Abstract

The Fresnel approximation is used in the design of a lens that turns a beam with one intensity distribution into a beam with a different distribution at the focal plane of a Fourier transform lens. In general, this cannot be done exactly, so one must be satisfied with approximate solutions to this problem. It is shown that the difficulty of this problem depends on a parameter β that is a dimensionless measure of how well the geometrical optics approximation holds. An analytical method is given for determining the lens when β is large. The sensitivity of this solution to various imperfections in the system alignment is also analyzed.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. M. Dickey, B. D. O’Neil, “Multifaceted laser beam integrators: general formulation and design concepts,” Opt. Eng. 27, 999–1007 (1988).
    [CrossRef]
  2. Wai-Hon Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 4691–471 (1981).
    [CrossRef]
  3. P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
    [CrossRef] [PubMed]
  4. C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
    [CrossRef]
  5. W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
    [CrossRef] [PubMed]
  6. J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1991 (1)

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

1988 (1)

F. M. Dickey, B. D. O’Neil, “Multifaceted laser beam integrators: general formulation and design concepts,” Opt. Eng. 27, 999–1007 (1988).
[CrossRef]

1982 (1)

1981 (1)

Wai-Hon Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 4691–471 (1981).
[CrossRef]

1980 (1)

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Dickey, F. M.

F. M. Dickey, B. D. O’Neil, “Multifaceted laser beam integrators: general formulation and design concepts,” Opt. Eng. 27, 999–1007 (1988).
[CrossRef]

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Goodman, J. W.

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Lee, Wai-Hon

Wai-Hon Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 4691–471 (1981).
[CrossRef]

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

O’Neil, B. D.

F. M. Dickey, B. D. O’Neil, “Multifaceted laser beam integrators: general formulation and design concepts,” Opt. Eng. 27, 999–1007 (1988).
[CrossRef]

Rhodes, P. W.

Shealy, D. L.

Veldkamp, W. B.

Appl. Opt. (2)

Opt. Commun. (1)

Wai-Hon Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 4691–471 (1981).
[CrossRef]

Opt. Eng. (2)

F. M. Dickey, B. D. O’Neil, “Multifaceted laser beam integrators: general formulation and design concepts,” Opt. Eng. 27, 999–1007 (1988).
[CrossRef]

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Other (1)

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Phase function ϕ(ξ) (in radians) that turns a Gaussian beam into a flattop beam for β = ∞.

Fig. 2
Fig. 2

Intensity of the beam obtained by modification of the phase of a Gaussian beam by the phase function in Fig. 1. Note that as β gets large, the intensity is approaching that of a flattop beam.

Fig. 3
Fig. 3

Deviation of the intensity (multiplied by β2) from its asymptotic value for different values of β. The plots are shown only away from the skirt region.

Fig. 4
Fig. 4

Intensity in the skirt region as a function of the rescaled variable α* = β0.9[(2/π0.9)α − 1].

Fig. 5
Fig. 5

Intensity obtained by truncation of the Gaussian at z = 21.5 2 before modification of its phase.

Fig. 6
Fig. 6

Effect of a positive and a negative value of γ on the intensity of the beam.

Fig. 7
Fig. 7

Effects of lateral misalignment on the intensity of the beam. These plots are for δ = 0.1.

Fig. 8
Fig. 8

Intensity when one uses the geometrical optics approximation to turn a circular Gaussian into a beam that is constant on a circular cross section.

Tables (2)

Tables Icon

Table 1 Demonstration of How Well Eq. (18) Predicts the Overshoot for γ > 0a

Tables Icon

Table 2 Demonstration of How Well Relation (18) Predicts the Value of γ for Which the Width of the Skirt Increases by a Factor of ma

Equations (100)

Equations on this page are rendered with MathJax. Learn more.

β = 2 π r i R o f λ
γ = δ z r i 2 R o f .
U ( x f , y f ) = exp ( i ψ 0 ) 1 i λ z 0 - - U l ( x , y ) × exp { - i 2 π λ z 0 [ x x f + y y f + 1 2 ( z 0 f - 1 ) ( x 2 + y 2 ) ] } d x d y ,
ψ 0 = π λ z 0 ( x f 2 + y f 2 + 2 z 0 2 ) .
z 0 = f + δ z .
1 f 1 z 0 + δ z f z 0 .
U ( x f , y f ) = exp ( i ψ 0 ) 1 i λ z 0 - - U l ( x , y ) × exp { - i 2 π λ z 0 [ x x f + y y f + δ z 2 z 0 ( x 2 + y 2 ) ] } d x d y .
U l ( x , y ) = E 0 g ( x / r i , y / r i ) exp [ i β ϕ ( x / r i , y / r i ) ] ,
U ( x f , y f ) = 4 π 2 E 0 exp ( i ψ 0 ) r i 2 i λ z 0 G ( x f / R o , y f / R o ) ,
R o = λ z 0 2 π r i , G ( ω x , ω y ) = 1 4 π 2 - - g ( ξ , η ) exp { i [ β ϕ ( ξ , η ) - β γ ( ξ 2 + η 2 ) - ξ ω x - η ω y ] } d ξ d η .
G ( ω x , ω y ) 2 = μ Q ( ω x / β , ω y / β ) .
G ( ω x , ω y ) = 1 4 π 2 - - g ( ξ , η ) exp { i [ β ϕ ( ξ , η ) - ξ ω x - η ω y ] } d ξ d η .
μ β 2 - - Q ( ω x , ω y ) d ω x d ω y = 1 4 π 2 - - g ( ξ , η ) 2 d ξ d η .
2 π r i R o λ f = 2 π r i R o λ f .
A 0 2 < 4 π 2 B A 2 ,
A 0 = - - g ( ξ , η ) 2 d ξ d η ,
A 2 = - - g ( ξ , η ) 2 ( ξ 2 + η 2 ) d ξ d η , B = - - G ( ω x , ω y ) 2 ( ω x 2 + ω y 2 ) d ω x d ω y .
β 2 A 0 B 0 A 2 B 2 ,
B 0 = - - Q ( ω x , ω y ) d ω x d ω y ,
B 2 = - - Q ( ω x , ω y ) ( ω x 2 + ω y 2 ) d ω x d ω y .
g ( ξ , η ) = g 1 ( ξ ) g 2 ( η ) , ϕ ( ξ , η ) = ϕ 1 ( ξ ) ϕ 2 ( η ) , Q ( ω x / β , ω y / β ) = Q 1 ( ω x / β ) Q 2 ( ω y / β ) .
G ( ω ) 2 = μ Q ( ω / β ) ,
G ( ω ) = 1 2 π - g ( ξ ) exp { i [ β ϕ ( ξ ) - ω ξ ] } d ξ
μ β - Q ( ω ) d ω = 1 2 π - g ( ξ ) 2 d ξ .
β > a 0 b 0 4 a 2 b 2 ,
a 0 = - g ( ξ ) 2 d ξ ,
a 2 = - g ( ξ ) 2 ξ 2 d ξ ,
b 0 = - Q ( ω ) d ω ,
b 2 = - Q ( ω ) ω 2 d ω .
Γ ( α ) = 1 2 π - g ( ξ ) exp { i β [ ϕ ( ξ ) - α ξ ] } d ξ ,
α = ω β = x f R o .
A ( α ) = Γ ( α ) 1 2 π g [ ξ ( α ) ] β ϕ [ ξ ( α ) ] ,
ϕ [ ξ ( α ) ] = α .
ϕ [ ξ ( α ) ] d ξ d α = 1.
g 2 [ ξ ( α ) ] d ξ d α = 2 π μ β Q ( α ) .
d ϕ d ξ = α ( ξ ) .
g ( ξ ) = exp ( - ξ 2 / 2 ) , Q ( α ) = { 1 for α < π 2 0 for α > π 2 .
d 2 ϕ d ξ 2 = exp ( - ξ 2 ) .
ϕ ( ξ ) = π 2 0 ξ erf ( s ) d s - 1 2 = ξ π 2 erf ( ξ ) + 1 2 exp ( - ξ 2 ) - 1 2 .
Γ ( α ) = 1 2 π - exp ( - ξ 2 / 2 ) exp { i β [ ϕ ( ξ ) - α ξ ] } d ξ .
2 π Δ α skirt 1.6 β 0.9 .
Γ T ( α ) = Γ ( α ) - 1 2 π ξ > ξ 0 exp ( - ξ 2 / 2 ) × exp { i β [ ϕ ( ξ ) - α ξ ] } d ξ .
T ( ξ 0 ) = max α < α sk | Γ ( α ) 2 - Γ T ( α ) 2 Γ ( α ) 2 | .
T ( ξ 0 ) < 2 β π exp ( - ξ 0 2 / 2 ) ( 1 ξ 0 + 1 ξ 0 2 + β 2 π / 4 ) .
ϕ ( ξ ) = 1 erf ( ξ 0 ) [ ξ π 2 erf ( ξ ) + 1 2 ( - ξ 2 ) - 1 2 ] .
Γ γ ( α ) = 1 2 π - exp ( - ξ 2 / 2 ) × exp { i β [ ϕ γ ( ξ ) - α ξ ] } d ξ ,
ϕ γ ( ξ ) = ϕ ( ξ ) - γ ξ 2 .
2 π β Γ γ ( α ) 2 = exp [ - ξ 2 ( α ) ] exp [ - ξ 2 ( α ) ] - 2 γ ,
π 2 erf [ ξ ( α ) ] - 2 γ ξ ( α ) = α .
0 ( β , γ ) 2.27 [ - β γ ln ( 2 γ ) ] 1 / 3 - 1             for - β γ ln ( 2 γ ) > 0.25 ,
0 ( β , γ ) 2 γ β ln ( 2 β )             for - β γ ln ( 2 γ ) < 0.25.
2 π Δ α - 3.88 γ - 3.53 γ 2 .
γ ( m , β ) = 0.54 ( - 1 + 1 - 1.5 m β 0.9 ) .
2 π Γ ( α ) 2 = exp { - [ ξ ( α ) - δ ] 2 } exp [ - ξ ( α ) 2 ] = exp [ 2 ξ ( α ) δ - δ 2 ] ,
π 2 erf [ ξ ( α ) ] = α .
2 π β Γ ( α ) 2 = 1 + 2 δ α             for α < π / 2.
Γ ( γ x , γ y ) = 1 4 π 2 - - g ( ξ , η ) exp { i β [ ϕ ( ξ , η ) - ξ γ x - η γ y ] } d ξ d η ,
Γ ( γ x , γ y ) 2 g 2 [ g ( γ x , γ y ) , y ( γ x , γ y ) ] β 2 4 π 2 J [ x ( γ x , γ y ) , y ( γ x , γ y ) ] ,
J ( x , y ) = 2 ϕ ( x , y ) x 2 2 ϕ ( x , y ) y 2 - [ 2 ϕ ( x , y ) x y ] 2
ϕ ( x , y ) x = γ x ,
ϕ ( x , y ) y = γ y .
2 ϕ ( x , y ) x 2 2 ϕ ( x , y ) y 2 - [ 2 ϕ ( x , y ) x y ] 2 = g 2 ( x , y ) β 2 4 π 2 μ Q ( ϕ / x , ϕ / x ) .
g 2 ( x , y ) d x d y = 4 π 2 β 2 μ Q ( γ x , γ y ) d γ x d γ y .
g ( x , y ) = g s ( r ) ,             ϕ ( x , y ) = ψ ( r ) ,
r 2 = x 2 + y 2 ,             Q ( γ x , γ y ) = R ( α ) ,
α 2 = γ x 2 + γ y 2 .
Γ ( α ) = 1 2 π 0 r g s ( r ) exp [ i β ψ ( r ) ] J 0 ( β α r ) d r .
2 ϕ ( x , y ) x 2 2 ϕ ( x , y ) y 2 - [ 2 ϕ ( x , y ) x y ] 2 = 1 r 2 ψ r 2 ψ r .
1 r d 2 ψ d r 2 d ψ d r = g s 2 ( r ) 4 π 2 β 2 μ R ( d ψ / d r ) .
μ β 2 0 α R ( α ) d α = 1 4 π 2 0 r g s 2 ( r ) d r .
g s ( r ) = exp ( - r 2 / 2 ) R ( α ) = { 1 for α < π 2 0 for α > π 2 .
ψ ( r ) = π 2 0 r 1 - exp ( - r 2 ) d r .
| 1 2 π ξ 0 exp ( - ξ 2 / 2 ) exp { i β [ ϕ ( ξ ) - α ξ ] } d ξ | 1 2 π ξ 0 exp ( - ξ 2 / 2 ) d ξ < exp ( - ξ 0 2 / 2 ) 2 π ξ 0 .
1 2 π - - ξ 0 exp ( - ξ 2 / 2 ) exp { i β [ ϕ ( ξ ) - α ξ ] } d ξ 1 2 π - - ξ 0 exp ( - ξ 2 / 2 ) exp [ - i β ( α + π / 2 ) ξ ] d ξ .
1 2 π - - ξ 0 exp ( - ξ 2 / 2 ) exp { i β [ - ( α + π / 2 ) ξ ] } d ξ 1 2 π exp [ - ξ 0 2 / 2 - i β ( α + π / 2 ) ξ 0 ] ξ 0 - i β ( α + π / 2 ) .
| 1 2 π - - ξ 0 exp ( - ξ 2 / 2 ) exp { i β [ ϕ ( ξ ) - α ξ ] } d ξ | < exp ( - ξ 0 2 / 2 ) 2 π ξ 0 2 + β 2 π / 4 .
- exp ( - ξ 0 2 / 2 ) 2 π ( 1 ξ 0 + 1 ξ 0 2 + β 2 π / 4 ) .
2 π β Γ T ( α ) 2 - Γ ( α ) 2 < 2 β π exp ( - ξ 0 2 / 2 ) ( 1 ξ 0 + 1 ξ 0 2 + β 2 π / 4 ) .
2 β π exp ( - ξ 0 2 / 2 ) ( 1 ξ 0 + 1 ξ 0 2 + β 2 π / 4 ) < .
ψ ( ξ ) = β [ ϕ ( ξ ) - α ξ - γ ξ 2 ] .
d ψ d ξ ( ξ c ) = d 2 ψ d 2 ξ ( ξ c ) = 0.
π 2 erfc ( ξ c ) = α + 2 γ ξ c ,             exp ( - ξ c 2 ) = 2 γ .
ξ c = - ln ( 2 γ ) , d 3 ψ d 3 ξ ( ξ c ) = - 2 β ξ c exp ( - ξ c 2 ) = - 4 β γ - ln ( 2 γ ) .
α = π 2 erfc ( ξ c ) - 2 γ ξ c .
G ( ω ) = exp [ - ξ c 2 / 2 + i ψ ( ξ c ) ] 2 π × - exp { i β [ 1 6 d 3 ϕ d 3 ξ ( ξ c ) ξ 3 - ( α - α c ) ξ ] } d ξ .
G ( ω ) = exp ( - ξ c 2 / 2 ) exp [ i ψ ( ξ c ) ] [ - β 2 d 3 ϕ d 2 ξ ( ξ c ) ] - 1 / 3 Ai ( α * ) ,
α * = β ( α - α c ) [ - β 2 d 3 ϕ d 3 ξ ( ξ c ) ] 1 / 3 .
Max ( 2 π β Γ 2 ) = 4 π [ γ β - 4 ln ( 2 γ ) ] 1 / 3 [ Ai ( - 1 ) ] 2 .
Max ( 2 π β Γ 2 ) = 2.27 [ - β γ ln ( 2 γ ) ] 1 / 3 .
β | 1 6 d 3 ϕ d 3 ξ ( ξ c ) | 1.
- 2 3 β γ ln ( 2 γ ) 1.
2 π β Γ 2 = 1 + 4 β γ - 0.43 ln ( 2 γ 0 ) ,
- β γ 0 ln ( 2 γ 0 ) = 1 / 4.
γ 0 = - 1 4 β ln ( 1 2 β ) .
2 π β Γ ( α ) 2 = exp [ - ξ 2 ( α ) ] exp [ - ξ 2 ( α ) ] - 2 γ ,
π 2 erf [ ξ ( α ) ] - 2 γ ξ ( α ) = α .
ξ 1 ( γ ) = ln ( 2 γ - 0.2 1.6 γ ) ,
ξ 2 ( γ ) = ln ( 0.4 γ 2 γ - 0.8 ) .
Δ α = π 2 erf [ ξ 2 ( γ ) ] - 2 γ ξ 2 ( γ ) - π 2 erf [ ξ 1 ( γ ) ] - 2 γ ξ 1 ( γ ) .
2 π Δ α = - 3.88 γ - 3.53 γ 2

Metrics