1838
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
Propagation of rotational field correlation through atmospheric turbulence Mazen Nairat* and David Voelz Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, New Mexico 88003, USA *Corresponding author: [email protected] Received December 10, 2013; revised February 13, 2014; accepted February 17, 2014; posted February 20, 2014 (Doc. ID 202721); published March 21, 2014 A general formulation is presented that describes the propagation of the rotational field correlation of an optical beam through atmospheric turbulence. The associated influence on the orbital angular momentum (OAM) of a single photon is described analytically. The analysis predicts the probability of change in the OAM state due to the process of propagating through turbulence. The probability of a change in an OAM state depends on the Fresnel number and on the ratio of the beam diameter to the Fried parameter. © 2014 Optical Society of America OCIS codes: (050.4865) Optical vortices; (010.1300) Atmospheric propagation; (030.1670) Coherent optical effects. http://dx.doi.org/10.1364/OL.39.001838
The circular symmetric second-order optical field moment is known as the rotational field correlation (RFC) and can indicate the spatial distribution that is characteristic of the orbital angular momentum (OAM) of a single photon [1]. Indeed, the circular harmonic transform of the RFC represents the probability distribution of certain OAM modes. Several studies have characterized the sensitivity of the RFC to the presence of atmospheric turbulence. For example, the decoherence effect on the Laguerre– Gaussian beam was determined [1] and later, the distributed power of OAM modes of a vortex optical field with constant amplitude was calculated [2]. Thus, the probability of change in a particular OAM mode due to turbulence can be calculated directly. This study presents a general model to describe the propagation of a light beam carrying a certain OAM state through atmospheric turbulence, as illustrated in Fig. 1. A paraxial light beam is employed to describe the turbulence influence on the RFC. The extended Huygens– Fresnel principle is applied to a light wave ψr⃗ traveling from a transmitter to a receiver located at the far-field distance z. We assume a circular symmetric source, j⃗r 1 j j⃗r 2 j, where ⃗r is the radial position vector at the transmitter plane, and Δθ is the azimuthal angle at the same plane, as shown in Fig 1. A homogeneous Kolmogorov spectrum is also considered for the turbulence. The received RFC, C ψ ⃗ρ; z, is given by Z C ψ ⃗ρ; z
⃗ 0Gr; ⃗ ⃗ρ; k; z exp−Dr; ⃗ ⃗ρ; (1) d2 rC ψ r;
where C ψ ⃗r; 0 is the RFC at the transmitter, and Gr⃗ ; ⃗ρ; k; z is an appropriate Green’s function for free propagation, which can be written at the far field using the cylindrical coordinates as ⃗ ⃗ρ; k; z Gr;
k 2πz
2 exp
ik rρ cosΔϕ − Δθ ; z
D ⃗r; ⃗ρ
3.44r 0−5∕3
Z 0
1
j1 − ν ⃗ρ − ⃗ρ 0 − ν ⃗r − ⃗r 0 j5∕3 dν; (3)
where r 0 is the Fried parameter. Equation (3) can be easily evaluated under circular symmetry and rewritten as follows: 5∕3 2 ρ sinΔϕ∕28∕3 − r sinΔθ∕28∕3 : D ⃗r; ⃗ρ 3.44 ρ sinΔϕ∕2 − r sinΔθ∕2 r0 (4) In the case of a perfect coherent source with a phase perturbation generated mainly by the atmosphere, the RFC at a certain beam radius is then given by ⃗ 0 hψr; 0; 0ψ r; Δθ; 0iδΔθ: C ψ r;
For a vortex wave, which is generally composed of radial and azimuthal parts, Eq. (5) can be simplified as C ψ ⃗r; 0 jRrj2 hexpimΔθiδΔθ∕2π:
0146-9592/14/071838-03$15.00/0
(6)
Substituting Eqs. (2), (3), and (4) into Eq. (1) and integrating over Δθ yields 5∕3 1 k 2 2ρ exp −3.44 sinΔϕ∕2 C ψ ρ; Δϕ; z 2π z r0 Z ik 2 rρ cos Δϕ : (7) × drjRrj r exp z Equation (7) can be further simplified by using the Jacobi–Anger expansion for the exponential inside the integral, which can be expressed briefly as
(2)
where k is the wave number. The function inside the ⃗ ⃗ρ, is the wave structure funcexponential in Eq. (1), Dr; tion, which is given for the horizontal propagation through Kolmogorov turbulence as [3]
(5)
Fig. 1. Schematic diagram for OAM analysis. © 2014 Optical Society of America
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
in einΔθ
∞ X n−∞
10
∆m = 0
0
10
α
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
Propagation of rotational field correlation through atmospheric turbulence Mazen Nairat* and David Voelz Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, New Mexico 88003, USA *Corresponding author: [email protected] Received December 10, 2013; revised February 13, 2014; accepted February 17, 2014; posted February 20, 2014 (Doc. ID 202721); published March 21, 2014 A general formulation is presented that describes the propagation of the rotational field correlation of an optical beam through atmospheric turbulence. The associated influence on the orbital angular momentum (OAM) of a single photon is described analytically. The analysis predicts the probability of change in the OAM state due to the process of propagating through turbulence. The probability of a change in an OAM state depends on the Fresnel number and on the ratio of the beam diameter to the Fried parameter. © 2014 Optical Society of America OCIS codes: (050.4865) Optical vortices; (010.1300) Atmospheric propagation; (030.1670) Coherent optical effects. http://dx.doi.org/10.1364/OL.39.001838
The circular symmetric second-order optical field moment is known as the rotational field correlation (RFC) and can indicate the spatial distribution that is characteristic of the orbital angular momentum (OAM) of a single photon [1]. Indeed, the circular harmonic transform of the RFC represents the probability distribution of certain OAM modes. Several studies have characterized the sensitivity of the RFC to the presence of atmospheric turbulence. For example, the decoherence effect on the Laguerre– Gaussian beam was determined [1] and later, the distributed power of OAM modes of a vortex optical field with constant amplitude was calculated [2]. Thus, the probability of change in a particular OAM mode due to turbulence can be calculated directly. This study presents a general model to describe the propagation of a light beam carrying a certain OAM state through atmospheric turbulence, as illustrated in Fig. 1. A paraxial light beam is employed to describe the turbulence influence on the RFC. The extended Huygens– Fresnel principle is applied to a light wave ψr⃗ traveling from a transmitter to a receiver located at the far-field distance z. We assume a circular symmetric source, j⃗r 1 j j⃗r 2 j, where ⃗r is the radial position vector at the transmitter plane, and Δθ is the azimuthal angle at the same plane, as shown in Fig 1. A homogeneous Kolmogorov spectrum is also considered for the turbulence. The received RFC, C ψ ⃗ρ; z, is given by Z C ψ ⃗ρ; z
⃗ 0Gr; ⃗ ⃗ρ; k; z exp−Dr; ⃗ ⃗ρ; (1) d2 rC ψ r;
where C ψ ⃗r; 0 is the RFC at the transmitter, and Gr⃗ ; ⃗ρ; k; z is an appropriate Green’s function for free propagation, which can be written at the far field using the cylindrical coordinates as ⃗ ⃗ρ; k; z Gr;
k 2πz
2 exp
ik rρ cosΔϕ − Δθ ; z
D ⃗r; ⃗ρ
3.44r 0−5∕3
Z 0
1
j1 − ν ⃗ρ − ⃗ρ 0 − ν ⃗r − ⃗r 0 j5∕3 dν; (3)
where r 0 is the Fried parameter. Equation (3) can be easily evaluated under circular symmetry and rewritten as follows: 5∕3 2 ρ sinΔϕ∕28∕3 − r sinΔθ∕28∕3 : D ⃗r; ⃗ρ 3.44 ρ sinΔϕ∕2 − r sinΔθ∕2 r0 (4) In the case of a perfect coherent source with a phase perturbation generated mainly by the atmosphere, the RFC at a certain beam radius is then given by ⃗ 0 hψr; 0; 0ψ r; Δθ; 0iδΔθ: C ψ r;
For a vortex wave, which is generally composed of radial and azimuthal parts, Eq. (5) can be simplified as C ψ ⃗r; 0 jRrj2 hexpimΔθiδΔθ∕2π:
0146-9592/14/071838-03$15.00/0
(6)
Substituting Eqs. (2), (3), and (4) into Eq. (1) and integrating over Δθ yields 5∕3 1 k 2 2ρ exp −3.44 sinΔϕ∕2 C ψ ρ; Δϕ; z 2π z r0 Z ik 2 rρ cos Δϕ : (7) × drjRrj r exp z Equation (7) can be further simplified by using the Jacobi–Anger expansion for the exponential inside the integral, which can be expressed briefly as
(2)
where k is the wave number. The function inside the ⃗ ⃗ρ, is the wave structure funcexponential in Eq. (1), Dr; tion, which is given for the horizontal propagation through Kolmogorov turbulence as [3]
(5)
Fig. 1. Schematic diagram for OAM analysis. © 2014 Optical Society of America
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
in einΔθ
∞ X n−∞
10
∆m = 0
0
10
α
