Abstract

A generalization of the Gaussian beam is obtained by introducing a complex-valued shift in the transverse dimension. The resulting beam has a Gaussian intensity distribution with width varying as an ordinary Gaussian beam, but whose peak traces an inclined linear trajectory. The wave fronts are displaced laterally in a sheared fashion. This generalized beam preserves its form after passing through arbitrary paraxial optical components, even if they are decentered. The peak-intensity line is modified by such systems as if it were a ray.

© 1995 Optical Society of America

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References

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  1. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
    [CrossRef]
  2. L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [CrossRef]
  3. L. W. Casperson, “Gaussian light beams in inhomogenous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  4. B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 3.
    [CrossRef]
  5. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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1976 (1)

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Figures (4)

Fig. 1
Fig. 1

(a) Intensity distribution of the decentered Gaussian beam at four locations. The peak-intensity axis is inclined at an angle θ I = tan−1(x 0/z 0). In this plot x 0 = W 0. (b) Wave fronts of the decentered Gaussian beam of (a) at five locations. The solid segments of the wave fronts indicate the region where the energy (intensity) is significant. The short-dashed curves are the trajectories of the center of curvature, and the dashed line is the peak-intensity axis [also shown in (a)].

Fig. 2
Fig. 2

Reflection of an off-axis Gaussian beam from a mirror.

Fig. 3
Fig. 3

Transmission of a decentered Gaussian beam through a Gaussian aperture. The incident beam has shear parameter x 0 = 4W 0, and the peak-intensity axis is inclined at θ = 14°. The transmitted beam is displaced downward, and its inclination is altered to θ′ ≅ 9°.

Fig. 4
Fig. 4

Parameters of a decentered Gaussian beam traveling through a graded-index slab with n 2/n 0 = 1/20. The propagation distance z is normalized to z 0, the Rayleigh length of the incident beam. The incident beam has the following parameters: z 0 = 1 m, λ0 = 514.5 nm, x d /W 0 = 7.4, and x 0/W 0 = 1.24. The first two plots are normalized to the waist radius W 0, and the last two are normalized to z 0.

Equations (40)

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z z + j z 0 ,
x x + j x 0 ,
U ( x , y , z ) = 1 z + j z 0 exp { - j k [ ( x + j x 0 ) 2 + y 2 ] 2 ( z + j z 0 ) } × exp ( - j k z ) .
1 q ( z ) = 1 z + j z 0 = 1 R ( z ) - j λ π W 2 ( z ) ,
W 2 ( z ) = W 0 2 [ 1 + ( z z 0 ) 2 ] , R ( z ) = z [ 1 + ( z 0 z ) 2 ] , W 0 = ( λ z 0 π ) 1 / 2 .
I ( x , y , z ) = [ W 0 W ( z ) ] 2 exp ( 2 x 0 2 W 0 2 ) × exp [ - 2 [ x - x I ( z ) ] 2 + y 2 W 2 ( z ) ] ,
φ ( x , y , z ) = k [ x - x p ( z ) ] 2 + y 2 2 R ( z ) + k z - k x 0 2 R ( z ) - tan - 1 ( z z 0 ) - z 0 z x 0 2 W 2 ( z ) ,
x I ( z ) = x 0 π W 2 ( z ) λ R ( z ) = x 0 z z 0 ,
x p ( z ) = - x 0 λ R ( z ) π W 2 ( z ) = - x 0 z 0 z .
θ I = tan - 1 ( x 0 z 0 )
U ( x , 0 , z ) = 1 q ( z ) exp [ - j k ( x - p ) 2 2 q ( z ) ] exp ( - j k z ) ,
U out ( x 2 ) = - K ( x 2 , x 1 ) U in ( x 1 ) d x 1 ,
K ( x 2 , x 1 ) = j / λ 0 B exp ( - j π L λ 0 ) × exp [ - j π λ 0 B ( A x 1 2 - 2 x 1 x 2 + D x 2 2 ) ]
U in ( x ) = exp [ - j k ( x - p in ) 2 2 q in ] ,
U out ( x ) = U 0 exp [ - j k ( x - p out ) 2 2 q out ] ,
q out = A q in + B C q in + D ,
p out = p in C q in + D ,
x I = A x I + B θ I
θ I = C x I + D θ I .
1 q out = 1 q in + 2 R m ,
p out = p in 2 R m q in + 1 .
θ I = tan - 1 ( - 2 Δ R m )
z c = - R m 2
Δ = x d + x 0 z c z 0 ,
t ( x ) = exp ( - x 2 / W a 2 ) .
p x d - j x 0 = [ ( 1 + W 0 2 W a 2 ) + j ( λ z a π W a 2 ) ( 1 + W 0 2 W a 2 ) 2 + ( λ z a π W a 2 ) 2 ] ( x d - j x 0 ) ,
1 z c + j z 0 = 1 q = 1 q - j λ π W a 2 ,
x I = x d + x 0 z a z 0 ,
x I = x d + x 0 z c z 0 ,
x d = ( 1 - α ) ( 2 α - 1 ) α 2 + ( 1 - α ) 2 x 0 ,
x 0 = α ( 2 α - 1 ) α 2 + ( 1 - α ) 2 x 0 ,
tan θ I = α tan θ I ,
α ( 1 + W 0 2 W a 2 ) ( 1 + 2 W 0 2 W a 2 ) .
x I = ( 2 α - 1 ) x I = x I 1 + 2 W 0 2 W a 2 .
n ( x ) = n 0 - 1 2 n 2 x 2 .
1 q ( z ) = 1 R ( z ) - j λ π W 2 ( z ) = A ( z ) q ( 0 ) + B ( z ) C ( z ) q ( 0 ) + D ( z ) ,
p ( z ) = x d ( z ) - j x 0 ( z ) = x d ( 0 ) - j x 0 ( 0 ) C ( z ) q ( 0 ) + D ( z ) .
x I ( z ) = x d ( z ) + x 0 ( z ) π W 2 ( z ) λ R ( z ) ,
x p ( z ) = x d ( z ) - x 0 ( z ) λ R ( z ) π W 2 ( z ) ,
n ( x ) n 0 - n 1 x - 1 2 n 2 x 2 = n 0 + 1 2 n 1 - 1 2 n 2 ( x + n 1 n 2 ) 2 .

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