Abstract

Interest has recently been shown in methods of converting the plane wave from a laser in the uniphase TEM0,0 mode to a plane wave having (a) uniform irradiance over a required cross section, and (b) all the power of the original beam. Two methods are proposed for accomplishing these aims: one employs two plano-aspheric lenses; the other requires a pair of selectively aberrated lens systems. A computer program has been written which determines the aspherics, and one example is presented. The aberrations required of the second method are expressed algebraically in terms of known quantities. These aberrations could conceivably be designed into a system of spherical lenses, by use of one of the automatic lens design programs now extant.

© 1965 Optical Society of America

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References

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  1. D. Dutton, M. P. Givens, R. E. Hopkins, Am. J. Phys. 32, 356 (1964).
    [CrossRef]
  2. P. J. Brownscombe, U. S. PatentNo. 3,076,377 (1963).
  3. Melvin H. Horman, Appl. Opt. 4, 336 (1965).
    [CrossRef]
  4. G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
  5. R. C. Rempel, Spectra-Physics Laser Technical Bulletin No. 1 (Spectra-Physics, Inc., Mountain View, Calif., 1963), p. 4.
  6. Institute of Optics, Summer School Notes, Vol. 2 (1962).

1965 (1)

Melvin H. Horman, Appl. Opt. 4, 336 (1965).
[CrossRef]

1964 (1)

D. Dutton, M. P. Givens, R. E. Hopkins, Am. J. Phys. 32, 356 (1964).
[CrossRef]

1962 (1)

Institute of Optics, Summer School Notes, Vol. 2 (1962).

1961 (1)

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Brownscombe, P. J.

P. J. Brownscombe, U. S. PatentNo. 3,076,377 (1963).

Dutton, D.

D. Dutton, M. P. Givens, R. E. Hopkins, Am. J. Phys. 32, 356 (1964).
[CrossRef]

Givens, M. P.

D. Dutton, M. P. Givens, R. E. Hopkins, Am. J. Phys. 32, 356 (1964).
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Hopkins, R. E.

D. Dutton, M. P. Givens, R. E. Hopkins, Am. J. Phys. 32, 356 (1964).
[CrossRef]

Horman, Melvin H.

Melvin H. Horman, Appl. Opt. 4, 336 (1965).
[CrossRef]

Rempel, R. C.

R. C. Rempel, Spectra-Physics Laser Technical Bulletin No. 1 (Spectra-Physics, Inc., Mountain View, Calif., 1963), p. 4.

Am. J. Phys. (1)

D. Dutton, M. P. Givens, R. E. Hopkins, Am. J. Phys. 32, 356 (1964).
[CrossRef]

Appl. Opt. (1)

Melvin H. Horman, Appl. Opt. 4, 336 (1965).
[CrossRef]

Bell System Tech. J. (1)

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Summer School Notes (1)

Institute of Optics, Summer School Notes, Vol. 2 (1962).

Other (2)

R. C. Rempel, Spectra-Physics Laser Technical Bulletin No. 1 (Spectra-Physics, Inc., Mountain View, Calif., 1963), p. 4.

P. J. Brownscombe, U. S. PatentNo. 3,076,377 (1963).

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram illustrating rectification problem.

Fig. 2
Fig. 2

Solution by plano-aspheric lenses, showing parameters involved.

Fig. 3
Fig. 3

Illustrative example showing aspheric surfaces S, S′ and the wavefront incident upon S′. For this case r0 = X = 2 mm, L = 20 mm, α = 1.4 mm, t = t′ = 5 mm, and n = n′ = 1.5. Note the extremely small and gradual departures from a plane of surfaces S and S′, conducive to fabrication.

Equations (25)

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E ( r ) = ( 2 π α 2 ) 1 [ 1 exp ( r 0 2 / 2 α 2 ) ] 1 P exp ( r 2 / 2 α 2 )
p A ( r ) = P [ 1 exp ( r 2 / 2 α 2 ) ] / [ 1 exp ( r 0 2 / 2 α 2 ) ] .
p B ( x ) = P ( x / X ) 2 .
x ( r ) = ± X [ 1 exp ( r 2 / 2 α 2 ) ] ½ / [ 1 exp ( r 0 2 / 2 α 2 ) ] ½ .
tan μ = [ x ( r ) r ] / ( L z ) ,
sin γ = n sin θ = n z ( r ) / [ 1 + z ( r ) 2 ] ½ .
tan γ = n z ( r ) / [ 1 ( n 2 1 ) z ( r ) 2 ] ½ .
tan ( γ θ ) = ( tan γ tan θ ) / ( 1 + tan γ tan θ )
n z z ( 1 α 0 2 z 2 ) ½ n z 2 + ( 1 α 0 2 z 2 ) ½ = x ( r ) r L z .
z ( r ) = X L ( n 1 ) ( 1 e r 0 2 / 2 α 2 ) 1 / 2 0 r ( 1 e r 0 2 / 2 α 2 ) ½ d ρ r 2 2 L ( n 1 ) + t .
r = r + [ L + t ( n 1 ) n z ( r ) ] sin μ ,
z = z ( r ) + [ L + t ( n 1 ) n z ( r ) ] cos μ .
sin μ = x ( r ) r { [ L z ( r ) ] 2 + [ x ( r ) r ] 2 } ½ , and
cos μ = L z ( r ) { [ L z ( r ) ] 2 + [ x ( r ) r ] 2 } ½ .
( n n ) 2 ( z ) 2 ( Z ) 2 + ( n z + n Z ) 2 = ( z + Z ) 2 ,
z ( r ) = X [ L ( n 1 ) ] 1 [ 1 exp ( r 2 / 2 α 2 ) ] ½ × [ 1 exp ( r 0 2 / 2 α 2 ) ½ r [ L ( n 1 ) ] 1 .
Z 0 = t y 0 = 0 Z i = Z i 1 + Z ( y i ) [ x ( r i ) y i 1 ] 1 + Z ( y i ) tan μ i y i = x ( r i ) Z i tan μ i tan μ i = [ x ( r i ) r i ] / [ L z ( r i ) ] ,
O P D = L + t ( n 1 ) + t ( n 1 ) n z ( r ) n Z ( y ) s .
x ( r ) = R ϕ ( r ) / r
ϕ ( r ) = a 1 r 2 + a 2 r 4 + . . . + a 5 r 10
x ( r ) = R r ( 2 a 1 + 4 a 2 r 2 + 6 a 3 r 4 + + . . . + + 10 a 5 r 8 ) .
x ( r ) = 2 ½ K ( r / α ) [ 1 8 1 ( r / α ) 2 + ( 5 / 384 ) ( r / α ) 4 + 0 ( r / α ) 6 ( 71 / 92 , 160 ) ( r / α ) 8 + . . . ] ,
a 1 = + 2 3 / 2 ( K / R ) α 1 , a 2 = 2 ½ ( 1 / 32 ) ( K / R ) α 3 , a 3 = + 2 ½ ( 5 / 2304 ) ( K / R ) α 5 , a 4 = 0 , a 5 = 2 ½ ( 71 / 921 , 600 ) ( K / R ) α 9 .
θ ( ρ ) = b 1 ρ 2 + b 2 ρ 4 + . . . + b 5 ρ 10
Δ ( ρ i ) = b 1 ρ i 2 + b 2 ρ i 4 + . . . + b 5 ρ i 10

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