Turbulence induced changes in spectrum and time shape of fully coherent Gaussian pulses propagating in atmosphere

Optics Communications 283 (2010) 2318–2323

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Turbulence induced changes in spectrum and time shape of fully coherent Gaussian pulses propagating in atmosphere D. Razzaghi a,*, F. Hajiesmaeilbaigi a, 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 4 October 2009 Accepted 22 January 2010

a b s t r a c t Propagation of fully coherent pulses with Gaussian spatial distribution through turbulent atmosphere is studied. Turbulence induced changes in spectrum of propagated pulse and its effect on temporal behavior of the signal is discussed and has been showed that signal is widened in time domain as propagating in turbulent atmosphere. It is also proved that various points in observation plane experience different effects. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Propagation induced changes in the spectrum of partially coherent light through atmospheric turbulence have attracted considerable attention recently [1–5]. It has also been realized that the broad spectrum of partially coherent light can lead to propagation through atmospheric turbulence induced changes in the shape of an optical beam [6,7]. It was shown in 1990 that a partially coherent, continuous-wave, Gaussian beam whose spot size at the beam waist is frequency independent does not remain Gaussian on propagation in atmospheric turbulence [3]. One may ask if that would also happen for a coherent Gaussian beam consisting of ultra short optical pulses such that the field spectrum is quite broad. Surprisingly, a definite answer to this question is not provided in the literatures. Of course, propagation of ultra short optical pulses in a linear optical medium has been studied extensively in recent years [8–13]. The propagation properties of ultra short pulsed beams with constant diffraction length in free space and dispersive media have been studied both analytically and numerically [14–18]. However, because of the mathematical difficulty in treating ultra short pulsed beams propagated through atmospheric turbulence, only numerical results or an approximate propagation expression in the far field are found. So the purpose of the present work is to study the propagation properties of the ultra short pulsed beams through atmospheric turbulence. Specially, a pulse with Gaussian spatial distribution is considered and some features of turbulence induced changes in temporal behavior of the pulse is predicted.

* Corresponding author. E-mail address: [email protected] (D. Razzaghi). 0030-4018/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.01.048

2. Basic theory Consider a pulse which has a spatial Gaussian form as below:

  r2 Eðr 0 ; z ¼ 0; tÞ ¼ A exp  02 ; w0

ð1Þ

where w0 is the beam waste, r0 is radial distance, and A is constant which is supposed to be one for simplicity. Moreover temporal part of the pulse is assumed to be Gaussian form at source plane so that spatiotemporal form of the pulse can be written as following form:

  r2 Eðr 0 ; z ¼ 0; tÞ ¼ f ðtÞ exp  02 ; w0

ð2Þ

where f(t), the temporal part, is:

"  2 # t cosðxc tÞ; f ðtÞ ¼ exp  ag Tc

ð3Þ

1

where ag ¼ ð2 ln 2Þ2 , xc is the carrier frequency, Tc is the pulse duration. We now consider pulse propagation through atmospheric turbulence using the paraxial form of the extended Huygens–Fresnel principle [2] so that each Fourier component of the pulse is propagated as below:

Eðr; z; xÞ ¼



k

2pz

2 Z Z

"

ðr  r 0 Þ2 d r0 Eðr 0 ; z ¼ 0; xÞ  exp ik 2z

#

2

 exp½wðr; r 0 ; z; xÞ;

ð4Þ

where r0 and r are position vector in source plane and observation plane, respectively, E(r0, z = 0,x) and E(r, z, x) represent Fourier components of input and propagated pulse, and finally w is a random phase factor representing the turbulence effect.

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Using Eq. (4), one can calculate the power spectrum at the observation plane as below:

hSðr; z; xÞi ¼ hEðr; z; xÞE ðr; z; xÞi  2 Z Z k 2 ¼ d r 01 2pz Z Z 2  d r02 hEðr 01 ; z ¼ 0; xÞE ðr 02 ; z ¼ 0; xÞi " # ðr  r 01 Þ2  ðr  r 02 Þ2  exp ik 2z  hexp½wðr; r01 ; z; xÞ þ w ðr; r 02 ; z; xÞi:

where Dw(r01  r02) is the wave structure function and U ¼ ð0:545C 2n k2 zÞ6=5 with C 2n being the structure constant of the refractive index of the media. Substituting Eqs. (6) and (2) into Eq. (5) and calculating related integral following relation is extracted for propagated power spectrum:

Sðr; z; xÞ ¼

   x 2 p2 x2 2 2 Sð0Þ ðxÞ exp ð q þ q Þ ; y 2pcz ac 4cc2 z2 x

where c is the speed of light in vacuum and other parameters are as below:

ð5Þ

a¼

1 ; 2w20

g¼

2 ix þ ; w20 cz

hexp½wðr; r 01 ; z; xÞ þ w ðr; r 02 ; z; xÞi ¼ exp½0:5Dw ðr01  r 02 Þ 2

¼ exp½Uðr 01  r02 Þ ;

" ð6Þ

S ðxÞ ¼ TF ½f ðtÞ ¼

1 g2  þ U: 2 4 a 8w0

ð8Þ

Z

1

ð2pÞ1=2

#2

1

½f ðtÞe

ixt

dt

ð9Þ

:

1

1

a

0.9

b

0.8

Normalized Spectrum

0.8

Normalized Spectrum

2

ð0Þ

1

0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0 -6

c¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Furthermore Sð0Þ ðxÞ is Fourier transform of f(t) which is derived directly from temporal shape of the pulse as below:

The last term in the integrand of Eq. (5) is [2]:

0.9

ð7Þ

-4

-2

0

2

4

Frequency (rad/s)

6

0 -6

-4

x 1015

-2

0

2

Frequency (rad/s)

4

6 x 1015

1

c Normalized Spectrum

0.8

0.6

0.4

0.2

0

-1

-0.5

0

Frequency(rad/s)

0.5

1 x 1016

Fig. 1. (a) Normalized spectrum of initial pulse. w0 = 1 mm, Tc = 1.66 fs, xc ¼ 1:77 rad fs1. (b) Normalized spectrum of the propagated pulse in free space w0 = 1 mm, 1 z = 10 km, Tc = 1.66 fs, xc ¼ 1:77 rad fs , r = 80 mm. (c) Normalized spectrum of the propagated pulse in turbulence. w0 = 1 mm, C 2n = 1.2 * 1013, z = 10 km, Tc = 1.66 fs, xc ¼ 1:777 rad fs1 , r = 80 mm.

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1

28

a

24 22 20 18 16 14 0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

b

0.9

relative turbulence indiced peak

Frequency (rad/fs)

26

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.8

2.6

0.85

0.9

0.95

1

1.05

1

Fig. 2. (a) Turbulence induced peaks position versus turbulence strength w0 = 1 mm, z = 5 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 70 mm and (b) relative turbulence induced 1 peak value versus turbulence strength. w0 = 1 mm, z = 5 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 70 mm.

1

1

a

0.9 0.8

Normalized Spectrum

Normalized Spectrum

0.8

b

0.6

0.4

0.2

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 -6

-4

-2

0

2

4

Frequency (rad/s)

0 -6

6

-4

-2

0

2

4

Frequency (rad/s)

x 1016

6 x 1016

1

c

0.9

Normalized Spectrum

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

-6

-4

-2

0

2

Frequency(rad/s)

4

6 x 1016 1

Fig. 3. (a) Normalized spectrum of the propagated pulse in turbulence w0 = 1 mm, C 2n = 1.2  1014, z = 6 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 20 mm. (b) Normalized 1 spectrum of the propagated pulse in turbulence w0 = 1 mm, C 2n = 1.2  1014, z = 6 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 20 mm. (c) Normalized spectrum of the propagated 1 pulse in turbulence. w0 = 1 mm, C 2n = 1.2  1014, z = 6 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 60 mm.

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To study spatio temporal behavior of the pulse, we characterize it by its autocorrelation magnitude which is inverse Fourier transform of the spectral density and use term intensity instead.

" hIðr; z; xÞi ¼ T1 F ½Sðr; z; xÞ ¼

1

ð2pÞ1=2

Z

1

ð10Þ

1

Although we tried to solve Eq. (10) analytically, only succeed to perform the inverse Fourier transform in free space which results are as below: x2 c 2p

 02    0  4w40 e x þ x0 þ x00 x þ 2b x4b02 x pffiffiffi e  þ erfc 2 3=2 15c2 z2 8p 2 b 4b 4b  2      2   2 2 x x x þ 2b 4b x x þ 2b 4b x e erfc pffiffiffi þ e erfc pffiffiffi þ 2 b 2 b 4b3=2 4b3=2  002   00  x þ 2b x4b002 x pffiffiffi ; erfc e ð11Þ  2 b 4b3=2

Iþ ðr; z; tÞ ¼

where ‘‘+” is used to show that only positive radian frequency is considered and the other parameters are as below:

15 1 þ ; 8w20 2p  2 ag p¼ : Tc

x1 ¼

ipt þ xc ; p

x2 ¼

ipt  xc ; p

x3 ¼ it

ð14Þ

in turbulence, only numerical calculation is carried out and free space solution is exploited, as a criteria, to check out validity of numerical calculations used to predict propagation of the pulse in turbulence.

1 0.9

b

0.8

Normalized Intensity

Normalized Intensity

ð13Þ

where

a

0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0 -2.5 -2

-1.5 -1

-0.5

0

0.5

1

t (s)

0 -2.5

1.5 2 x 10-15

-2

-1.5

-1

-0.5

0

0.5

1

1.5 2 x 10-15

0.5

1

1.5 2 x 10-15

t (s)

1

1

c

0.9

0.8 0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0 -2.5 -2

d

0.8

Normalized Intensity

Normalized Intensity

ð12Þ

" pffiffiffiffi x2 rffiffiffiffi 2 # c 3 x2 i 4w40 e 2p X p p xi 4b ; Iðr; z; tÞ ¼ e þ 15c2 z2 8p i¼1 b 2b 2b3=2

0.8

0.9

x00 ¼ it;

In free space by considering the whole frequency, a more simple relation for intensity will result as below:

1 0.9

ipt  xc ; p q2x þ q2y ; b¼ 4z2 c2 c x0 ¼

c¼

# ½Sðr; z; xÞeixt dt

ipt þ xc ; p

x¼

-1.5 -1

-0.5

t (s)

0

0.5

1

1.5 2 x 10-15

0 -2.5

-2

-1.5

-1

-0.5

0

t (s) 1

Fig. 4. (a) Normalized intensity of the propagated pulse in free space w0 = 1 mm, C 2n = 1.2  1014, z = 8 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 60 mm. (b) Normalized 1 intensity of the propagated pulse in turbulence w0 = 1 mm, C 2n = 1.2  1014, z = 8 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 30 mm. (c) Normalized intensity of the propagated 1 pulse in turbulence w0 = 1 mm, C 2n = 1.2  1014, z = 8 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 60 mm. (d) Normalized Intensity of the propagated pulse in turbulence. 1 w0 = 1 mm, C 2n = 1.2  1014, z = 8 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 90 mm.

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3. Numerical results and discussion

25

20

T (fs)

15

10

5

0

0

10

20

30

40

50

60

70

80

r/w0 Fig. 6. propagated pulse duration in various observation points. w0 = 1 mm, 1 C 2n = 1.2 * 1014, z = 10 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs .

0.35

turbulese induced signal intensity

To investigate turbulence induced variation in pulse shape, beginning from the spectrums of the initial and propagated pulse in turbulence and free space, seems to be convenient. Fig. 1, showing these spectrums, clearly exhibits steady nature of the pulse spectrum when propagated in free space. Furthermore it indicates that turbulence media, can considerably affects propagated spectrum. Deformation of the pulse spectrum, as is evident from previous section, is mainly depended on beam and atmosphere parameters such as beam waist, carrier frequency, pulse duration, turbulence strength, propagation distance, detection point etc. Although, in theory, it is possible to change turbulence induced term in the overall spectrum formula, by changing any parameters mentioned above, However nearly same behavioral changes is observed. According to Fig. 1c, a narrow sideband peak is appeared in pulse spectrum which grows by increasing turbulence induced term and shifts simultaneously towards central portion. To explain this behavior more clearly, turbulence induced peak position and its relative value is plotted in Fig. 2. Fig. 2a shows the turbulence induced peak position and Fig. 2b its relative size versus turbulence strength. In Fig. 3, propagated pulse spectrum is plotted for various detection points. We see that various positions in receiver plane, senses different spectrums and as a result different signals are expected to be generated. Comparing these figures with those generated by changing turbulence strength, shows that both have similar behavior. On the other hand, far from center portion of the receiver experiences turbulence more sever. In the time domain, turbulence changed spectrum shows itself as distortion of the main signal in its real duration and also generating side signals over the real signal duration. Fig. 4 shows the propagated signal as can be seen in different receiver point. As discussed, signal expansion in time domain is evident and is more serious in off center portions. Investigation showed that side signal is a nearly sinusoidal wave modulated with a damping signal as shown in Fig. 5. By use of second moment formula for pulse duration [19], in Fig. 6 a plot of pulse duration, is presented showing pulse widening in turbulence media in which parts of the signal below 0.1 * Imax, is ignored. Also it is apparent that ratio of turbulence induced signal power to main signal power will increase by increasing observation point distance from center. For example the ratio is calculated to be 0.120 for r = 10w0 and 0.138 for

0.3

0.25

0.2

0.15

0.1

0.05

0

-6

-4

-2

0

2

t (s) Fig. 7. turbulence induced signal. w0 = 1 mm, 1 Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 10 mm.

4

6 x 10-15

C 2n = 1.21014,

z = 10 km,

r = 50w0. This has been computed by driving turbulence induced signal, Fig. 7, and calculating time integral of it and the main signal.

1 0.9

4. Conclusion

Normalized Intensity

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -1.5

-1

-0.5

0

t (s)

0.5

1

1.5 x 10-14

Fig. 5. Normalized Intensity of the propagated pulse in turbulence w0 = 1 mm, 1 C 2n = 1014, z = 10 km, Tc = 0.266 fs, xc ¼ 2:36 rad fs , r = 10 mm.

We have studied turbulence induced changes in the spectrum of a pulse and noticed that magnifying the turbulence term, will cause side peaks to appear in the propagated spectrum. These peaks grows by increasing turbulence induced term and shifts simultaneously towards central portion. In addition temporal behavior of the propagated pulse has been studied in different point of receiving plane. It was shown that off center points senses turbulence more severely and as a result wider signal is expected to reach to these point than central portion. Numerical graphs also showed that, under equal conditions, turbulence induced signal is more powerful in off center parts of observation plane. It is important to note that signal distortion as a result of propagation in turbulence, shows a regular predictable pattern so is different from wideband noises. On the other hand, inverse filter or narrow band pass filter can be designed to avoid turbulence induced signal in receiver. At last authors have not found a

D. Razzaghi et al. / Optics Communications 283 (2010) 2318–2323

noticeable change for pulses even with several femtoseconds duration, by choosing typical beam and atmosphere parameters, and only below femtosecond pulses showed sensitivity to atmospheric turbulence which implies that for pulses longer than several femtoseconds atmospheric turbulence can be safely unnoticed by designers, the procedure which will not be true in more strong turbulent media. References [1] [2] [3] [4] [5]

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