave front is plane gaussian. These investigations are applicable to propagation of laser beams in self-focusing fibers/rods.

© 1972 Optical Society of America

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References

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  1. S. A. Akhamanov, A. P. Sukhorukov, and R. V. Khokhlov, Zh. Eksperim. i Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].
  2. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Usp. Fiz. Nauk 93, 19 (1968) [Sov. Phys. Usp. 10, 609 (1968)].
  3. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
    [Crossref]
  4. H. Kita and T. Uchida, in Fiber Optics: Applications and Technology, SPIE Seminar Proc., Vol. 21 (Soc. Photo-optical Instrumentation Engrs., Redondo Beach, California, 1970).
  5. M. S. Sodha, A. K. Ghatak, and D. P. S. Mailk, J. Opt. Soc. Am. 61, 1492 (1971).
    [Crossref]
  6. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [Crossref]
  7. P. J. Sands, J. Opt. Soc. Am. 61, 777 (1971).
    [Crossref] [PubMed]
  8. P. J. Sands, J. Opt. Soc. Am. 61, 879 (1971).
    [Crossref]
  9. P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1971).
    [Crossref]
  10. D. T. Moore and P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971).
    [Crossref]

1971 (5)

1970 (2)

P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
[Crossref]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
[Crossref]

1968 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Usp. Fiz. Nauk 93, 19 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

1966 (1)

S. A. Akhamanov, A. P. Sukhorukov, and R. V. Khokhlov, Zh. Eksperim. i Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

Akhamanov, S. A.

S. A. Akhamanov, A. P. Sukhorukov, and R. V. Khokhlov, Zh. Eksperim. i Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Usp. Fiz. Nauk 93, 19 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
[Crossref]

Ghatak, A. K.

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Usp. Fiz. Nauk 93, 19 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

S. A. Akhamanov, A. P. Sukhorukov, and R. V. Khokhlov, Zh. Eksperim. i Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

Kita, H.

H. Kita and T. Uchida, in Fiber Optics: Applications and Technology, SPIE Seminar Proc., Vol. 21 (Soc. Photo-optical Instrumentation Engrs., Redondo Beach, California, 1970).

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
[Crossref]

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
[Crossref]

Mailk, D. P. S.

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
[Crossref]

Moore, D. T.

Sands, P. J.

Sodha, M. S.

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Usp. Fiz. Nauk 93, 19 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

S. A. Akhamanov, A. P. Sukhorukov, and R. V. Khokhlov, Zh. Eksperim. i Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
[Crossref]

H. Kita and T. Uchida, in Fiber Optics: Applications and Technology, SPIE Seminar Proc., Vol. 21 (Soc. Photo-optical Instrumentation Engrs., Redondo Beach, California, 1970).

IEEE J. (1)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. QE-6, 606 (1970).
[Crossref]

J. Opt. Soc. Am. (6)

Usp. Fiz. Nauk (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Usp. Fiz. Nauk 93, 19 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

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

S. A. Akhamanov, A. P. Sukhorukov, and R. V. Khokhlov, Zh. Eksperim. i Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

Other (1)

H. Kita and T. Uchida, in Fiber Optics: Applications and Technology, SPIE Seminar Proc., Vol. 21 (Soc. Photo-optical Instrumentation Engrs., Redondo Beach, California, 1970).

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Figures (1)

Fig. 1
Fig. 1

The variation of A02 cos2ξ with ρ for σ = 6×10−3 and for an incident laser beam with gaussian irradiance distribution. The solid curves show the calculated values obtained by using the first three terms of the series solution [Eq. (23)]. The dashed curves correspond to the WKB approximation.

Equations (45)

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2 E = c 2 2 t 2 E ,
E = A exp [ - i ( k z - ω t ) ] .
N ( r ) = N 0 + N 1 r + N 2 r 2 + ,
= 0 - 2 ( z ) r 2 .
A = A 0 exp ( - i k S ) ,
( S z ) 2 + 2 S z + ( S r ) 2 = - 2 ( z ) 0 r 2 + 1 k 2 A 0 ( 2 A 0 z 2 + 2 A 0 r 2 + 1 r A 0 r ) ,
( S z + 1 ) A 0 2 z + S r A 0 2 r + ( 2 S z 2 + 2 S r 2 + 1 r S r ) A 0 2 = 0.
S = ϕ ( z ) + r 2 2 β ( z ) + r 4 4 η ( z ) + r 6 6 Ψ ( z ) + ,
A 0 2 = α 0 ( z ) + r 2 α 2 ( z ) + r 4 α 4 ( z ) + .
( ϕ ) 2 + 2 ϕ = 0 ,
( ϕ + 1 ) β + β 2 = - [ 2 ( z ) / 0 ] ,
1 4 ( β ) 2 + 1 2 η ( 1 + ϕ ) + 2 β η = 0 ,
1 3 ( 1 + ϕ ) Ψ + 2 β Ψ + η 2 + 1 4 β η = 0 ,
( 1 + ϕ ) α 0 + ( 2 β + ϕ ) α 0 = 0 ,
( 1 + ϕ ) α 2 + ( 4 β + ϕ ) α 2 + 1 2 β α 0 + ( 1 2 β + 4 η ) α 0 = 0 ,
( 1 + ϕ ) α 4 + ( 6 β + ϕ ) α 4 + 1 2 β α 2 + 1 4 η α 0 + 2 η α 2 + ( 1 2 β + 4 η ) α 2 + ( 1 4 η + 6 Ψ ) α 0 = 0 ,
ϕ = 0             or             - 2.
β = ( 1 / f ) ( d f / d z ) ,
f + [ 2 ( z ) / 0 ] f = 0.
η = η d d z ( ln 1 f 4 ) - 1 2 ( β ) 2 ,
Ψ = Ψ d d z ( ln 1 f 6 ) - 3 ( η 2 + 1 4 β η ) .
η = 1 f 4 [ C 1 - 1 2 0 z ( β ) 2 f 4 d z ] ,
Ψ = 1 f 6 { C 2 - 3 0 z f 6 [ η 2 - β β η - 1 8 ( β ) 3 ] d z } .
α 0 = 1 / f 2 .
α 2 = α 2 d d z ( ln 1 f 4 ) + B 2 ( z ) ,
α 4 = α 4 d d z ( ln 1 f 6 ) + B 4 ( z ) ,
B 2 ( z ) = 1 f 2 ( β β - 4 η - 1 2 β ) ,
B 4 ( z ) = - [ 1 2 β α 2 + 1 4 η α 0 + ( 1 2 β + 6 η ) α 2 + ( 1 4 η + 6 Ψ ) α 0 ] .
α 2 ( z ) = 1 f 4 [ D 1 + 0 z f 4 B 2 ( z ) d z ] ,
α 4 ( z ) = 1 f 6 [ D 2 + 0 z f 6 B 4 ( z ) d z ] ,
A 0 2 ( r , z = 0 ) = exp ( - r 2 a 2 ) = 1 - r 2 a 2 + r 4 2 a 4 - .
α 2 ( z ) z = 0 = - 1 a 2 = D 1 , a 4 ( z ) z = 0 = 1 2 a 4 = D 2 , etc .
f = cos p z ,
p = ( 2 / 0 ) 1 2 .
η = - p 4 z 2 cos 4 p z ,
Ψ = - 3 p 6 z 4 cos 6 p z ( p z tan p z + 1 2 ) ;
α 2 ( z ) = 1 cos 4 p z ( - 1 a 2 + 2 p 3 z tan p z )
α 4 ( z ) = - 1 2 cos 6 p z { 1 a 4 - 6 p 3 z a 2 tan p z + p 4 [ - 9 ( p z ) 2 + ( 9 / 2 ) p z tan p z + ( 21 / 2 ) ( p z ) 2 sec 2 p z ] } .
A 0 2 = 1 cos 2 ξ - ρ 2 cos 4 ξ [ 1 - 2 σ ξ tan ξ ] + ρ 4 2 cos 6 ξ [ 1 - 6 σ ξ tan ξ + σ 2 { - 9 ξ 2 + ( 9 / 2 ) ξ tan ξ + ( 21 / 2 ) ξ 2 sec 2 ξ } ] ,
ρ = r / a , ξ = p z ,
σ = ( a p ) 2 .
A 0 2 = 1 cos 2 ξ exp ( - ρ 2 cos 2 ξ )
= 1 cos 2 ξ - ρ 2 cos 4 ξ + ρ 4 2 cos 6 ξ - .
= 0 - 2 ( z ) r 2 + 4 ( z ) r 4 - 6 ( z ) r 6 + ,
η = 1 f 4 [ c 1 - 1 2 0 z { β 2 - 4 4 ( z ) 0 } f 4 d z ] , Ψ = 1 f 6 [ c 2 - 3 0 z { η 2 - β β η - 1 8 β 3 + β 4 ( z ) 2 0 + 6 ( z ) 0 } f 6 d z ] .