Abstract

We derive explicit analytical relations to describe paraxial light beams that represent a particular case of the hypergeometric (HyG) laser beams [J. Opt. Soc. Am. A 25, 262–270 (2008)]. Among these are modified quadratic Bessel–Gaussian beams, hollow Gaussian optical vortices, modified elegant Laguerre– Gaussian beams, and gamma-HyG beams. Using e-beam microlithography, a binary diffractive optical element capable of producing near-HyG beams is synthesized. Theory and experiment are in sufficient agreement. We experimentally demonstrate the ability to rotate dielectric microparticles using the bright diffraction ring of a HyG beam.

© 2008 Optical Society of America

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2008

2007

2006

2005

2004

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, “Rotation of microparticles with Bessel beams generated by diffractive elements,” J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

J. Enderlein and F. Pampaloni, “Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes,” J. Opt. Soc. Am. A 21, 1553-1558 (2004).
[CrossRef]

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29, 144-146 (2004).
[CrossRef] [PubMed]

M. A. Bandres and J. C. Gutiérrez-Vega, “Elegant Ince-Gaussian beams,” Opt. Lett. 29, 1724-1726 (2004).
[CrossRef] [PubMed]

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

1999

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83-93 (1999).
[CrossRef]

1997

1996

L. Allen, M. P. Padgett, and N. B. Simpson, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

1987

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1986

1985

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

Allen, L.

L. Allen, M. P. Padgett, and N. B. Simpson, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

L. Allen, S. M. Barnet, and M. J. Padgett, Orbital Angular Momentum (Institute of Optical Publishing, 2003).
[CrossRef]

Almazov, A. A.

Bandres, M. A.

Barnet, S. M.

L. Allen, S. M. Barnet, and M. J. Padgett, Orbital Angular Momentum (Institute of Optical Publishing, 2003).
[CrossRef]

Bernet, S.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka Publishers, 1983).

Caron, C. F. R.

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83-93 (1999).
[CrossRef]

Christodoulides, D. N.

Ding, J.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Elfstrom, H.

Enderlein, J.

Fukumitsu, O.

Furhapter, S.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guo, C.

Gutiérrez-Vega, J. C.

Han, Y.

Jefimovs, K.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, “Rotation of microparticles with Bessel beams generated by diffractive elements,” J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

Jesacher, A.

Karimi, E.

Khonina, S. N.

Kotlyar, V. V.

Kovalev, A. A.

Law, C. T.

Marrucci, L.

Marychev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka Publishers, 1983).

Menon, R.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Padgett, M. J.

L. Allen, S. M. Barnet, and M. J. Padgett, Orbital Angular Momentum (Institute of Optical Publishing, 2003).
[CrossRef]

Padgett, M. P.

L. Allen, M. P. Padgett, and N. B. Simpson, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Pampaloni, F.

Piccirillo, B.

Potvliege, R. M.

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83-93 (1999).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka Publishers, 1983).

Ritsch-Marte, M.

Rozas, D.

Santamato, E.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Simonen, J.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, “Rotation of microparticles with Bessel beams generated by diffractive elements,” J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

Simpson, N. B.

L. Allen, M. P. Padgett, and N. B. Simpson, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

Siviloglou, G. A.

Skidanov, R. V.

Smith, H. I.

Soifer, V. A.

Swartzlander, G. A.

Takenaka, T.

Torre, A.

A. Torre, “A note on the general solution of the paraxial wave equation: a Lie algebra view,” J. Opt. A Pure Appl. Opt. 10, 055006 (2008).
[CrossRef]

Tossavainen, N.

Turunen, J.

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

Xu, J.

Yokota, M.

Zauderer, E.

Zito, G.

Appl. Opt.

J. Mod. Opt.

L. Allen, M. P. Padgett, and N. B. Simpson, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43, 2485-2492 (1996).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, “Rotation of microparticles with Bessel beams generated by diffractive elements,” J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

J. Opt. A Pure Appl. Opt.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

A. Torre, “A note on the general solution of the paraxial wave equation: a Lie algebra view,” J. Opt. A Pure Appl. Opt. 10, 055006 (2008).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83-93 (1999).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Other

A. E. Siegman, Lasers (University Science, 1986).

M. Abramovitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Applied Math Series (National Bureau of Standards, 1965).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka Publishers, 1983).

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

L. Allen, S. M. Barnet, and M. J. Padgett, Orbital Angular Momentum (Institute of Optical Publishing, 2003).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Radial intensity profile of the HyG mode ( n = 4 , γ = 10 , and m = 1 ) at (a)  z = 0 and (b)  z = 100 mm : (1) analytical HyG mode and (2) calculated aperture-bounded mode.

Fig. 2
Fig. 2

(a) DOE binary phase ( c = 10 mm 1 ) and (b) calculated diffraction pattern at distance z = 700 mm .

Fig. 3
Fig. 3

Radial intensity distributions of the ideal HyG modes (Curves 1) and the HyG beams produced by the binary DOE of Fig. 2a (Curves 2) at a distance z = 2000 mm : (a)  n = 7 , γ = 10 and (b)  n = 7 , γ = 10 .

Fig. 4
Fig. 4

Central fragment of the binary DOE [Fig. 2a] microrelief of size 353 μm × 265 μm generated in the fused silica substrate.

Fig. 5
Fig. 5

Diffraction patterns produced by the DOE of Fig. 4 illuminated by a plane wave of diameter 4 mm ( λ = 532 nm ) recorded by a CCD camera at distances of (a)  2000 mm , (b)  2300 mm , and (c)  3000 mm .

Fig. 6
Fig. 6

(a) Experimental and (c) calculated diffraction patterns of a plane wave diffracted from the DOE for the HyG mode ( n = 7 , γ = 10 ) and (b), (d) their respective radial intensity distributions.

Fig. 7
Fig. 7

Optical setup for the rotation of microbeads in water by the HyG beam generated by DOE.

Fig. 8
Fig. 8

Rotation of a polystyrene microbead of diameter 5 μm in water (locations are indicated with white arrows) in the major diffraction ring of the HyG beam ( n = 7 , γ = 10 ) resulting from the diffraction of a plane wave by the DOE of Fig. 4. The time between frames is 15 s .

Equations (32)

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E γ , n , m ( r , φ , z = 0 ) = 1 2 π ( r ω ) m exp [ r 2 2 σ 2 + i γ ln ( r ω ) + i n φ ] ,
E γ , n , m ( ρ , θ , z ) = ( i ) n + 1 2 π n ! ( z 0 z q 2 ) ( 2 σ ω q ) m + i γ ( k σ ρ 2 q z ) n × exp ( i k ρ 2 2 z + i n θ ) Γ ( n + m + 2 + i γ 2 ) × F 1 1 [ n + m + 2 + i γ 2 , n + 1 , ( k σ ρ 2 q z ) 2 ] ,
F 1 1 ( n + 1 2 , n + 1 , x ) = Γ ( n 2 + 1 ) exp ( x 2 ) ( i x 4 ) n 2 J n 2 ( i x 2 ) ,
E i ( m + 1 ) , n , m ( ρ , θ , z ) = E 0 , n , 1 ( ρ , θ , z ) = ( i ) n + 1 2 π ( k σ ω 2 z q ) exp [ i n θ + i k ρ 2 2 R 1 ( z ) ρ 2 2 σ 2 ( z ) ] I n 2 [ ρ 2 2 σ 2 ( z ) + i k ρ 2 2 R ( z ) ] ,
σ 2 ( z ) = 2 σ 2 ( 1 + z 2 z 0 2 ) , R ( z ) = 2 z ( 1 + z 2 z 0 2 ) , R 1 ( z ) = R ( z ) ( 1 + 2 z 2 z 0 2 ) 1 ,
0 J n 2 ( a r 2 ) exp ( b r 2 ) J n ( c r ) r d r = 0.5 ( a 2 + b 2 ) 1 2 J n 2 [ c 2 a 4 ( a 2 + b 2 ) ] exp [ c 2 b 4 ( a 2 + b 2 ) ] .
E 0 , n , 1 ( r , φ , z = 0 ) = 1 2 π ( ω r ) exp ( r 2 2 σ 2 + i n φ ) ,
E ˜ 0 , n , 1 ( ρ , θ , z ) = ω 2 i k 2 π z ( i ) n 2 + 1 exp ( i n θ + i k ρ 2 4 z ) J n 2 ( k ρ 2 4 z ) .
F 1 1 ( a , b , x ) = exp ( x ) F 1 1 ( b a , b , x ) .
E γ , n , m ( ρ , θ , z ) = ( i ) n + 1 2 π n ! ( z 0 z q 2 ) ( 2 σ ω q ) m + i γ ( k σ ρ 2 q z ) n × exp [ i n θ + i k ρ 2 2 z ( k σ ρ 2 q z ) 2 ] Γ ( n + m + 2 + i γ 2 ) × F 1 1 [ n m i γ 2 , n + 1 , ( k σ ρ 2 q z ) 2 ] .
F 1 1 ( n 2 , n + 1 , x ) = Γ ( n + 1 2 ) 2 n 1 2 ( x 2 ) ( n 1 2 ) exp ( x 2 ) [ I n 1 2 ( x ) I n + 1 2 ( x ) ] .
E i m , n , m ( ρ , θ , z ) = E 0 , n , 0 ( ρ , θ , z ) = ( i ) n + 1 4 π ( z 0 z q 2 ) ( k σ ρ 2 q z ) exp [ i n θ + i k ρ 2 2 R 1 ( z ) ρ 2 2 σ 2 ( z ) ] [ I n 1 2 ( y ) I n + 1 2 ( y ) ] ,
y = 1 2 ( k σ ρ 2 q z ) 2 = ρ 2 2 σ 2 ( z ) + i k ρ 2 2 R ( z ) .
E 0 , n , 0 ( r , φ , z = 0 ) = 1 2 π exp ( r 2 2 σ 2 + i n φ ) ,
E 0 , n , 1 ( r , φ , z = 0 ) = 1 2 π ( r ω ) exp ( r 2 2 σ 2 + i n φ ) .
0 r 2 exp ( p r 2 ) J n ( c r ) d r = c n p ( n + 3 ) / 2 2 ( n + 1 ) Γ ( n + 3 2 ) × Γ 1 ( n + 1 ) F 1 1 ( n + 3 2 , n + 1 , c 2 4 p ) .
0 exp ( p r 2 ) J n ( c r ) d r = 1 2 π p exp ( c 2 8 p ) I n 2 ( c 2 8 p ) ,
0 r 2 exp ( p r 2 ) J n ( c r ) d r = π 2 p 3 2 exp ( c 2 8 p ) × [ ( 1 n 2 c 2 8 p ) I n 2 ( c 2 8 p ) + ( c 2 8 p ) I n 2 2 ( c 2 8 p ) ] .
F 1 1 ( n + 3 2 , n + 1 , x ) = x n 2 exp ( x 2 ) π n ! Γ 1 ( n + 3 2 ) × [ ( 1 n 2 x 2 ) I n 2 ( x 2 ) + x 2 I n 2 2 ( x 2 ) ] .
E i ( m 1 ) , n , m ( ρ , θ , z ) = E 0 , n , 1 ( ρ , θ , z ) = ( i ) n + 1 2 π ( k σ 3 z ω q 3 ) exp [ i n θ + i k ρ 2 2 R 1 ( z ) ρ 2 2 σ 2 ( z ) ] × [ ( 1 n 2 y ) I n 2 ( y ) + y I n 2 2 ( y ) ] ,
E γ , n , 2 p + n ( r , φ , z = 0 ) = 1 2 π ( r ω ) 2 p + n exp ( r 2 2 σ 2 + i n φ ) .
F 1 1 ( p , n + 1 , x ) = p ! n ! ( n + p ) ! L p n ( x ) .
E 0 , n , 2 p + n ( ρ , θ , z ) = ( i ) n + 1 p ! 2 π ( z 0 z q 2 ) ( 2 σ ω q ) n + 2 p t n 2 exp ( i n θ + i k ρ 2 2 z t ) L p n ( t ) ,
t = 2 y = ( k σ ρ 2 q z ) 2 .
E eLG ( ρ , θ , z ) = ( i ) p + 1 ( z 0 z q 2 ) p + 1 ( 2 i σ 2 z 0 ω 2 q 2 z ) n 2 s n 2 exp ( i n θ s ) L p n ( s ) ,
E eLG ( r , φ , z = 0 ) = ( r ω ) n exp ( r 2 2 σ 2 + i n φ ) L p n ( r 2 2 σ 2 ) .
F 1 1 ( n , n + 1 , x ) = n x n γ ( n , x ) ,
γ ( ν , x ) = 0 x ξ ν 1 exp ( ξ ) d ξ
E i ( m + 2 n ) , n , m ( ρ , θ , z ) = E 0 , n , n 2 ( ρ , θ , z ) = ( i ) n + 1 2 π ( k ω 2 2 z ) ( k ρ ω 2 z ) n exp ( i k ρ 2 2 z + i n θ ) γ [ n , ( k σ ρ 2 q z ) 2 ] .
E 0 , n , n 2 ( r , φ , z = 0 ) = 1 2 π ( r ω ) n 2 exp ( r 2 2 σ 2 + i n φ ) .
E γ , n , 1 ( r , φ , z = 0 ) = 1 2 π ( ω r ) exp [ i γ ln ( r ω ) + i n φ ] .
τ γ , n ( r , φ ) = sgn { cos [ γ ln ( r ω ) + n φ + c r cos φ ] } ,

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