Abstract

We compare the iterative angular spectrum (IAS) and the optimal rotation angle (ORA) methods in designing two-dimensional finite aperture diffractive optical elements (FADOEs) used as beamfanners. The transfer functions of both methods are compared analytically in the spatial frequency domain. We have designed several structures of 1-to-4 and 1-to-6 beamfanners to investigate the differences in the performance of the beamfanners designed by ORA method for near field operation. Using the three-dimensional finite difference time-domain (3-D FDTD) method with perfect matched layer (PML) absorbing boundary condition (ABC), the diffraction efficiency is calculated for each designed FADOE and the corresponding values are compared.

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References

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  1. J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express 7(6), 237–242 (2000).
    [CrossRef] [PubMed]
  2. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14(1), 34–43 (1997).
    [CrossRef]
  3. D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15(6), 1599–1607 (1998).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005).
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    [CrossRef]
  6. J. Stigwall and J. Bengtsson, “Design of array of diffractive optical elements with inter-element coherent fan-outs,” Opt. Express 12(23), 5675–5683 (2004).
    [CrossRef] [PubMed]
  7. R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–250 (1972).
  8. F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5(7), 1058–1065 (1988).
    [CrossRef]
  9. F. Di, Y. Yingbai, J. Guofan, T. Qiaofeng, and H. Liu, “Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm,” J. Opt. Soc. Am. A 20(9), 1739–1746 (2003).
    [CrossRef]
  10. T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
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  11. S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express 8, 706–722 (2001).
    [CrossRef]
  12. D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave and Guided Wave Letters 7(7), 184–186 (1997).
    [CrossRef]
  13. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press series on RF and microwave technology, 2000.

2008 (1)

2004 (1)

2003 (1)

2001 (1)

S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express 8, 706–722 (2001).
[CrossRef]

2000 (1)

1998 (1)

1997 (3)

1988 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–250 (1972).

Bengtsson, J.

Bryngdahl, O.

Collins, J. P.

Di, F.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–250 (1972).

Guofan, J.

Jabbour, T. G.

Jiang, J.

Kuebler, S. M.

Liu, H.

Mait, J. N.

Mellin, S. D.

S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express 8, 706–722 (2001).
[CrossRef]

Mirotznik, M. S.

Nordin, G. P.

S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express 8, 706–722 (2001).
[CrossRef]

J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express 7(6), 237–242 (2000).
[CrossRef] [PubMed]

Prather, D. W.

Qiaofeng, T.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–250 (1972).

Stigwall, J.

Sullivan, D. M.

D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave and Guided Wave Letters 7(7), 184–186 (1997).
[CrossRef]

Wyrowski, F.

Yingbai, Y.

Appl. Opt. (1)

IEEE Microwave and Guided Wave Letters (1)

D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave and Guided Wave Letters 7(7), 184–186 (1997).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Express (4)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–250 (1972).

Other (2)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005).

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press series on RF and microwave technology, 2000.

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Figures (5)

Fig. 1
Fig. 1

Geometry of 2D finite aperture DOE and the two regions used in the IAS method.

Fig. 2
Fig. 2

Geometry of DOE and the spots used in the ORA method.

Fig. 3
Fig. 3

The middle cross sections of the transfer functions for IAS (...) and ORA (___) methods at zobs=10λ, 30λ and 100λ

Fig. 4
Fig. 4

The middle cross sections of typical DOE profiles designed by (a) IAS and (b) ORA methods for zobs=10λ case.

Fig. 5
Fig. 5

Cross section of the analyzed region.

Tables (1)

Tables Icon

Table 1 The diffraction efficiency (DE) obtained by FDTD method for designed 1-to-4 and 1-to-6 beamfanners.

Equations (10)

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t(x,y)=τexp[jϕ(x,y)]Π(x/Lx,y/Ly),
U0(fx,fy)=P0u0(x,y)exp[j2π(fxx+fyy)]dxdy.
u1(x,y)=U0exp[jk2zobs1(λ2fx)2(λ2fy)2]exp[j2π(fxx+fyy)]dfxdfy,
H(fx,fy)=exp[jk2zobs1(λ2fx)2(λ2fy)2]
u0,k=Akincexp[j(ϕkinc+ϕk)],
u1,m=ku0,khk,m,
hk,m=14π[jk2zobs(jk21/rkmc)/rkmc]4exp(jk2rkmc)rkmcsin(k˜xa2)sin(k˜yb2)k˜xk˜y,
h(x,y;zobs)=14π[jk2zobsr(jk21r)]exp(jk2r)r,
u1(x,y)=P0u0(η,ξ)h(xη,yξ;zobs)dηdξ.
DE=WI(x,y)dxdyP1I(x,y)dxdy

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