Abstract

The electromagnetic field associated with a generalized beam is analyzed theoretically, where the beam may take on an irradiance cross section described by a gaussian function with arbitrary elliptical symmetry. For this analysis, the field is represented by an expansion into an angular spectrum of plane waves. Expressions for the field components throughout a medium that is free of sources are found by use of an asymptotic approximation that is common in gaussian-beam analysis. In addition, more-precise expressions for these components are found, which are valid outside the neighborhood of focus. Near focus, gaussian beams may have only approximately gaussian cross sections, and in this region the behavior of beams without circular symmetry is greatly complicated. The effects of noncircular symmetry are discussed in some detail, and a method for correcting a beam to produce circular symmetry is described.

© 1972 Optical Society of America

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References

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  1. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
    [Crossref]
  2. D. R. Rhodes, IEEE Trans. AP14, 676 (1966).
    [Crossref]
  3. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 268.
  4. G. Borgiotti, Alta Frequenza 32, 196 (1963).
  5. W. H. Carter, Opt. Commun. 2, 142 (1970).
    [Crossref]
  6. The next section shows that the field behaves in this limit like a magnetic-dipole field. The Ex and Ey components of such a field have amplitudes that each behave as a scalar-dipole field.
  7. For a related analysis, see the appendix in K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
    [Crossref]
  8. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 754.
  9. G. Goubau and F. Schwering, IRE Trans. AP9, 248 (1961).
  10. G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948), p. 85.

1970 (1)

W. H. Carter, Opt. Commun. 2, 142 (1970).
[Crossref]

1966 (2)

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

D. R. Rhodes, IEEE Trans. AP14, 676 (1966).
[Crossref]

1963 (1)

G. Borgiotti, Alta Frequenza 32, 196 (1963).

1962 (1)

1961 (1)

G. Goubau and F. Schwering, IRE Trans. AP9, 248 (1961).

Borgiotti, G.

G. Borgiotti, Alta Frequenza 32, 196 (1963).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 754.

Campbell, G. A.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948), p. 85.

Carter, W. H.

W. H. Carter, Opt. Commun. 2, 142 (1970).
[Crossref]

Foster, R. M.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948), p. 85.

Goubau, G.

G. Goubau and F. Schwering, IRE Trans. AP9, 248 (1961).

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Li, T.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Miyamoto, K.

Rhodes, D. R.

D. R. Rhodes, IEEE Trans. AP14, 676 (1966).
[Crossref]

Schwering, F.

G. Goubau and F. Schwering, IRE Trans. AP9, 248 (1961).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 268.

Wolf, E.

For a related analysis, see the appendix in K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 754.

Alta Frequenza (1)

G. Borgiotti, Alta Frequenza 32, 196 (1963).

IEEE Trans. (1)

D. R. Rhodes, IEEE Trans. AP14, 676 (1966).
[Crossref]

IRE Trans. (1)

G. Goubau and F. Schwering, IRE Trans. AP9, 248 (1961).

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

W. H. Carter, Opt. Commun. 2, 142 (1970).
[Crossref]

Proc. IEEE (1)

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Other (4)

The next section shows that the field behaves in this limit like a magnetic-dipole field. The Ex and Ey components of such a field have amplitudes that each behave as a scalar-dipole field.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 268.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 754.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948), p. 85.

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Figures (2)

F. 1
F. 1

An oblique projection showing the boundary of an elliptical gaussian beam near the focus in the z = 0 plane. This boundary is symmetrical about the focal plane in this figure.

F. 2
F. 2

The radiation pattern for a magnetic-dipole field. The surface formed by the loci of points of constant |S| in Eq. (19) is drawn here in the asymptotic limit of small x and y for which G is constant. The x and y axes have been chosen so that Hy = 0 in order to simplify the figure.

Equations (61)

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f ( x , y ) 1 σ x σ y exp [ 1 2 ( x 2 σ x 2 + y 2 σ y 2 ) ] .
E x ( x , y , z ) = A x ( p , q ) e i k ( p x + q y + m z ) d p d q ,
E y ( x , y , z ) = A y ( p , q ) e i k ( p x + q y + m z ) d p d q ,
E z ( x , y , z ) = [ p m A x ( p , q ) + q m A y ( p , q ) ] × e i k ( p x + q y + m z ) d p d q ,
H x ( x , y , z ) = 1 Z 0 [ p q m A x ( p , q ) + 1 p 2 m A y ( p , q ) ] × e i k ( p x + q y + m z ) d p d q ,
H y ( x , y , z ) = 1 Z 0 [ 1 q 2 m A x ( p , q ) + p q m A y ( p , q ) ] × e i k ( p x + q y + m z ) d p d q ,
H z ( x , y , z ) = 1 Z 0 [ q A x ( p , q ) p A y ( p , q ) ] × e i k ( p x + q y + m z ) d p d q ,
m = ( 1 p 2 q 2 ) 1 2 if p 2 + q 2 1 ,
= i ( p 2 + q 2 1 ) 1 2 if p 2 + q 2 > 1 ,
A x ( p , q ) = ( 1 / λ 2 ) E x ( x , y , 0 ) e i k ( p x + q y ) d x d y
A y ( p , q ) = ( 1 / λ 2 ) E y ( x , y , 0 ) e i k ( p x + q y ) d x d y .
l = ( λ / 2 π ) ( p 2 + q 2 1 ) 1 2 .
E x ( x , y , 0 ) = e x 2 π σ x σ y exp [ 1 2 ( x 2 σ x 2 + y 2 σ y 2 ) ] ,
E y ( x , y , 0 ) = e y 2 π σ x σ y exp [ 1 2 ( x 2 σ x 2 + y 2 σ y 2 ) ] .
A x ( p , q ) = e x λ 2 exp [ k 2 2 ( σ x 2 p 2 + σ y 2 q 2 ) ] ,
A y ( p , q ) = e y λ 2 exp [ k 2 2 ( σ x 2 p 2 + σ y 2 q 2 ) ] ,
A x ( p , q ) = e x λ 2 exp [ k 2 2 ( σ x 2 p 2 + σ y 2 q 2 ) ] if p 2 + q 2 1 , = 0 if p 2 + q 2 > 1 ,
A y ( p , q ) = e y λ 2 exp [ k 2 2 ( σ x 2 p 2 + σ y 2 q 2 ) ] if p 2 + q 2 1 , = 0 if p 2 + q 2 > 1 .
E x ( x , y , z ) ~ e x | z r | G ( x , y , z ) as k r ,
E y ( x , y , z ) ~ e y | z r | G ( x , y , z ) as k r ,
E z ( x , y , z ) ~ [ x r e x + y r e y ] G ( x , y , z ) as k r ,
H x ( x , y , z ) ~ 1 Z 0 [ x y r 2 e x + y 2 + z 2 r 2 e y ] G ( x , y , z ) as k r ,
H y ( x , y , z ) ~ 1 Z 0 [ x 2 + y 2 r 2 e x + x y r 2 e y ] G ( x , y , z ) as k r ,
H z ( x , y , z ) ~ 1 Z 0 | z r | [ y r e x x r e y ] G ( x , y , z ) as k r ,
G ( x , y , z ) = i k e ± i k r 2 π r exp [ k 2 2 r 2 ( σ x 2 x 2 + σ y 2 y 2 ) ]
E x ( x , y , z ) ~ e x G ( x , y , z ) ,
E y ( x , y , z ) ~ e y G ( x , y , z ) ,
E z ( x , y , z ) ~ i [ x k γ x 2 e x + y k γ y 2 e y ] G ( x , y , z ) ,
H x ( x , y , z ) ~ 1 Z 0 [ x y k 2 γ x 2 γ y 2 e x e y ] G ( x , y , z ) ,
H y ( x , y , z ) ~ 1 Z 0 [ e x x y k 2 γ x 2 γ y 2 e y ] G ( x , y , z ) ,
H z ( x , y , z ) ~ i Z 0 [ y k γ y 2 e x x k γ x 2 e y ] G ( x , y , z ) ,
G ( x , y , z ) = k 2 2 π γ x γ y exp [ i k z + i k 3 z x 2 2 ( k 4 σ x 4 + k 2 z 2 ) + i k 3 z y 2 2 ( k 4 σ y 4 + k 2 z 2 ) ] exp [ k 2 2 ( σ x 2 x 2 k 2 σ x 4 + z 2 + σ y 2 y 2 k 2 σ y 4 + z 2 ) ]
γ x = ( k 2 σ x 2 + i k z ) 1 2 ,
γ y = ( k 2 σ y 2 + i k z ) 1 2
G ( x , y , 0 ) = 1 2 π σ x σ y exp [ 1 2 ( x 2 σ x 2 + y 2 σ y 2 ) ] ,
G ( x , y , z ) ~ i k 2 π z exp [ i k ( z + x 2 + y 2 2 z ) ] × exp [ k 2 2 z 2 ( σ x 2 x 2 + σ y 2 y 2 ) ] as k z .
z ( x 2 + y 2 ) 1 2 .
2 π σ x 2 λ z 2 π σ x 2 λ for d x ( z ) ,
2 π σ y 2 λ z 2 π σ y 2 λ for d y ( z ) .
d d | z | ( d x ( z ) ) ~ λ 2 π σ x as | z | ,
d d | z | ( d y ( z ) ) ~ λ 2 π σ y as | z | ,
| z | = z c 2 π σ x σ y λ ,
G ( x , y , z c ) = 1 / { 2 π σ x σ y [ ( 1 ± i σ x / σ y ) ( 1 ± i σ y / σ x ) ] 1 2 } × exp { ± i [ k 2 σ x σ y + ( σ y / σ x ) x 2 + ( σ x / σ y ) y 2 2 ( σ x 2 + σ y 2 ) ] } exp { x 2 + y 2 2 ( σ x 2 + σ y 2 ) } .
S ( x , y , z ) ~ ± c 4 π Z 0 G 2 ( x , y , z ) ( e x 2 + e y 2 ) r r sin 2 ϕ as k r ,
U ( p , q ) = S M ( p , q , p , q ) e i h g ( p , q , p , q ) d p d q ,
U ( p , q ) ~ 2 π i h j j ( | α j β j γ j 2 | ) 1 2 M ( p j , q j , p , q ) × e i h g ( p j , q j , p , q ) as h .
[ g ( p , q , p , q ) p ] p = p j q = q j = 0 , [ g ( p , q , p , q ) q ] p = p j q = q j = 0 ,
α j = [ 2 g ( p , q , p , q ) p 2 ] p = p j q = q j , β j = [ 2 g ( p , q , p , q ) q 2 ] p = p j q = q j , γ j = [ 2 g ( p , q , p , q ) p q ] p = p j q = q j ,
j = 1 if α j β j > γ j 2 and α j > 0 , = 1 if α j β j > γ j 2 and α j < 0 , = i if α j β j < γ j 2 .
E x ( x , y , z ) = U ( x r , y r ) ,
M ( p , q , x r , y r ) = e x λ 2 exp [ k 2 2 ( σ x 2 p 2 + σ y 2 q 2 ) ] ,
h = k r ,
g ( p , q , x r , y r ) = x p r + y q r + z r ( 1 p 2 q 2 ) 1 2 .
p 1 = ± x r , q 1 = ± y r , m 1 = ± z r ,
α 1 = ( 1 + x 2 / z 2 ) , β 1 = ( 1 + y 2 / z 2 ) , γ 1 = x y / z 2 ,
1 = 1 .
E x ( x , y , z ) ~ i k e x 2 π | z r | exp [ k 2 2 r 2 ( σ x 2 x 2 + σ y 2 y 2 ) ] e ± i k r r as k r .
H x ( x , y , z ) = 1 Z 0 λ 2 [ p q e x m + ( 1 p 2 ) e y m ] × exp [ k 2 2 ( σ x 2 p 2 + σ y 2 q 2 ) ] × exp [ i k ( p x + q y + { 1 p 2 q 2 } 1 2 z ) ] d p d q .
σ x λ / 2 π , σ y λ / 2 π ,
( 1 p 2 q 2 ) 1 2 = 1 p 2 + q 2 2 + O ( ( p 2 + q 2 ) 2 )
H x ( x , y , z ) ~ e i k z Z 0 λ 2 { e x p exp [ 1 2 ( k 2 σ x 2 + i k z ) p 2 ] × e i k p x d p q exp [ 1 2 ( k 2 σ y 2 + i k z ) q 2 ] e i k q y d q + e y exp [ 1 2 ( k 2 σ x 2 + i k z ) p 2 ] e i k p x d p × exp [ 1 2 ( k 2 σ y 2 + i k z ) q 2 ] e i k q y d q } as k σ x and k σ y .