Abstract

An algorithm is reported for the design of a phase-only diffractive optical element (DOE) that reshapes a beam focused using a high numerical aperture (NA) lens. The vector diffraction integrals are used to relate the field distributions in the DOE plane and focal plane. The integrals are evaluated using the chirp-z transform and computed iteratively within the Method of Generalized Projections (MGP) to identify a solution that simultaneously satisfies the beam shaping and DOE constraints. The algorithm is applied to design a DOE that transforms a circularly apodized flat-top beam of wavelength λ to a square irradiance pattern when focused using a 1.4-NA objective. A DOE profile is identified that generates a 50λ×50λ square irradiance pattern having 7% uniformity error and 74.5% diffraction efficiency (fraction of focused power). The diffraction efficiency and uniformity decrease as the size of the focused profile is reduced toward the diffraction limited spot size. These observations can be understood as a manifestation of the uncertainty principle.

© 2008 Optical Society of America

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References

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2007 (1)

2006 (4)

2004 (2)

2003 (1)

D. L. Shealy and S. H. Chao, "Geometric optics-based design of laser beam shapers," Opt. Eng. 42, 3123-3138 (2003).
[CrossRef]

2002 (3)

2001 (1)

2000 (1)

M. Johansson and J. Bengtsson, "Robust design method for highly efficient beam-shaping diffractive optical elements using iterative-Fourier-transform algorithm with soft operations," J. Mod. Opt. 47, 1385-1398 (2000).
[CrossRef]

1998 (1)

V. V. Kotlyar, P. G. Seraphimovich, and V. A. Soifer, "An iterative algorithm for designing diffractive optical elements with regularization," Opt. Laser Eng. 29, 261-268 (1998).
[CrossRef]

1997 (2)

1995 (1)

1993 (1)

R. Kant, "An analytical solution of vector diffraction for focusing optical systems with Seidal aberrations I. Spherical aberration, curvature of field, and distortion," J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

1990 (2)

1984 (1)

1974 (1)

1959 (1)

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
[CrossRef]

Aagedal, H.

Bakx, J. L.

Bengtsson, J.

M. Johansson and J. Bengtsson, "Robust design method for highly efficient beam-shaping diffractive optical elements using iterative-Fourier-transform algorithm with soft operations," J. Mod. Opt. 47, 1385-1398 (2000).
[CrossRef]

Beth, T.

Bryngdhal, O.

Caley, A. J.

J. S. Liu, A. J. Caley, and M. R. Taghizadeh, "Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms," Opt. Commun. 267, 347-355 (2006).
[CrossRef]

Chan, Y. C.

Chao, S. H.

D. L. Shealy and S. H. Chao, "Geometric optics-based design of laser beam shapers," Opt. Eng. 42, 3123-3138 (2003).
[CrossRef]

Dickey, F. M.

Dorrer, C.

Dowd, P.

Egner, S.

Honkanen, M.

Jabbour, T. G.

Johansson, M.

M. Johansson and J. Bengtsson, "Robust design method for highly efficient beam-shaping diffractive optical elements using iterative-Fourier-transform algorithm with soft operations," J. Mod. Opt. 47, 1385-1398 (2000).
[CrossRef]

Kant, R.

R. Kant, "An analytical solution of vector diffraction for focusing optical systems with Seidal aberrations I. Spherical aberration, curvature of field, and distortion," J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

Kim, H.

Kotlyar, V. V.

V. V. Kotlyar, P. G. Seraphimovich, and V. A. Soifer, "An iterative algorithm for designing diffractive optical elements with regularization," Opt. Laser Eng. 29, 261-268 (1998).
[CrossRef]

Kuebler, S. M.

Kuittinen, M.

Lam, Y. L.

Lasser, T.

Lee, B.

Leitgeh, R. A.

Leutenegger, M.

Levi, A.

Li, Y.-P.

Liu, J. S.

J. S. Liu, A. J. Caley, and M. R. Taghizadeh, "Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms," Opt. Commun. 267, 347-355 (2006).
[CrossRef]

J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
[CrossRef]

Müller-Quade, J.

Piestun, R.

R. Piestun, and J. Shamir, "Synthesis of three-dimensional light fields and applications," Proceedings of the IEEE 90, 222-244 (2002).
[CrossRef]

Piquero, G.

Ramírez-Sánchez, V.

Rao, R.

Romero, L. A.

Schmid, M.

Seraphimovich, P. G.

V. V. Kotlyar, P. G. Seraphimovich, and V. A. Soifer, "An iterative algorithm for designing diffractive optical elements with regularization," Opt. Laser Eng. 29, 261-268 (1998).
[CrossRef]

Shamir, J.

R. Piestun, and J. Shamir, "Synthesis of three-dimensional light fields and applications," Proceedings of the IEEE 90, 222-244 (2002).
[CrossRef]

Shealy, D. L.

D. L. Shealy and S. H. Chao, "Geometric optics-based design of laser beam shapers," Opt. Eng. 42, 3123-3138 (2003).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, P. G. Seraphimovich, and V. A. Soifer, "An iterative algorithm for designing diffractive optical elements with regularization," Opt. Laser Eng. 29, 261-268 (1998).
[CrossRef]

Stark, H.

Taghizadeh, M. R.

J. S. Liu, A. J. Caley, and M. R. Taghizadeh, "Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms," Opt. Commun. 267, 347-355 (2006).
[CrossRef]

J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
[CrossRef]

Turunen, J.

Vahimaa, P.

Wolf, E.

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
[CrossRef]

Wyrowski, F.

Yang, B.

Yuan, X.

Zhao, Y.

Zhou, G.

Zhou, Q.-G.

Zuegel, J. D.

Appl. Opt. (3)

J. Mod. Opt. (2)

M. Johansson and J. Bengtsson, "Robust design method for highly efficient beam-shaping diffractive optical elements using iterative-Fourier-transform algorithm with soft operations," J. Mod. Opt. 47, 1385-1398 (2000).
[CrossRef]

R. Kant, "An analytical solution of vector diffraction for focusing optical systems with Seidal aberrations I. Spherical aberration, curvature of field, and distortion," J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

J. S. Liu, A. J. Caley, and M. R. Taghizadeh, "Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms," Opt. Commun. 267, 347-355 (2006).
[CrossRef]

Opt. Eng. (1)

D. L. Shealy and S. H. Chao, "Geometric optics-based design of laser beam shapers," Opt. Eng. 42, 3123-3138 (2003).
[CrossRef]

Opt. Express (2)

Opt. Laser Eng. (1)

V. V. Kotlyar, P. G. Seraphimovich, and V. A. Soifer, "An iterative algorithm for designing diffractive optical elements with regularization," Opt. Laser Eng. 29, 261-268 (1998).
[CrossRef]

Opt. Lett. (2)

Proc. Roy. Soc. A (1)

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
[CrossRef]

Proceedings of the IEEE (1)

R. Piestun, and J. Shamir, "Synthesis of three-dimensional light fields and applications," Proceedings of the IEEE 90, 222-244 (2002).
[CrossRef]

Other (2)

F. M. Dickey and D. L. Shealy, Laser Beam Shaping (Marcel Dekker, Inc., New York, 2000).
[CrossRef]

F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications (Taylor & Francis Group, Boca Raton, 2006).

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Figures (5)

Fig. 1.
Fig. 1.

Optical setup of the beam shaping problem. The aperture represents the input pupil of the objective lens. The focal plane is divided into two regions. Ω represents the region of interest that contains and bounds the targeted beam shape. Its complement Ωc represents the remainder of the focal plane.

Fig. 2.
Fig. 2.

(A) Calculated irradiance distribution resulting when a circularly apodized flat-top input beam of radius R is passed through the phase-only DOE shown in Fig. 3 and focused using a 1.4-NA objective. The DOE was designed to reshape the beam into a flat-top square irradiance pattern of area 50λ×50λ (D=100λ). (B)–(D) Irradiances of the constituent x-, y-, and z-polarized components of the total field. Each profile is normalized to the peak of If .

Fig. 3.
Fig. 3.

DOE phase profile that generates the focal irradiance distributions shown in Fig. 2. The phase is plotted in units of radians.

Fig. 4.
Fig. 4.

(A) Evolution of the beam shaping diffraction efficiency and (B) uniformity error versus iteration number for D=100λ.

Fig. 5.
Fig. 5.

(Left) Normalized focused irradiance distributions and (right) corresponding DOE phase profiles obtained when the vector diffraction algorithm was used to reshape the input beam to a focused square flattop irradiance profile for which (A, B) D=50λ, (C, D) D=25λ, (E, F) D=10λ, (G, H) and D=5λ. The DOE phase is plotted in units of radians.

Tables (1)

Tables Icon

Table 1. Performance parameters for the reshaped beams shown in Figs. 2 and 5 as function of the focused beam parameter D.

Equations (13)

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[ E x E y E z ] = i E in f 2 π k x 2 + k y 2 k max T ( k x , k y ) e i Φ ( k x , k y ) k z k t 1 k z [ k x 2 ( k z k t ) + k y 2 k x 2 + k y 2 k x k y ( k z k t 1 ) k x 2 + k y 2 k x k t ] × e i ( k x x f + k y y f ) d k x d k y .
x a R = k x k max = u and y a R = k y k max = v ,
ξ = x f D = x f m λ and η = y f D = y f m λ .
β = 2 π D R λ f = 2 π D ( N A ) λ n .
[ E x E y E z ] = i E in f λ ( N A n ) 2 u 2 + v 2 1 T ( u , v ) e i Φ ( u , v ) 1 q [ u 2 q + v 2 u 2 + v 2 u v ( q 1 ) u 2 + v 2 N A n u ] × e i ( u ω x + v ω y ) d u d v ,
κ = Ω I f ( ξ , η ) d ξ d η Ω + Ω c I f ( ξ , η ) d ξ d η
δ = max [ I f ( ξ , η ) ] min [ I f ( ξ , η ) ] max [ I f ( ξ , η ) ] + min [ I f ( ξ , η ) ] for ( ξ , η ) Ω .
I t = { α for ( ξ , η ) Ω I f for ( ξ , η ) Ω c ,
α = I f d ξ d η d ξ d η for ( ξ , η ) Ω .
E x ( ξ , η ) = 2 n c ε 0 [ γ I t ( ξ , η ) I y ( ξ , η ) I z ( ξ , η ) ] exp [ i ϕ x ( ξ , η ) ] for ( ξ , η ) Ω ,
I f = exp [ ( x f a ) 2 N ( y f a ) 2 N ] .
β 2 Δ I in Δ I t 1 ,
Δ I in = ( u 2 + v 2 ) I in ( u , v ) d u d v I in ( u , v ) d u d v and Δ I t = ( ξ 2 + η 2 ) I t ( ξ , η ) d ξ d η I t ( ξ , η ) d ξ d η .

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