Abstract

This paper proposes the concept of low dimensional optical beam and operator. In low dimensional space, beam (or operator) is decomposed into a limited number of orthogonalized low dimensional beams (or operators) through the singular value decomposition. It is possible to generate an unconventional beam by these low dimensional beams. Low dimensional operator allows independent operation of orthogonal dimensions which may produce greater freedoms. Storage space and computation resource are saved dramatically by using this method. Experimental realization of this scheme is briefly discussed at the end.

© 2015 Optical Society of America

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References

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2014 (1)

2012 (1)

2011 (2)

2009 (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

2007 (1)

2002 (1)

1975 (1)

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Biener, G.

Bomzon, Z.

Cao, J. X.

Chen, Y. B.

Couairon, A.

Courvoisier, F.

Dudley, J.

Duparré, M.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Forbes, A.

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Guo, D. D.

Hasman, E.

Itina, T.

Jukna, V.

Kleiner, V.

Leng, H. Y.

Li, L.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref] [PubMed]

Li, T.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref] [PubMed]

Lv, X. J.

Milián, C.

Ngcobo, S.

Schulze, C.

Siegman, A. E.

Sziklas, E. A.

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Wang, S. M.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref] [PubMed]

Wu, F. T.

Xie, C.

Xie, Z. D.

Xu, P.

Yin, Y. L.

Yuan, Y.

Zhang, C.

Zhao, L. N.

Zhu, S. N.

Appl. Opt. (2)

Nat. Photonics (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

Low dimensional (LD) lensing and propagation of optical beam.

Fig. 2
Fig. 2

(a) Original Laguerre-Guassian mode with p = 2 and s = 2. (b) Composition of the first 2 ROBs. (c)-(f) First 4 ROBs.

Fig. 3
Fig. 3

Singular values of beam shown in Fig. 2(a).

Fig. 4
Fig. 4

Difference between the original beam and the composed beam.

Fig. 5
Fig. 5

(a) and (b) Modulus and phase of Bessel-Gaussian beam. (c) and (d) Modulus and phase of composed beam.

Fig. 6
Fig. 6

Singular values of vortex Bessel-Gauss beam.

Fig. 7
Fig. 7

Phase of the Kirchhoff operator.

Fig. 8
Fig. 8

Singular values of Kirchhoff operator.

Fig. 9
Fig. 9

Strip shaped beam generated from a slab laser. Upper: the beam in actual proportion. Lower: beam zoomed in.

Fig. 10
Fig. 10

Singular values of strip shaped beam.

Fig. 11
Fig. 11

Beam after lensing and free space propagation. Left: the beam in actual proportion. Right: beam zoomed in.

Fig. 12
Fig. 12

Difference between low dimensional focusing and Kirchhoff integral.

Equations (17)

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M m × n = U m × m S m × n V n × n * = k = 1 r U : , k S k , k V : , k * = k = 1 r M k .
E ( x , y ) = k = 1 r Ě k ( x , y ) = k = 1 r S k · E k ( x ) E k ( y ) ,
H ( A + B ) = H ( A ) + H ( B ) .
H ( s A ) = s H ( A ) .
H ( A ( x ) B ( y ) ) = H ( A ( x ) ) H ( B ( y ) ) ,
H = q = 1 s H q .
H E ( x , y ) = k = 1 k = r q = 1 q = s H q Ě k = k = 1 k = r q = 1 q = s S k H q E k ( x ) H q E k ( y ) .
E m n ( x , y ) = E 0 H m ( ( 2 ) x ω ) H n ( ( 2 ) y ω ) exp ( x 2 + y 2 ω ) .
E p s ( x , y ) = E 0 ρ s L p s ( ρ 2 ) exp ( ρ 2 / 2 ) cos ( s θ ) ,
E j k = E ( x = j L x m , y = k L y n ) ,
k = 1 r S k 2 = j , k M j k 2 = M F .
S m × n .
E ( ρ ) = J 0 ( α ρ ) exp [ ( ρ ω 0 ) 2 ] exp ( i θ ) ,
F ( f x , f y ) = f ( x , y ) exp ( i 2 π f x x i 2 π f y y ) d x d y .
F ( x , y ) = f ( x , y ) exp [ k ( x 2 2 f x + y 2 2 f y ) ] ,
F ( x , y ) = f ( f x , f y ) exp [ 2 π z λ 1 λ 2 ( f x 2 + f y 2 ) ] ,
x = α z x y = α z y z = α 2 z z 0 ( z z 0 ) ,

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