Abstract

A new family of paraxial laser beams that form an orthogonal basis is discussed. When propagated in uniform space, these beams preserve their structure to scale. The intensity distribution profile for such beams is similar to that for the Bessel modes, representing a set of alternating bright and dark concentric rings. The complex amplitude of these beams is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric (HyG) modes. The HyG modes are generated with a liquid-crystal microdisplay.

© 2007 Optical Society of America

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References

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2006

2005

2004

2000

1987

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, 5157 (2004).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Almazov, A. A.

Bandres, M. A.

Bentley, J. B.

Brichkov, V. A.

A. P. Prudnikov, V. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Chavez-Cerda, S.

Davis, J. A.

Durnin, J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Elfstrom, H.

Gutierrez-Vega, J.

Gutierrez-Vega, J. C.

Iturbe-Castillo, M. D.

Khonina, S. N.

Kotlyar, V. V.

Marichev, O. I.

A. P. Prudnikov, V. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Miller, W.

W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, 1977).

Prudnikov, A. P.

A. P. Prudnikov, V. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Schwarz, U. T.

Siegman, A. E.

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

Soifer, V. A.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Turunen, J.

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, 5157 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, 5157 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Other

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

A. P. Prudnikov, V. A. Brichkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, 1977).

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Figures (3)

Fig. 1
Fig. 1

(a) Intensity E γ , n ( r , φ , z ) 2 , (b) phase, and (c) intensity radial section derived from formula (2) for a HyG mode with the numbers ( γ , n ) = ( 2 , 3 ) at a distance z = 1000 mm . The frame size in (a) and (b) is 8 mm × 8 mm .

Fig. 2
Fig. 2

(a) Binary phase generated with a LCD and (b) intensity distribution for the HyG mode ( γ , n ) = ( 5 , 10 ) recorded with a CCD camera at a distance z = 700 mm from the display.

Fig. 3
Fig. 3

Intensity distribution for the HyG mode ( γ , n ) = ( 3 , 3 ) recorded with a CCD camera at a distance z of (a) 2 m , (b) 2.1 m , (c) 2.2 m .

Equations (8)

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( 2 i k z + 2 r 2 + 1 r 2 2 φ 2 ) E ( r , φ , z ) = 0 ,
E γ , n ( r , φ , z ) = 1 2 π n ! ( z 0 z ) 1 2 Γ ( n + 1 i γ 2 ) × exp [ i π 4 ( 3 n + i γ 1 ) + i γ 2 ln z z 0 + i n φ ] × x n 2 F 1 1 ( n + 1 i γ 2 , n + 1 , i x ) ,
F 1 1 ( a , b , y ) = Γ ( b ) Γ ( a ) Γ ( b a ) 0 1 t a 1 ( 1 t ) b a 1 exp ( y t ) d t ,
0 t α 1 exp ( p t 2 ) J v ( c t ) d t = c v p ( α + v ) 2 2 v 1 Γ ( α + v 2 ) Γ 1 ( v + 1 ) × F 1 1 ( α + v 2 , v + 1 , c 2 4 p ) .
E γ , n ( r , φ , z = 0 ) = 1 2 π ( w r ) exp [ i γ ln ( r w ) + i n φ ] ,
0 0 2 π E γ , n ( r , φ , z ) E μ , m * ( r , φ , z ) r d r d φ = δ n , m δ ( γ μ ) ,
x n 2 F 1 1 ( n + 1 i γ 2 , n + 1 , i x ) 1 x .
E 0 , 0 ( r , φ , z ) = ( 1 i ) 2 ( z 0 2 π z ) 1 2 J 0 ( x 2 ) exp ( i x 2 ) .

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