Abstract

Many laser applications require specific irradiance distributions to ensure optimal performance. Geometric optical design methods based on numerical calculation of two plano-aspheric lenses have been thoroughly studied in the past. In this work, we present an alternative new design approach based on functional differential equations that allows direct calculation of the rotational symmetric lens profiles described by two-point Taylor polynomials. The formalism is used to design a Gaussian to flat-top irradiance beam shaping system but also to generate a more complex dark-hollow Gaussian (donut-like) irradiance distribution with zero intensity in the on-axis region. The presented ray tracing results confirm the high accuracy of both calculated solutions and emphasize the potential of this design approach for refractive beam shaping applications.

© 2014 Optical Society of America

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References

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2013

2012

2011

2008

2007

2003

Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
[CrossRef] [PubMed]

J. A. Hoffnagle, C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[CrossRef]

2000

1996

1980

Benítez, P.

Cai, Y.

Dickey, F.

Dickey, F. M.

F. M. Dickey, S. C. Holswade, D. L. Shealy, Laser Beam Shaping Applications (CRC Press, 2005).
[CrossRef]

Du, S.

Duerr, F.

Estes, R. H.

R. H. Estes, E. R. Lancaster, “Two-point Taylor series expansions,” NASA TMX-55759 (N67-23965) (1966).

Feng, Z.

Glückstad, J.

Gong, M.

Hoffnagle, J. A.

D. L. Shealy, J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE 8490, 849003 (2012).
[CrossRef]

J. A. Hoffnagle, C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[CrossRef]

J. A. Hoffnagle, C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488–5499 (2000).
[CrossRef]

Holswade, S. C.

F. M. Dickey, S. C. Holswade, D. L. Shealy, Laser Beam Shaping Applications (CRC Press, 2005).
[CrossRef]

Huang, L.

Jefferson, C. M.

Jiang, P.

Jin, G.

Lancaster, E. R.

R. H. Estes, E. R. Lancaster, “Two-point Taylor series expansions,” NASA TMX-55759 (N67-23965) (1966).

Lin, Q.

Liu, Z.

Lu, X.

Ma, H.

Meuret, Y.

Miñano, J. C.

Oliker, V.

Palima, D.

Rhodes, P. W.

Romero, L.

Rubinstein, J.

Shealy, D. L.

D. L. Shealy, J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE 8490, 849003 (2012).
[CrossRef]

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]

F. M. Dickey, S. C. Holswade, D. L. Shealy, Laser Beam Shaping Applications (CRC Press, 2005).
[CrossRef]

Thienpont, H.

Wolansky, G.

Xu, X.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

J. A. Hoffnagle, C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

D. L. Shealy, J. A. Hoffnagle, “Review: design and analysis of plano-aspheric laser beam shapers,” Proc. SPIE 8490, 849003 (2012).
[CrossRef]

Other

R. H. Estes, E. R. Lancaster, “Two-point Taylor series expansions,” NASA TMX-55759 (N67-23965) (1966).

F. M. Dickey, S. C. Holswade, D. L. Shealy, Laser Beam Shaping Applications (CRC Press, 2005).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Typical configuration of a refractive laser beam shaper consisting of two planoaspheric lenses. The 1st surface redistributes while the 2nd surface recollimates the rays.

Fig. 2
Fig. 2

(a) In comparison to the one-point Taylor series expansion about r1 that does not converge everywhere, the two-point Taylor series expansion about r1 and r2 approximates the mapping function R(r) (dotted line) for all values r. (b) Introduction of all necessary initial values and functions to derive the conditional equations from Fermat’s principle.

Fig. 3
Fig. 3

The convex shape of the merit function makes it an easy task to reach an optimal value for z2 which adjusts the relative position in z-direction of the local solutions about r1 and r2 using a local optimization algorithm.

Fig. 4
Fig. 4

Demonstration of the convergence of the analytic solution: (a) the variation in OPL and (b) the spatial mapping error clearly reduce with an increasing approximation order.

Fig. 5
Fig. 5

Illustration of a laser beam shaping system based on two plano-aspheric lenses that transforms a Gaussian into a dark-hollow Gaussian (DHG) beam with zero intensity at the optical axis.

Fig. 6
Fig. 6

(a) The mapping function R(r) is given as a Taylor series expansion of the inverse function of the mapping function r(R). (b) Introduction of all necessary initial values and functions to derive the conditional equations from Fermat’s principle.

Fig. 7
Fig. 7

Demonstration of the convergence of the analytic solution: (a) the variation in OPL and (b) the spatial mapping error clearly reduce with an increasing approximation order.

Equations (25)

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I in ( r ) = 2 / ( π w 0 2 ) exp [ 2 ( r w 0 ) 2 ]
I F L ( R ) = 1 π R F L 2 ( 1 + ( R / R F L ) q ) 1 + q 2
A ( r ) = 0 r I in ( r ) 2 π r d r = 1 exp [ 2 ( r w 0 ) 2 ]
B ( R ) = 0 R I F L ( R ) 2 π R d R = [ 1 + ( R / R F L ) q ] ( 2 / q )
R ( r ) = ε R F L 1 exp [ 2 ( r w 0 ) 2 ] 1 [ 1 exp [ 2 ( r w 0 ) 2 ] ] q / 2 q
V 1 = n 2 n 0 ( p v 0 ) + ( ( r R ( r ) ) 2 + ( f ( r ) g ( R ( r ) ) ) 2
D 1 = r V 1 = 0
D 2 = f ( r ) r g ( r ) | r = R ( r ) = 0
lim r r 1 n r n D i = 0 ( i = 1 , 2 ) , { n 0 } .
T ( r ) = i = 0 k [ a i ( r r 1 ) + b i ( r r 2 ) ] [ ( r r 1 ) ( r r 2 ) ] i
f ( r ) = i = 0 k [ p i ( r r 1 ) + q i ( r r 2 ) ] [ ( r r 1 ) ( r r 2 ) ] i
g ( r ) = i = 0 k [ u i ( r r 3 ) + v i ( r r 4 ) ] [ ( r r 3 ) ( r r 4 ) ] i
f ( r 1 ) = z 1 f ( r 2 ) = z 2 f ( r 1 ) = m 1 f ( r 2 ) = m 2 g ( r 3 ) = z 3 g ( r 4 ) = z 4 g ( r 3 ) = m 3 g ( r 4 ) = m 4 R ( r 1 ) = r 3 R ( r 2 ) = r 4
lim r r 1 n r n D i = 0 , lim r r 2 n r n D i = 0 for ( i = 1 , 2 ) , { n 1 }
I DHG ( R ) = H 0 ( R 2 w 1 2 ) 2 n exp [ 2 ( R w 1 ) 2 ]
B ( R ) = 0 R I DHG ( R ) 2 π R d R = π H 0 ( w 1 4 exp [ 2 ( R w 1 ) 2 ] ( 2 R 4 + 2 R 2 w 1 2 + w 1 4 ) ) 4 w 1 2
r ( R ) = w 0 2 2 ln [ 1 B ( R ) ]
R ( r ) = i = 0 k c i r ( 2 i + 1 ) / 3 = c 0 r 1 / 3 + c 1 r + c 2 r 5 / 3 + + c k r ( 2 k + 1 ) / 3
V 2 = n 2 n 0 ( p v 0 ) + ( R r ( R ) ) 2 + ( g ( R ) f ( r ( R ) ) ) 2
D 3 = R V 2 = 0
D 4 = g ( R ) R f ( R ) | R = r ( R ) = 0
g ( R ) = i = 0 k [ u i ( R R 1 ) + v i ( R R 2 ) ] [ ( R R 1 ) ( R R 2 ) ] i
lim R R 1 n R n D i = 0 , lim R R 2 n R n D i = 0 for ( i = 3 , 4 ) , { n 1 }
f ( R ) = 0 R g ( R ( r ) ) d r
f ( R ) = f 0 + i = 1 k f i R ( i + 2 ) / 3 = f 0 + f 1 R 1 + f 2 R 4 / 3 + f 3 R 5 / 3 + f 4 R 2 + + f k R ( k + 2 ) / 3

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