Abstract

Hermite–Gauss and Laguerre–Gauss modes of a continuous optical field in two dimensions can be obtained from each other through paraxial optical setups that produce rotations in (four-dimensional) phase space. These transformations build the SU(2) Fourier group that is represented by rigid rotations of the Poincaré sphere. In finite systems, where the emitters and the sensors are in N×N square pixellated arrays, one defines corresponding finite orthonormal and complete sets of two-dimensional Kravchuk modes. Through the importation of symmetry from the continuous case, the transformations of the Fourier group are applied on the finite modes.

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References

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2007

2006

2005

G. F. Calvo, "Wigner representation and geometric transformations of optical orbital angular momentum spatial modes," Opt. Lett. 30, 1207-1209 (2005).
[CrossRef] [PubMed]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, Suppl. 3, 521-555 (2005).

2003

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

2001

L. Barker, "Continuum quantum systems as limits of discrete quantum systems: II. State functions," J. Phys. A 34, 4673-4682 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, "Finite two-dimensional oscillator: I. The Cartesian model," J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

2000

R. Simon and K. B. Wolf, "Fractional Fourier transforms in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000).
[CrossRef]

L. Barker, Ç. Çandan, T. Hakioglu, and H. M. Ozaktas, "The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform," J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

1999

N. M. Atakishiyev, L. M. Vicent, and K. B. Wolf, "Continuous vs. discrete fractional Fourier transforms," J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

1997

1994

N. M. Atakishiyev and K. B. Wolf, "Approximation on a finite set of points through Kravchuk functions," Rev. Mex. Fís. 40, 366-377 (1994).

1991

N. M. Atakishiyev and S. K. Suslov, "Difference analogs of the harmonic oscillator," Theor. Math. Phys. 85, 1055-1062 (1991).
[CrossRef]

1929

M. Krawtchouk, "Sur une généralization des polinômes d'Hermite," C. R. Acad. Sci. Paris 189, 620-622 (1929).

C. R. Acad. Sci. Paris

M. Krawtchouk, "Sur une généralization des polinômes d'Hermite," C. R. Acad. Sci. Paris 189, 620-622 (1929).

Int. J. Mod. Phys. A

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

J. Comput. Appl. Math.

N. M. Atakishiyev, L. M. Vicent, and K. B. Wolf, "Continuous vs. discrete fractional Fourier transforms," J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, "Finite two-dimensional oscillator: I. The Cartesian model," J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

L. Barker, Ç. Çandan, T. Hakioglu, and H. M. Ozaktas, "The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform," J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

L. Barker, "Continuum quantum systems as limits of discrete quantum systems: II. State functions," J. Phys. A 34, 4673-4682 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Part. Nucl.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, Suppl. 3, 521-555 (2005).

Rev. Mex. Fís.

N. M. Atakishiyev and K. B. Wolf, "Approximation on a finite set of points through Kravchuk functions," Rev. Mex. Fís. 40, 366-377 (1994).

Theor. Math. Phys.

N. M. Atakishiyev and S. K. Suslov, "Difference analogs of the harmonic oscillator," Theor. Math. Phys. 85, 1055-1062 (1991).
[CrossRef]

Other

L. C. Biedenharn and J. D. Louck, "Angular momentum in quantum mechanics," in Encyclopedia of Mathematics and Its Applications, Vol. 8, G.-C.Rota, ed. (Addison-Wesley, 1981), Sect. 3.6.

L. E. Ruiz-Vicent, "Análisis de señales discretas finitas mediante el modelo de oscilador finito de su(2)," Ph. D. thesis (Universidad Autónoma del Estado de Morelos, Cuernavaca, 2007).

A. Frank and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (Wiley, 1998); Sect. 1.1.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light" Progress in Optics, Vol. XXXIX, E.Wolf, ed. (Elsevier, 1999), pp. 294-374.

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Figures (5)

Fig. 1
Fig. 1

Rotations of the Poincaré sphere generated by the su ( 2 ) algebra of operators L ¯ 1 (antisymmetric Fourier), L ¯ 2 (gyration), and L ¯ 3 (rotation). As will be defined below, the rotation angles on the sphere are 2 β , 2 γ , and 2 α , respectively.

Fig. 2
Fig. 2

Two-dimensional Kravchuk modes Φ n x , n y ( N ) ( q x , q y ) of the finite oscillator in a 17 × 17 -point array ( j = 8 ) classified by n x , n y { 0 , 1 , , 16 } . Each horizontal line of modes is characterized by the principal quantum number n = n x + n y . At each level n in the lower triangle 0 n 2 j = N 1 there are n + 1 modes classified by n x n y ; in the upper triangle 2 j = N 1 n 4 j = 2 N 2 there are 4 j n + 1 = 2 N n 1 modes. The functions in the upper triangle of the rhombus equal their reflected partners in the lower triangle, but for a checkerboard of alternating signs.

Fig. 3
Fig. 3

Rotation of the five n = 4 ( λ = 2 ) two-dimensional Kravchuk modes (top row), classified by 1 2 ( n x n y ) { 2 , 1 , 0 , 1 , 2 } , in a 23 × 23 pixellated array ( N = 23 , j = 11 ). Succesive rows show their rotations according to relation (12) by α = 0 , π 16 , π 8 , 3 π 16 , π 4 , corresponding to rotations of the Poincaré sphere by 2 α . The functions are real. Midgray pixels are 0, black and white are minima and maxima, respectively.

Fig. 4
Fig. 4

Gyration according to relation (15) of the five n = 4 ( λ = 2 ) two-dimensional Kravchuk modes, for angles γ = 0 , π 16 , π 8 , 3 π 16 , π 4 . The top row corresponds to that in Fig. 3 but here shows absolute values (instead of real parts); the two bottom rows show the absolute values and phases of the n + 1 Laguerre–Kravchuk modes classified by angular momentum m { 4 , 2 , 0 , 2 , 4 } . Black is 0, white are maxima; in the last row, the range from black to white is ( π , π ) , except for the m = 0 mode, which is real, but whose phase at ± π is numerically unstable.

Fig. 5
Fig. 5

Two-dimensional Laguerre–Kravchuk modes Λ n , m ( N ) ( q x , q y ) for N = 2 j + 1 = 17 points ( j = 8 ) , classified by n and m. Since ( Λ n , m ( N ) ) * = Λ n , m ( N ) , the real parts are shown for m 0 and the imaginary parts for m < 0 . The highest angular momenta occur for ± m = 2 j = n as if a quantum particle circulated around the periphery of the square; the m = 0 modes have that particle moving along the diagonals.

Equations (23)

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N ¯ k 1 2 ( P ¯ k 2 + Q ¯ k 2 1 ) , k { x , y } ,
[ Q ¯ k , P ¯ k ] = i δ k , k 1 , [ N ¯ k , Q ¯ k ] = i δ k , k P ¯ k ,
[ N ¯ k , P ¯ k ] = i δ k , k Q ¯ k ,
symmetric FT L ¯ 0 1 4 ( P ¯ x 2 + P ¯ y 2 + Q ¯ x 2 + Q ¯ y 2 2 1 ) = 1 2 ( N ¯ x + N ¯ y ) ,
antisymmetric FT L ¯ 1 1 4 ( P ¯ x 2 P ¯ y 2 + Q ¯ x 2 Q ¯ y 2 ) = 1 2 ( N ¯ x N ¯ y ) ,
gyration L ¯ 2 1 2 ( P ¯ x P ¯ y + Q ¯ x Q ¯ y ) ,
rotation L ¯ 3 1 2 ( Q ¯ x P ¯ y Q ¯ y P ¯ x ) 1 2 M ¯ ,
[ L ¯ 0 , L ¯ k ] = 0 , [ L ¯ i , L ¯ j ] = i L ¯ k ,
[ Q k , P k ] = i δ k , k ( N k j 1 ) , [ N k , Q k ] = i δ k , k P k ,
[ N k , P k ] = i δ k , k Q k ,
Φ n ( N ) ( q ) = d n j , q j ( 1 2 π ) ( 1 ) n 2 j ( 2 j n ) ( 2 j j + q ) K n ( j + q ; 1 2 , 2 j ) ,
K n ( s ; 1 2 , 2 j ) = F 1 2 ( n , j s ; 2 j ; 2 ) = K s ( n ; 1 2 , 2 j ) ,
Φ n x , n y ( N ) ( q x , q y ) Φ n x ( N ) ( q x ) Φ n y ( N ) ( q y ) ,
q x , q y , n x j , n y j { j , j + 1 , , j } .
R ( α ) : Φ n x , n y ( N ) ( q x , q y ) n x + n y = n d 1 2 ( n x n y ) , 1 2 ( n x n y ) n 2 ( 2 α ) Φ n x , n y ( N ) ( q x , q y ) ,
K A ( β ) : Φ n x , n y ( N ) ( q x , q y ) = exp [ i β ( n x n y ) ] Φ n x , n y ( N ) ( q x , q y ) .
G ( γ ) F A ( 1 4 π ) R ( γ ) F A ( 1 4 π ) .
Φ n x , n y ( N ) ( q x , q y ; γ ) G ( γ ) : Φ n x , n y ( N ) ( q x , q y )
= n x + n y = n e i π ( n x n y ) 4 d 1 2 ( n x n y ) , 1 2 ( n x n y ) n 2 ( 2 γ ) e + i π ( n x n y ) 4 Φ n x , n y ( N ) ( q x , q y ) .
Λ n , m ( N ) ( q x , q y ) e i π m 2 Φ n x , n y ( N ) ( q x , q y ; 1 4 π )
= n x + n y = n e i π ( n x n y ) 4 d 1 2 m , 1 2 ( n x n y ) n 2 ( 1 2 π ) d n x n 2 , q x n 2 ( 1 2 π ) d n y n 2 , q y n 2 ( 1 2 π ) .
D ( ψ , θ , ϕ ) = exp ( i ψ L ¯ 3 ) exp ( i θ L ¯ 2 ) exp ( i ϕ L ¯ 3 ) .
D ( ψ , θ , ϕ ) = K A ( 1 2 ψ ) G ( 1 2 θ ) K A ( 1 2 ϕ ) = K A ( 1 2 ψ + 1 4 π ) R ( 1 2 θ ) K A ( 1 2 ϕ 1 4 π ) .

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