Spectrum changes of phase-locked partially coherent flat topped laser beam array propagating in turbulent atmosphere

Optik 124 (2013) 2135–2139

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Spectrum changes of phase-locked partially coherent flat topped laser beam array propagating in turbulent atmosphere D. Razzaghi a , F. Hajiesmaeilbaigi b , M. Alavinejad b,∗ a b

Laser and Optics Research School, P.O. Box 14155-1339, Tehran, Iran Photonics Laboratory, Physics Department, Iran University of Science and Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 8 February 2012 Accepted 20 June 2012

Keywords: Turbulent atmosphere Propagation Flat topped laser beam Phase-locked array beam

a b s t r a c t Propagation of the phase-locked partially coherent flat topped laser beam array trough turbulent media is studied in details. Concentrating on critical radius as an important designing factor, effects of coherence length, beamlet distance and flat topped degree on this parameter are discussed. Also geometrical shape of the array is shown to be important by comparing circular and rectangular shapes. Crown Copyright © 2012 Published by Elsevier GmbH. All rights reserved.

1. Introduction

2. Propagation formula for flat-topped laser beam array

Turbulence induced changes in spectrum of the laser beam have been studied intensively in recent years [1–5]. Among various effects of propagation, the spectral shift is of noticeable importance in optical communications. Behavior of propagated spectrum shows rapid transition for different parts of the receiver [2–4] which is depended on beams specifications, propagation distance and turbulence strength. Many applications need high irradiance on receiver surface to perform safely so that combining numerous light sources is demanding in those applications. Flat topped array beams in which numerous flat topped sources are combined, have received many attention in recent years [6–8]. Researchers have studied some features of propagation of phased locked partial coherent flat topped beams in turbulent atmosphere [8–12]. Phase locking of laser array beams has been extensively studied to obtain better beam quality [14–17]. Lü and Ma have studied properties of elliptical Gaussian array beams propagating in ABCD optical system [18] and propagation of these kind of array beams in turbulent atmosphere, have been investigated by Cai et al. [10]. They have also studied propagation of rectangular stochastic Gaussian–Schell model array beams. In the present paper, we will focus on propagation of phase-locked partially coherent flat topped beam array in turbulent atmosphere. Spectral changes are investigated and different parameters affecting critical radius value are determined.

Consider an array of coherently combined flattened laser beam so that each laser beam propagates along the z-axis in the Cartesian coordinate system. The optical field of such an array can be expressed as bellow:

∗ Corresponding author. Tel.: +98 09123845647. E-mail addresses: [email protected] (D. Razzaghi), [email protected] (F. Hajiesmaeilbaigi), m [email protected] (M. Alavinejad).

E(x, y, z = 0) =

M 

Ei (x, y, z = 0)

(1)

i=1

where Ei (x, y, z = 0) denotes the ith flattened beamlet in the laser array and can be expressed as [6–11]: Ei (x, y, z = 0) =

  N  (−1)n−1 N N n=1



× exp

n 2

−n[(x − ai )2 + (y − bi ) ] w02

 (2)

In Eq. (2), w0 is the beam waist of the Gaussian beam, r = (x, y) denotes a two dimensional transverse vector  perpendicular to the N direction of the beam propagation, denotes the binomial n coefficient, “N” is the order of the circular flat-topped beams, and (ai , bi ) is the center of the ith beamlet located at the source plane. The complex degree of the spatial coherence for a partial beam can be expressed as [19–21]:

N

g(x, y) =

t=1

0030-4026/$ – see front matter. Crown Copyright © 2012 Published by Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.08.003

exp(−t(x2 + y2 )/202 ) N

(3)

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D. Razzaghi et al. / Optik 124 (2013) 2135–2139

where 0 denotes transverse coherence width of the laser beam. The degree of the spatial coherence depends on the position coordinates of two points at the source plane. The cross-spectral density in the space-frequency domain of the whole beam array at the source plane can be written as [12]: W (0) (x, y, , , z = 0, ω) = E(x, y, z = 0, ω)E ∗ (, , z = 0, ω) × g(x − , y − , ω) =

N M M   i

d

d

α = 45

S (0) (ω)Ei (x, y, z = 0, ω)

t

j

2

× Ej∗ (, , z = 0, ω)

exp{−t[(x − ) + (y − )2 ]/202 (ω)} N

(4)

where S (0) (ω) is the normalized source spectrum and 0 (ω) is the spatial correlation length. Assuming a statistically homogeneous and isotropic medium and using the paraxial form of extended Huygens–Fresnel principle, the cross spectral density function of partial coherent flat topped (PCFT) beams propagating in turbulent atmosphere reads as [3]: W (p1 , q1 ; p2 , q2 ; z, ω) =



∞



2

 k

∞

∞

dx dy

2 z

−∞

−∞

∞

d d W (0) (x, y, , , z = 0, ω) × exp

× −∞

d

−∞

 ik 2z

[(p1 − x)2



2

+ (q1 − y)2 − (p2 − ) − (q2 − )2 ] ∗

× exp[ (p1 , q1 , x, y, ω) +

(p2 , q2 , , , ω)]

In Eq. (5), (p1 , q1 ) and (p2 , q2 ) denote the transverse coordinates at the receiver plane, is a random phase factor representing the effect of turbulence on the propagation of a spherical wave, k = ω/c is the wave number and the sharp brackets represents time averaging operation. It follows from Eq. (5) that the spectrum of the field S(p, q, z, ω) = W (p, q; p, q; z, ω) at a point specified by the position vector R = (p, q, z), is given by the following expression: S(p, q; z, ω) =

 ω 2

2cz

∞



−(q − )2 ]

∞



−∞

−∞

 iω 2cz

where



−∞

(p, q, , , ω)]

× exp

(6)

× exp

 

= exp

−



1



2

02

[(x − ) − (y − )2 ]

(7)

where D (x − , y − ) is the wave structure function in Rytov’s representation which is approximated by the phase structure function −3/5

and 0 = (0.545Cn2 k2 z) is the coherence length of a spherical wave propagating in the turbulent medium with Cn2 being the structure constant. Using Eqs. (1)–(7), one can express the spectrum of a phase-locked partially coherent flat topped laser beam array at the receiving z-plane in the following form: S(p, q, z, ω) =

M M N N N  ω 2    

2cz i

j

m

n

t

ijmnt

∞

(8)



−∞

−∞

∞

(−1)n+m−2 S (0) (ω) N3 −∞

w02

 −

 N

N

n

m



2

2

2z

(p, q, , , ω)]

= exp[−0.5 D (x − , y − )]



−n[(x − ai )2 + (y − bi ) ]

 ik

× exp ∗

∞

2

The last term in the integrand of Eq. (6) can be expressed as [12–21]: exp[ (p, q, x, y, ω) +



−∞

[(p − x)2 + (q − y)2 − (p − ) ∗

∞

ijmnt =

d d W (0) −∞

× exp[ (p, q, x, y, ω) +

Fig. 1. Schematic diagram of the circular and rectangular laser beam array.

∞

dx dy

× (x, y, , , z = 0, ω) × exp





∞

d

(5)

2

−m[( − aj ) + ( − bj ) ]

−



w02



2

[(p − x)2 + (q − y)2 − (p − ) − (q − )2 ]



1

2

02

2

[(x − ) − (y − ) ]

dx dy d d

(9)

Calculating the related integral in Eq. (9) we can obtain the following expression for propagated spectrum:

S(p, q, z, ω) =

M N M  N  N  ω   

2cz i

  ×

N

N

n

m

2 × exp ˛1 ˛2



 × exp



j

ij2 + b2ij 4˛1

m

n

t

S (0) (ω)

(−1)n+m−2 N3

−n(a2i + b2i ) − m(a2j + b2j )



ω02



 exp

x2 + y2 4˛2

 (10)

D. Razzaghi et al. / Optik 124 (2013) 2135–2139

2137

Fig. 2. Normalized spectra S(ω) of a phase-locked partially coherent flat topped beam, 0 = 0.02 m, N = 1, d = 2w0 , z = 5 km, solid line S (0) (ω); dashed lines S(ω) in turbulence; dot lines S(ω) in free space.

where

˛1 =

3. Turbulence induced changes in spectrum

m+n w02

bij = 2

˛2 =

vx =

+

w02

−

iω , zc

ij = 2

w02

,

w02

w02 ω02

m−n

nbi + mbj

m+n

nai

ˇ1 =

,

nai + maj

−

+

t 202 (ω)

maj

ω02 2ˇ1 bij 4˛1

+ −

−

2ˇ1 ij 4˛1

ˇ12 4˛1 −

+

ik p, z

1

(11)

02

vy =

To investigate turbulence induced changes in spectrum, We begin from Eq. (10). Our motivation is to study the influence of the beam order, transverse coherence width, and turbulence strength on spectrum. First of all it is necessary to consider a functional form for the source spectrum. Supposing a Lorentzian line shape centered at frequency ω0 , normalized spectrum in the source plane can be written as bellow: S (0) (ω) =

nbi ω02

−

mbj ω02

ik q z

Eq. (10) is the final analytical expression derived, for investigating turbulence induced changes in the spectrum. For a phase-locked partially coherent flat topped laser beam array, It (Eq. (10)) enables us predicting phenomenon occurring as a result of propagation in turbulent atmosphere as well as quantifying effects of source and turbulent medium characteristics on spectrum content variations.

2 (ω − ω0 )2 +  2

,

(12)

where  is the half-width at half-maximum (HWHM). 0 (ω) is intentionally adjusted so that the spectrum of the propagated beams to becomes different from the source spectrum. Next, a geometrical structure should be supposed for the array. Among various possible configurations, we considered a circular and rectangular laser beam array located symmetrically at the source plane with 6 equal beamlets, as shown in Fig. 1. Using Eqs. (8)–(12), numerical calculations have been performed which clarified behavior of the spectrum for a phase-locked partially coherent flat topped laser beam array in turbulent atmosphere. In the following numerical calculations, we take

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D. Razzaghi et al. / Optik 124 (2013) 2135–2139 1

1.6 d=2w0 d=3w0 d=4w0 d=5w0

ρ C (m) 1.4 1.2

0.9

ρ C (m ) 0.8

1

0.7

0.8

0.6 0.5

0.6

0.4

0.4 0.2

circular rectangular

4000

5000

6000

7000

8000

3000

9000

3500

4000

4500

5000

5500

6000

6500 7000

Propagaon distance (m) Propagaon distance (m) Fig. 3. The behavior of the C versus propagation distance for various beamlet distances, with N = 2.

ω0 = 3.2 × 1015 rad s−1 ,  = 0.6 × 1015 rad s−1 , w0 = 0.05 m, and Cn2 = 10−14 for atmospheric turbulence and Cn2 = 0 for free space. Normalized spectrum at source and observation planes for free and turbulence atmosphere are shown in Fig. 2. Fig. 2a shows propagated normalized spectrum for  = 0.33 m,  denotes for distance from center of observation plane. Comparing source and propagated spectrum in this figure, it can be seen that in addition to the main peak, another peak appears in the spectrum. The position and strength of this additional peak is depended on  so that for a special C , called critical radius, its magnitude will be as same as the main peak. Beyond critical radius(Fig. 2d), dominant peak interchanges and coincides on newly appeared peak. On the other hand, spectral transition occurs for a special distance from receiver center which is known as rapid transition. This behavior permits us to state that for non central portion of the receiver, turbulence has no significant effects on spectrum and is important mainly for radius smaller than critical value. It is worth mentioning that, rapid transition only exists in turbulent media and in free space propagation is not seen. So experts should consider this famous phenomenon, when optical designing of the receiver is a matter of subject. Based on importance of rapid transition, critical radius dependence on geometrical structure of the array beam, finds a vital aspect, so it will be discussed in what follows. At first, supposing a circular structure, critical radius dependency on beam lets distance is investigated. Fig. 3 shows critical radius variation versus propagation distance for various beamlet 1.2

ρ c (m) 1

N=1 N=2 N=3 N=4

0.6

0.4

4000

0.55 circular rectangular

ρ C (m) 0.5

0.45

0.4

0.35

0

0.005

0.01

0.015

0.02

0.025

Correlaon length (m) Fig. 6. The behavior of the C versus the correlation length.

distances. Although all curves are progressive, but there is a fact here, the more the beam let distance the more the critical radius. Secondly, flat topped degree effects on critical radius has been investigated and is shown that this parameter has an effect similar to beam let distance as is evident in Fig. 4. Next, we were interested to see whether geometrical shape had any effects on critical radius, so a rectangular structure for array beam has been supposed. We found that between a circular shape of radius “d” and a rectangular shape of diameter “d”, circular shape exhibits a larger critical radius. This point is shown clearly in Fig. 5. Finally we investigated; the effects of coherence length on critical radius value for two proposed geometrical shapes in Fig. 6. It has been shown that a nonlinear decreasing in critical radius exists when coherence length increases for both shapes, but decreasing rate is more severe for rectangular shape. 4. Conclusion

0.8

0.2 3000

Fig. 5. The behavior of the C versus propagation distance for a circular shape of radius d and a rectangular shape, with d = 4w0 , N = 2.

5000

6000

7000

8000

9000

10000

Propagaon distance (m) Fig. 4. The behavior of the C versus propagation distance for different order of circular flat topped beam, with d = 3w0 .

In this article, an analytic expression for the spectrum of phaselocked partially coherent flat topped laser beam array propagating through turbulent atmosphere has been derived and used to study spectral changes of these beams. It has been shown that, there exists a rapid transition of the spectrum which takes place at critical radius C . It has been found that increasing the beam let distance, causes the critical radius to increase. It is also proved that order of flat topped beam, has a similar effect on the critical radius. We noticed that the circular shape poses a larger critical radius in compare with rectangular shape. Finally effects of coherence length on critical radius is studied. Although, decrease of critical radius for larger amount of coherence length is observed for both circular

D. Razzaghi et al. / Optik 124 (2013) 2135–2139

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