Abstract

Laser beam shaping by phase-only transformations, often referred to as field mapping, has for a long time been considered wavelength dependent. In this Letter we outline a simple mathematical argument that shows how the problem may be formulated in a wavelength tunable manner, requiring only a minor adjustment in the observation plane. We verify the theoretical prediction by experiment using the example of a Gaussian-to-flattop-beam transformation, and we show that the shaping is valid across a wide range of wavelengths for a single diffractive optical element.

© 2012 Optical Society of America

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References

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  1. F. M. Dickey and S. C. Holswade, Laser Beam Shaping—Theory and Techniques (Marcel Dekker, 2000).
  2. L. A. Romero and F. M. Dickey, J. Opt. Soc. Am. A 13, 751 (1996).
    [CrossRef]
  3. F. M. Dickey and S. C. Holswade, Opt. Eng. 35, 3285 (1996).
    [CrossRef]
  4. J. A. Hoffnagle and C. M. Jefferson, Appl. Opt. 39, 5488 (2000).
    [CrossRef]
  5. T. Stone and N. George, Appl. Opt. 27, 2960 (1988).
    [CrossRef]
  6. I. A. Litvin and A. Forbes, Opt. Lett. 34, 2991 (2009).
    [CrossRef]

2009

2000

1996

L. A. Romero and F. M. Dickey, J. Opt. Soc. Am. A 13, 751 (1996).
[CrossRef]

F. M. Dickey and S. C. Holswade, Opt. Eng. 35, 3285 (1996).
[CrossRef]

1988

Dickey, F. M.

L. A. Romero and F. M. Dickey, J. Opt. Soc. Am. A 13, 751 (1996).
[CrossRef]

F. M. Dickey and S. C. Holswade, Opt. Eng. 35, 3285 (1996).
[CrossRef]

F. M. Dickey and S. C. Holswade, Laser Beam Shaping—Theory and Techniques (Marcel Dekker, 2000).

Forbes, A.

George, N.

Hoffnagle, J. A.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Opt. Eng. 35, 3285 (1996).
[CrossRef]

F. M. Dickey and S. C. Holswade, Laser Beam Shaping—Theory and Techniques (Marcel Dekker, 2000).

Jefferson, C. M.

Litvin, I. A.

Romero, L. A.

Stone, T.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

F. M. Dickey and S. C. Holswade, Opt. Eng. 35, 3285 (1996).
[CrossRef]

Opt. Lett.

Other

F. M. Dickey and S. C. Holswade, Laser Beam Shaping—Theory and Techniques (Marcel Dekker, 2000).

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Figures (3)

Fig. 1.
Fig. 1.

Experimental results for the conversion of the Gaussian beam into a flattop beam for (a) the design wavelength and (b)–(d) nondesign wavelengths. The wavelengths used are shown as text insets.

Fig. 2.
Fig. 2.

Experimental results for the propagation of the 9.6 µm beam from the Gaussian to the flattop beam after the DOE and lens combination, for distances from the DOE of 230–300 mm. The flattop beam is found in a region 265–275 mm after the lens, as predicted by theory.

Fig. 3.
Fig. 3.

Numerical simulation of the flattop beams generated after the DOE of Eq. 6a, showing that indeed the beam shaping problem is wavelength independent.

Equations (8)

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u(x,y,z)=1izλexp(ikz)exp(iπ(x2+y2)zλ)×G(ξ,η)exp(i2πr0w0fλϕ(ξ,η))×exp(iπfλ(ξ2+η2)+iπzλ(ξ2+η2))×exp(i2π(xξ+yη)fλ)dξdη,
fλ=zλorz=fλλ,
u(x,y,z)=1ifλexp(ikf)exp(iπ(x2+y2)fλ)×G(ξ,η)exp(i2πr0w0fλϕ(ξ,η))×exp(i2π(xξ+yη)fλ)dξdη.
u(x,y,z)=1izλexp(ikz)exp(iπ(x2+y2)zλ)×G(ξ,η)exp(iψ(ξ,η))exp(iπzλ[ξ2+η2])×exp(i2π(xξ+yη)zλ)dξdη.
ψ(ξ,η)=ψ(ξ,η)iπ(ξ2+η2)/zλ
ϕ(ξ,η)=βπ20ξ2+η2w01exp(ρ2)dρ
r0=βfλ2πw0.
n(λ)1n(λ)1.

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