Abstract

The design of a two-mirror optical system for reshaping the irradiance distribution of a laser beam is presented. The second mirror is decentered relative to the first to eliminate the obscuration inherent in an axially symmetric design. A geometric-optics approach is used to derive a set of equations that describe the surface figures of each of the mirrors. (In general, the mirror surfaces are not rotationally symmetric.) The special case of a system to convert a Gaussian input beam into a uniform output distribution is considered. The expressions for the surface figures are evaluated numerically for several specific systems to provide illustrative examples. It is observed that in some cases rotationally symmetric aspheres may be used to construct the beam-shaping system.

© 1992 Optical Society of America

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References

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  1. P. W. Scott, W. H. Southwell, “Reflective optics for irradiance redistribution of laser beams: design,” Appl. Opt. 20, 1606–1610 (1981).
    [CrossRef] [PubMed]
  2. J. W. Ogland, “Mirror system for uniform beam transformation in high-power annular lasers,” Appl. Opt. 17, 2917–2923 (1978).
    [CrossRef] [PubMed]
  3. W. L. Hales, D. Korsch, “Design and analysis of afocal, two-mirror systems for arbitrary intensity transformations,” Tech. Rep. RH-81-2 (U. S. Army Missile Command, Redstone Arsenal, Alabama, government accession AD-A094439, 1980).
  4. W. B. Wetherell, “Afocal lenses,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987).
  5. D. F. Cornwell, “Intensity redistribution,” J. Opt. Soc. Am. 69, 1456 (1979).
  6. 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]
  7. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
    [CrossRef]
  8. D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
    [CrossRef]
  9. S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
    [CrossRef]
  10. M. A. Karim, A. K. Cherri, A. A. S. Awwal, A. Basit, “Refracting system for annular laser beam transformation,” Appl. Opt. 26, 2446–2449 (1987).
    [CrossRef] [PubMed]
  11. 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]
  12. M. T. Eismann, A. M. Tai, J. M. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. 28, 2641–2650 (1989).
    [CrossRef] [PubMed]
  13. N. C. Roberts, “Beamshaping by holographic filters,” Appl. Opt. 28, 31–32 (1989).
    [CrossRef] [PubMed]
  14. C. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  15. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

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]

1989 (3)

1987 (1)

1983 (1)

1982 (1)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

1981 (1)

1980 (1)

1979 (1)

D. F. Cornwell, “Intensity redistribution,” J. Opt. Soc. Am. 69, 1456 (1979).

1978 (1)

1965 (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]

Awwal, A. A. S.

Basit, A.

Cederquist, J. M.

Cherri, A. K.

Cornwell, D. F.

D. F. Cornwell, “Intensity redistribution,” J. Opt. Soc. Am. 69, 1456 (1979).

Eismann, M. T.

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]

Frieden, B. R.

Hales, W. L.

W. L. Hales, D. Korsch, “Design and analysis of afocal, two-mirror systems for arbitrary intensity transformations,” Tech. Rep. RH-81-2 (U. S. Army Missile Command, Redstone Arsenal, Alabama, government accession AD-A094439, 1980).

Han, C.

Ishii, Y.

Jahan, S. R.

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

Karim, M. A.

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

M. A. Karim, A. K. Cherri, A. A. S. Awwal, A. Basit, “Refracting system for annular laser beam transformation,” Appl. Opt. 26, 2446–2449 (1987).
[CrossRef] [PubMed]

Korsch, D.

W. L. Hales, D. Korsch, “Design and analysis of afocal, two-mirror systems for arbitrary intensity transformations,” Tech. Rep. RH-81-2 (U. S. Army Missile Command, Redstone Arsenal, Alabama, government accession AD-A094439, 1980).

Murata, K.

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]

Ogland, J. W.

Rhodes, P. W.

Roberts, N. C.

Scott, P. W.

Shafer, D.

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Shealy, D. L.

Southwell, W. H.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

Tai, A. M.

Wetherell, W. B.

W. B. Wetherell, “Afocal lenses,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987).

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

D. F. Cornwell, “Intensity redistribution,” J. Opt. Soc. Am. 69, 1456 (1979).

Opt. Eng. (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]

Opt. Laser Technol. (2)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

Other (3)

W. L. Hales, D. Korsch, “Design and analysis of afocal, two-mirror systems for arbitrary intensity transformations,” Tech. Rep. RH-81-2 (U. S. Army Missile Command, Redstone Arsenal, Alabama, government accession AD-A094439, 1980).

W. B. Wetherell, “Afocal lenses,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987).

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

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Figures (7)

Fig. 1
Fig. 1

Optical schematic of the two-mirror beam-shaping system: (a) side view, (b) perspective view.

Fig. 2
Fig. 2

Surface profile of the first mirror of the beam-shaping system for parameter set 1.

Fig. 3
Fig. 3

Surface profile of the second mirror of the beam-shaping system for parameter set 1.

Fig. 4
Fig. 4

Surface profile of the first mirror of the beam-shaping system for parameter set 2. (The corresponding plot for parameter set 3 is essentially identical.)

Fig. 5
Fig. 5

Surface profile of the second mirror of the beam-shaping system for parameter set 2. (The corresponding plot for parameter set 3 is essentially identical.)

Fig. 6
Fig. 6

Plot of the difference between the horizontal profile and the vertical profile of the first surface of the beam-shaping system. This plot shows the amount of rotational asymmetry for all three parameter sets.

Fig. 7
Fig. 7

Plot of the difference between the horizontal profile and the vertical profile of the second surface of the beam-shaping system. This plot shows the amount of rotational asymmetry for all three parameter sets.

Tables (1)

Tables Icon

Table 1 Parameter Sets Used for Illustrative Examples of a Beam-Shaping System that Transforms a Gaussian Input Beam into a Uniform Output Beam

Equations (23)

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B ^ = ( u - x ) i + ( v - y - h ) j + ( L + F - G ) k [ ( u - x ) 2 + ( v - y - h ) 2 ( L + F - G ) 2 ] 1 / 2 .
B ^ = - 2 ( A ^ · n ^ ) n ^ + A ^ ,
n ^ = - G x i - G y j + k [ 1 + G x 2 + G y 2 ] 1 / 2 ,
B ^ = - 2 G x i - 2 G y j + ( 1 - G x 2 - G y 2 ) k [ 1 + G x 2 + G y 2 ] .
G x u - x = G y v - y - h .
- 2 G x u - x = 1 - G x 2 - G y 2 L + F - G .
G x = ( u - x ) ( L + F - G ) ( u - x ) 2 + ( v - y - h ) 2 ± ( u - x ) [ ( L + F - G ) 2 + ( u - x ) 2 + ( v - y - h ) 2 ] 1 / 2 ( u - x ) 2 + ( v - y - h ) 2 .
OPL = L 2 + h 2 .
OPL = [ ( L - G + F ) 2 + ( h + y - v ) 2 + ( x - u ) 2 ] 1 / 2 + F - G .
F - G = - ( x - u ) 2 - ( y - v + h ) 2 + h 2 2 ( L + l 0 ) ,
l 0 = L 2 + h 2 .
G x = x - u L + l 0 ,
G y = y - v + h L + l 0 .
G r = G x x r + G y y r = 1 L + l 0 [ r - p + h r sin θ ] ,
G θ = G x x θ + G y y θ = 1 L + l 0 [ h r cos θ ] .
G ( r , θ ) = 1 L + l 0 { 0 r [ r - ρ ] d r + h r sin θ } .
F ( ρ , θ ) = 1 L + l 0 { 0 ρ [ r - ρ ] d ρ + h ρ sin θ } .
0 ρ = f ( r ) I out ( ρ ) ρ d ρ = 0 r I in ( r ) r d r ,
0 ρ 0 I out ( ρ ) ρ d ρ = 0 r 0 I in ( r ) r d r ,
f ( r ) = ρ = 2 σ [ 1 - exp ( - r 2 2 σ 2 ) 1 - exp ( - 4.5 ) ] 1 / 2 .
G ( r , θ ) = σ 2 L + l 0 { 1 2 ( r σ ) 2 - 2 [ 1 - exp ( - 4.5 ) ] 1 / 2 × 0 r / σ [ 1 - exp ( - r 2 2 ) ] 1 / 2 d r + h σ ( r σ ) sin θ } ,
F ( ρ , θ ) = σ 2 L + l 0 × ( 2 0 ρ / σ { - ln [ 1 - 1 - exp ( - 4.5 ) 4 ρ 2 ] } 1 / 2 × d p - 1 2 ( ρ σ ) 2 + h σ ( ρ σ ) sin θ ) .
α = tan - 1 ( h L + l 0 ) .

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