Abstract

A beam shaping method is presented where a diffractive optical element (DOE) is designed by optimizing the complex mode coefficient weights of a set of Gaussian beam modes. This method is compared with the more standard unidirectional approach. Differential evolution is used for the optimization in both the unidirectional and Gaussian beam mode optimization methods. For the particular transforms carried out, the Gaussian beam mode set optimization (GBMSO) approach achieved more optimal solutions. The GBMSO approach is extended to design DOEs that control the amplitude distribution of a beam at multiple planes, rather than at just a single plane (i.e., the far field).

© 2010 Optical Society of America

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References

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    [CrossRef]
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2007

C. Rocha-Alicanoa, D. Covarrubias-Rosalesa, C. Brizuela-Rodrigueza, and M. Panduro-Mendozab, “Differential evolution algorithm applied to sidelobe level reduction on a planar array,” AEU, Int. J. Electron. Commun. 61, 286-290 (2007).
[CrossRef]

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913-1922 (2007).
[CrossRef] [PubMed]

2005

N. Trappe, R. Mahon, W. Lanigan, J. A. Murphy, and S. Withington, “The quasi-optical analysis of Bessel beams in the far infrared,” Infrared Phys. Technol. 46, 233-247 (2005).
[CrossRef]

2004

1999

J. Murphy, C. O'Sullivan, W. Lanigan, S. Withington, and N. Trappe, “Modal analysis of the quasi-optical performance of phase gratings,” Int. J. Infrared Millim. Waves 20, 1469-1486 (1999).
[CrossRef]

S. Monk, J. Arlt, D. Robertson, J. Courtial, and M. Padgett, “The generation of Bessel beams at millimetre-wave frequencies by use of an axicon,” Opt. Commun. 170, 213-215 (1999).
[CrossRef]

1995

1987

Abushagur, M.

Arlt, J.

S. Monk, J. Arlt, D. Robertson, J. Courtial, and M. Padgett, “The generation of Bessel beams at millimetre-wave frequencies by use of an axicon,” Opt. Commun. 170, 213-215 (1999).
[CrossRef]

Brizuela-Rodrigueza, C.

C. Rocha-Alicanoa, D. Covarrubias-Rosalesa, C. Brizuela-Rodrigueza, and M. Panduro-Mendozab, “Differential evolution algorithm applied to sidelobe level reduction on a planar array,” AEU, Int. J. Electron. Commun. 61, 286-290 (2007).
[CrossRef]

Courtial, J.

S. Monk, J. Arlt, D. Robertson, J. Courtial, and M. Padgett, “The generation of Bessel beams at millimetre-wave frequencies by use of an axicon,” Opt. Commun. 170, 213-215 (1999).
[CrossRef]

Covarrubias-Rosalesa, D.

C. Rocha-Alicanoa, D. Covarrubias-Rosalesa, C. Brizuela-Rodrigueza, and M. Panduro-Mendozab, “Differential evolution algorithm applied to sidelobe level reduction on a planar array,” AEU, Int. J. Electron. Commun. 61, 286-290 (2007).
[CrossRef]

Di Leonardo, R.

Durnin, J.

Goldsmith, P.

P. Goldsmith, Quasioptical Systems (IEEE, 1997).

Ianni, F.

Johnson, E.

Kathman, A.

D. O'Shea, T. Suleski, A. Kathman, and D. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

Kim, H.

Lampinend, J.

K. Price, R. Storn, and J. Lampinend, Differential Evolution (Springer, 2005).

Lanigan, W.

N. Trappe, R. Mahon, W. Lanigan, J. A. Murphy, and S. Withington, “The quasi-optical analysis of Bessel beams in the far infrared,” Infrared Phys. Technol. 46, 233-247 (2005).
[CrossRef]

J. Murphy, C. O'Sullivan, W. Lanigan, S. Withington, and N. Trappe, “Modal analysis of the quasi-optical performance of phase gratings,” Int. J. Infrared Millim. Waves 20, 1469-1486 (1999).
[CrossRef]

Lavelle, J.

J. Lavelle, “The design and optimisation of quasioptical telescopes,” Ph.D. dissertation (NUI Maynooth, 2008).

Lee, B.

Mahon, R.

N. Trappe, R. Mahon, W. Lanigan, J. A. Murphy, and S. Withington, “The quasi-optical analysis of Bessel beams in the far infrared,” Infrared Phys. Technol. 46, 233-247 (2005).
[CrossRef]

Monk, S.

S. Monk, J. Arlt, D. Robertson, J. Courtial, and M. Padgett, “The generation of Bessel beams at millimetre-wave frequencies by use of an axicon,” Opt. Commun. 170, 213-215 (1999).
[CrossRef]

Murphy, J.

J. Murphy, C. O'Sullivan, W. Lanigan, S. Withington, and N. Trappe, “Modal analysis of the quasi-optical performance of phase gratings,” Int. J. Infrared Millim. Waves 20, 1469-1486 (1999).
[CrossRef]

Murphy, J. A.

N. Trappe, R. Mahon, W. Lanigan, J. A. Murphy, and S. Withington, “The quasi-optical analysis of Bessel beams in the far infrared,” Infrared Phys. Technol. 46, 233-247 (2005).
[CrossRef]

O'Shea, D.

D. O'Shea, T. Suleski, A. Kathman, and D. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

O'Sullivan, C.

J. Murphy, C. O'Sullivan, W. Lanigan, S. Withington, and N. Trappe, “Modal analysis of the quasi-optical performance of phase gratings,” Int. J. Infrared Millim. Waves 20, 1469-1486 (1999).
[CrossRef]

Padgett, M.

S. Monk, J. Arlt, D. Robertson, J. Courtial, and M. Padgett, “The generation of Bessel beams at millimetre-wave frequencies by use of an axicon,” Opt. Commun. 170, 213-215 (1999).
[CrossRef]

Panduro-Mendozab, M.

C. Rocha-Alicanoa, D. Covarrubias-Rosalesa, C. Brizuela-Rodrigueza, and M. Panduro-Mendozab, “Differential evolution algorithm applied to sidelobe level reduction on a planar array,” AEU, Int. J. Electron. Commun. 61, 286-290 (2007).
[CrossRef]

Prather, D.

D. O'Shea, T. Suleski, A. Kathman, and D. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

Price, K.

K. Price, R. Storn, and J. Lampinend, Differential Evolution (Springer, 2005).

Robertson, D.

S. Monk, J. Arlt, D. Robertson, J. Courtial, and M. Padgett, “The generation of Bessel beams at millimetre-wave frequencies by use of an axicon,” Opt. Commun. 170, 213-215 (1999).
[CrossRef]

Rocha-Alicanoa, C.

C. Rocha-Alicanoa, D. Covarrubias-Rosalesa, C. Brizuela-Rodrigueza, and M. Panduro-Mendozab, “Differential evolution algorithm applied to sidelobe level reduction on a planar array,” AEU, Int. J. Electron. Commun. 61, 286-290 (2007).
[CrossRef]

Ruocco, G.

Soifer, V.

V. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

Storn, R.

K. Price, R. Storn, and J. Lampinend, Differential Evolution (Springer, 2005).

Suleski, T.

D. O'Shea, T. Suleski, A. Kathman, and D. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

Trappe, N.

N. Trappe, R. Mahon, W. Lanigan, J. A. Murphy, and S. Withington, “The quasi-optical analysis of Bessel beams in the far infrared,” Infrared Phys. Technol. 46, 233-247 (2005).
[CrossRef]

J. Murphy, C. O'Sullivan, W. Lanigan, S. Withington, and N. Trappe, “Modal analysis of the quasi-optical performance of phase gratings,” Int. J. Infrared Millim. Waves 20, 1469-1486 (1999).
[CrossRef]

Turunen, J.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Vch Verlagsgesellschaft Mbh, 1998).

Withington, S.

N. Trappe, R. Mahon, W. Lanigan, J. A. Murphy, and S. Withington, “The quasi-optical analysis of Bessel beams in the far infrared,” Infrared Phys. Technol. 46, 233-247 (2005).
[CrossRef]

J. Murphy, C. O'Sullivan, W. Lanigan, S. Withington, and N. Trappe, “Modal analysis of the quasi-optical performance of phase gratings,” Int. J. Infrared Millim. Waves 20, 1469-1486 (1999).
[CrossRef]

Wyrowski, F.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Vch Verlagsgesellschaft Mbh, 1998).

Yang, B.

AEU, Int. J. Electron. Commun.

C. Rocha-Alicanoa, D. Covarrubias-Rosalesa, C. Brizuela-Rodrigueza, and M. Panduro-Mendozab, “Differential evolution algorithm applied to sidelobe level reduction on a planar array,” AEU, Int. J. Electron. Commun. 61, 286-290 (2007).
[CrossRef]

Infrared Phys. Technol.

N. Trappe, R. Mahon, W. Lanigan, J. A. Murphy, and S. Withington, “The quasi-optical analysis of Bessel beams in the far infrared,” Infrared Phys. Technol. 46, 233-247 (2005).
[CrossRef]

Int. J. Infrared Millim. Waves

J. Murphy, C. O'Sullivan, W. Lanigan, S. Withington, and N. Trappe, “Modal analysis of the quasi-optical performance of phase gratings,” Int. J. Infrared Millim. Waves 20, 1469-1486 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

S. Monk, J. Arlt, D. Robertson, J. Courtial, and M. Padgett, “The generation of Bessel beams at millimetre-wave frequencies by use of an axicon,” Opt. Commun. 170, 213-215 (1999).
[CrossRef]

Opt. Express

Other

J. Lavelle, “The design and optimisation of quasioptical telescopes,” Ph.D. dissertation (NUI Maynooth, 2008).

D. O'Shea, T. Suleski, A. Kathman, and D. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

V. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Vch Verlagsgesellschaft Mbh, 1998).

K. Price, R. Storn, and J. Lampinend, Differential Evolution (Springer, 2005).

P. Goldsmith, Quasioptical Systems (IEEE, 1997).

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Figures (9)

Fig. 1
Fig. 1

For a DOE designed using unidirectional optimization, the global optimization algorithm optimizes the depth of each element of the DOE.

Fig. 2
Fig. 2

The results of the optimizations to transform a 100 mm radius Gaussian input field to a 1° radius rect ( ) function far-field distribution. (a) The phase that gives the DOE profile designed by unidirectional optimization (dotted line) and GBMSO (solid line). (b) The amplitude distribution given by the GBMs at the DOE plane (solid line) and the target amplitude distribution (dashed line). (c) The far-field target amplitude distribution (dashed line), the far-field amplitude distribution given by the GBMs at the DOE plane (solid line), the far-field amplitude distribution from the DOE designed using GBMSO with the Gaussian amplitude input distribution (thick dotted-dashed line), and the far-field amplitude distribution from the DOE designed by the unidirectional method (dotted line).

Fig. 3
Fig. 3

The results of the optimizations to transform a 100 mm radius Gaussian input field to a 10° radius rect ( ) function far-field distribution. (a) The phase that gives the DOE profile designed by unidirectional optimization (dotted line) and GBMSO (solid line). (b) The amplitude distribution given by the GBMs at the DOE plane (solid line) and the target amplitude distribution (dashed line). (c) The far-field target amplitude distribution (dashed line), the far-field amplitude distribution given by the GBMs at the DOE plane (solid line), the far-field amplitude distribution from the DOE designed using GBMSO with the Gaussian amplitude input distribution (thick dotted-dashed line), and the far-field amplitude distribution from the DOE designed by the unidirectional method (dotted line).

Fig. 4
Fig. 4

The upper and lower limits of the random values for the radius of the mode coefficients with which the optimization is initialized were chosen to encompass the likely optimal value (see text). The 20th order mode, the highest order mode used in the optimizations, is shown for a radius at the lower and upper limits (thick solid and dotted-dashed lines, respectively), along with the target amplitude distributions (dashed line). In the far field, the mode radius is given a divergence angle, whereas at the DOE plane, the mode radius is given as a distance. (a) The target (input) Gaussian amplitude distribution for the 1° target far-field amplitude distribution. (b) The target (output) rect ( ) function amplitude distribution for the 1° target far-field amplitude distribution. (c) The target (input) Gaussian amplitude distribution for the 10° target far-field amplitude distribution. (d) The target (output) rect ( ) function amplitude distribution for the 10° target far-field amplitude distribution.

Fig. 5
Fig. 5

The merit function of Eq. (16) calculates the overlap of the GBM field at the target line and target Gaussian distributions to achieve a non-diffracting beam of constant amplitude at the center of the beam. The propagation distance z b from the DOE plane to the non-diffracting region and the radius of the input Gaussian beam incident on the DOE are optimized. The reference plane, z = 0 , of the GBMs is located equidistant between the two target Gaussians at ± z a .

Fig. 6
Fig. 6

The results of the optimization to achieve a non-diffracting beam. (a) The amplitude of the mode coefficients found by the optimization. (b) The target amplitude distribution at the DOE plane (dashed line) shown with the amplitude distribution of the optimized solution at this plane (solid line). (c) The phase that gives the shape of the DOE.

Fig. 7
Fig. 7

The amplitude at the center of the beam as a function of distance of the non-diffracting beam. The solid line shows the optimized field given by Eq. (14) and the dashed line shows the field from the DOE with the Gaussian input field.

Fig. 8
Fig. 8

The intensity of the field from the DOE which produces a non-diffracting beam as a function of distance.

Fig. 9
Fig. 9

Plots of the amplitude of the field from the axicon (thick solid line) and the field from the DOE designed by optimizing GBMs (thin solid line). These fields are shown with the target 5 mm radius target Gaussian distribution (dashed line).

Tables (2)

Tables Icon

Table 1 Results of the Optimization to Convert a λ = 3   mm 100 mm Gaussian Field to Target 1° and 10° Radii rect ( ) Function Amplitude Distributions, Using the Unidirectional and GBMSO Methods a

Tables Icon

Table 2 Range of z Values after Optimization Was Initialized with Random Values w and w 0, and Values of w 0 and z Found by the Optimization

Equations (19)

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v i , g = x r 0 , g + F ( x r 1 , g x r 2 , g ) ,
u i , g = { v i , g if   ( rand ( 0 , 1 ) C ) r x i , g otherwise , }
E ( k x , k y ) = F { T i n ( x , y ) exp [ i ϕ DOE ( x , y ) ] } ,
MSE = 1 N i = 1 N ( E ( k i , 0 ) T o u t ( k i , 0 ) ) 2 ,
PC = ( i = 1 N | E ( k i , 0 ) | T o u t ( k i , 0 ) Δ k ) 2 ,
E ( x , y , z ) = M N a n , m   exp ( i ϕ n , m ) ψ n , m ( w 0 , x , y , z ) M N a n , m 2 ,
ψ n , m ( w 0 , x , y , z ) = 2 ( 1 / 2 ) n π w n ! H m ( 2 x w ) H n ( 2 y w ) exp [ x 2 + y 2 w 2 i k z i π ( x 2 + y 2 ) R ( z ) λ + i ϕ 0 ( m + n + 1 ) 2 ] ,
w ( z ) = ( z 2 λ 2 π 2 w 0 4 + 1 ) w 0 2 ,     R ( z ) = z ( π 2 w 0 4 z 2 λ 2 + 1 ) ,
ϕ 0 ( z ) = tan 1 ( z λ π w 0 2 ) .
η = | E ( x , y , z ) | T i n ( x , y ) d x d y + | E ( θ x , θ y , z + z ¯ ) | T o u t ( θ x , θ y ) d θ x d θ y ,
ϕ DOE ( x , y ) = arg [ E ( x , y , z o ) ] ,
E DOE ( x , y ) = T i n ( x , y ) exp [ i ϕ DOE ( x , y ) ] .
z ( w 0 , w ) = ± π w 0 2 λ w 2 w 0 2 1 .
E f ( k x , k y ) = F { T i n ( x , y ) exp ( i   arg [ E DOE ( x , y ) ] ) } .
E ( r , z ) = E 0   exp ( i k z z ) J 0 ( k r r ) ,
E L ( x , y , z ) = P M a p , m   exp ( i ϕ p , m ) ψ p , m ( w 0 , x , y , z ) M N a p , m 2 ,
ψ p m = 2 π p ! ( p + | m | ) ! 2 | m | / 2 ( r w ) | m | w L p | m | ( 2 θ 2 w 2 ) exp ( i m ϕ r 2 w 2 ) ,
η ¯ = 2 π ψ 0 , 0 ( w spot , r , 0 ) ( | E L ( r , 0 , z a ) | + | E L ( r , 0 , + z a ) | ) r d r + W z 1 z 2 ψ 0 , 0 ( w , x = 0 , z = 0 ) | E L ( r , 0 , + z ) | d z z 1 z 2 ψ 0 , 0 ( w , x = 0 , z = 0 ) 2 d z z 1 z 2 | E L ( r , 0 , + z ) | 2 d z + 2 π ψ 0 , 0 ( w t , r , 0 ) | E L ( r , 0 , + z ) | r d r ,
E DOE ( r , θ ) = ψ 0 , 0 ( w t , r , 0 ) exp ( i   arg [ E ( r , θ , z b ) ] ) .

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