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Optical pulse propagation through a slab of randommediumCynthia Young Hopen aa Department of Mathematics and the Florida Space Institute , University of CentralFlorida , Orlando, FL, 32816, USAPublished online: 19 Aug 2006.

To cite this article: Cynthia Young Hopen (1999) Optical pulse propagation through a slab of random medium, Waves inRandom Media, 9:4, 551-560, DOI: 10.1088/0959-7174/9/4/307

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Waves Random Media9 (1999) 551–560. Printed in the UK PII: S0959-7174(99)00835-6

Optical pulse propagation through a slab of random medium

Cynthia Young HopenDepartment of Mathematics and the Florida Space Institute, University of Central Florida,Orlando, FL 32816, USA

Received 12 January 1999, in final form 5 July 1999

Abstract. Tractable analytic expressions are developed for the expected broadening that aGaussian space–time optical pulse experiences as it propagates through a slab of random mediumconfined to a portion of the propagation path between the input and output planes. The analyticresults developed in this paper are based on the modified von Karman spectrum for refractive indexfluctuations and the Rytov method which restricts the validity of the results to theweakopticalturbulence regime. It is further assumed that the Gaussian beam is collimated and that the Gaussianpulse is narrowband with respect to the carrier frequency which, for optical frequencies, requirespulse widths greater than 20 fs. The results in this paper are given for both near and far fields andit is shown that when the thickness of the slab of random medium is the total propagation distance,the results given in this paper agree with the results for an extended random medium. Examplesare given for unmanned air vehicle (UAV) to low Earth orbit (LEO) satellite and LEO-to-LEOcrosslinks.

1. Introduction

The study of short optical pulsed signals through the atmosphere has received considerableattention over the years [1–8]. This interest is mainly due to the promising large bandwidthsat high frequencies that will enable higher data rate satellite communication over that ofconventional radio frequency systems. The atmosphere can cause a distortion of the opticalpulse shape during propagation. The received pulse is a wider (broadened) pulse, particularlyfor narrow signals in the femtosecond–picosecond (10−15–10−12 s) regime.

Recently, Younget al [8] obtained an analytic expression governing the broadening that anoptical pulse experiences as it propagates through the Earth’s atmosphere (extended medium)in theweakfluctuation regime. However, there are applications in which the random mediumexists only over a portion of the propagation path and the remaining portions of the path arethrough free space. For example, in some satellite-to-satellite optical communication links onlythe middle portion of the propagation path is through the Earth’s upper atmosphere. When therandom medium is confined to only a portion of the path it can be modelled as a random phasescreen ofarbitrary thickness. Although theoretical analyses of pulse broadening throughrandom phase screens (weakor strong) do not appear in the literature, many investigatorshave studied general beam wave (CW) propagation through random phase screens [9–14].Parryet al [9, 11] measured the intensity fluctuations of a beam as it propagates through arandom phase screen. In those experiments the phase screen was produced in the laboratoryby turbulent mixing of hot and cold air and they found that the intensity fluctuations developedthrough two distinct mechanisms: focusing and speckle. It is well known that the focusingregime is classified as a moderate turbulence phenomenon and the speckle effect occurs in

0959-7174/99/040551+10$30.00 © 1999 IOP Publishing Ltd 551

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INPUTPLANE

OUTPUTPLANE

PHASESCREEN

L2L1 L3

z = 0

TurbulenceCells

z = L

LaserReceiver

Figure 1 : Phase screen geometry

Figure 1. Phase screen geometry.

strong optical turbulence known as the saturation regime. Jakemanet al [10, 12] developedtheoretical expressions for scintillation and compared the analytic and numerical results withexperimental measurements of a beam propagating above a propane flame. They classifiedphase screens characterized by focusing and speckle asdeeprandom phase screens whichcorresponds tomoderateto strongoptical turbulence. For an unmanned air vehicle (UAV)-to-satellite communication channel, the optical turbulence is classified asweakso the theory ofdeepphase screens does not apply. We classify a random phase screen asthin when the opticalwave experiences only phase fluctuations upon leaving the phase screen, whereas a phase screenof arbitrary thickness can cause both amplitude and phase fluctuations. Bookeret al [13]showed that athin phase screen placed midway between the source and receiver can be used torepresent plane wave propagation through an extended random medium. Andrewset al [14]developed analytic expressions for the statistical quantities of a beam wave (CW) propagatingthrough athin random phase screen and formulated integral expressions forarbitrarily thickphase screens based on the second order Rytov approximation. Although the Rytov method forplane waves and spherical waves is adequate in studying phase fluctuations under all conditionsof irradiance fluctuations, this is not true in general for beam waves. For Gaussian beam wavesthe Rytov method is valid only in the weak fluctuation regime. However, in this study we areconsidering both near and far fields which behave like plane and spherical waves respectively,so the Rytov method may apply beyond the weak irradiance regime.

In this paper, an analytic closed-form expression is developed for the expected temporalbroadening that an optical pulse experiences as it propagates through aweakrandom phasescreen ofarbitrary thickness. This involves the formulation in section 2 of a two-frequencymutual coherence function (MCF) for a Gaussian space–time pulse propagating through aslab of random medium. Based on the two-frequency MCF, the implied temporal broadeningof an optical pulse propagating through a random slab ofarbitrary thickness is calculated insection 3. The implied pulse broadening depends on the strength of optical turbulence along thepropagation path which can be deduced from the scintillation that the optical wave experiencesas it propagates through the channel. Thus, an expression for the scintillation of a continuouswave propagating through an arbitrarily thick slab is obtained in section 4, and applications of

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Optical pulse propagation 553

the theoretical results to UAV–LEO (low Earth orbit) and LEO–LEO crosslinks are given insection 5.

The general model for the phase screen consists of a slab of random medium arbitrarilylocated between source and receiver, as illustrated in figure 1.

It is assumed the transmitter is located atz = 0 and propagation is along the positivez axis. It is further assumed that the random medium exists only between the planesz = L1

andz = L1 +L2 and that the receiver is located atz = L, whereL = L1 +L2 +L3.

2. Random slab model for the two-frequency MCF

The two-frequency MCF is defined by

0(r1, k1, r2, k2, L) = 〈U(r1, k1, L)U∗(r2, k2, L)〉 (1)

where〈·〉 denotes the ensemble average,U(r, k, z) is the optical field,r is a vector in thetransverse plane at propagation distancez, andk denotes the optical wavenumber at the point(r, z). Using Rytov theory it can be shown that in the weak fluctuation regime, the two-frequency MCF of a Gaussian beam propagating through a weakly fluctuating random mediumis given [5] by

0(r1, k1, r2, k2, L) = 00(r1, k1, r2, k2, L)exp[2E1(0, k1, 0, k2) +E2(r1, k1, r2, k2)] (2)

where00(r1, k1, r2, k2, L) is the two-frequency MCF in free space given by

00(r1, k1, r2, k2, L) = 〈U0(r1, k1, L)U∗0 (r2, k2, L)〉 (3)

with

U0(r1, k1, L) = (21− i31) exp(ik1L) exp

[ik1

2L(21 + i31)r

21

]U ∗0 (r2, k2, L) = (22 + i32) exp(−ik2L) exp

[− ik2

2L(22 − i32)r

22

] (4)

and the effect of the optical turbulence is given by

E1(0, k1, 0, k2) = −π2(k21 + k2

2)

∫ L

0

∫ ∞0κ8n(κ, z)dκ dz

E2(r1, k1, r2, k2) = 4π2k1k2

∫ L

0

∫ ∞0κ8n(κ, z)exp

[− iκ2L

2

(1− z

L

)(γ1

k1− γ

∗2

k2

)](5)

× J0{κ|(γ1r1− γ ∗2 r2)|

}dκ dz

where8n(κ, z) is the spatial power spectrum of refractive index fluctuations,J0(x) is thezero-order Bessel function, andk = 2π/λ, whereλ is the optical wavelength. The Gaussianbeam parameters are defined by

γ1 = 1− (21 + i31)(1− z

L

)γ ∗2 = 1− (22 − i32)

(1− z

L

)(6)

21 = �0

�20 +�2

1

31 = �1

�20 +�2

1

21 = 1−21

22 = �0

�20 +�2

2

32 = �2

�20 +�2

2

22 = 1−22

�0 = 1− L

R0�1 = 2L

k1W20

�2 = 2L

k2W20

(7)

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where the quantitiesW0 andR0 are respectively the beam radius and phase front radius ofcurvature at the transmitter. For a collimated beam (�0 = 1) in the near field (�1,2 � 1), wesee that

21,2 = 1 21,2 = 0 31,2 = �1,2 γ1 = γ ∗2 = 1 (8)

and in the far field (�1,2� 1) we see that

21,2 = 0 21,2 = 1 31,2 = 1

�1,2γ1 = γ ∗2 =

z

L. (9)

Introducing the parametersωd = (k1− k2)/c andωc = (k1 + k2)/2c wherec is the speedof light, we can now write the two-frequency MCF (2) of a collimated Gaussian beam whenr1 = r2 = r as

0NF(r, ωc, r, ωd, L) = exp

[−2r2

W 20

]exp

[−αω2

d +iL

cωd

](10)

0FF(r, ωc, r, ωd, L) =(W 2

0ωc

2Lc

)2

exp

[−αω2

d + i

(L

c+r2

2Lc

)ωd − 2

(W0r

2Lc

)2

ω2c

](11)

where we have made the narrowband assumption,ωd � ωc. NF and FF refer to near field andfar field, respectively, andα is given by

α = 2π2

c2

∫ L

0

∫ ∞0κ8n(κ, z)dκ dz. (12)

The random medium in figure 1 is assumed to exist only over the intervalL1 6 z 6 L1+L2

along the propagation path. Thus, it is convenient to introduce the normalized distance variablegiven by

1− z

L= d3(1 +d2η) 06 η 6 1 (13)

whered2 = L2/L3 andd3 = L3/L. This effectively changesα, equation (12), to

α = 2π2

c2Ld2d3

∫ 1

0

∫ ∞0κ8n(κ, η)dκ dη. (14)

Using the modified von Karman spectrum for refractive-index fluctuations given by

8n(κ) = 0.033C2n

exp[−κ2/κ2m]

[κ2 + κ20]11/6

(15)

where C2n is the refractive-index structure constant of the phase screen,κm = 5.92/l0,

κ0 = 1/L0, and l0 and L0 are the inner and outer scale sizes of the random slabs, we findthat the parameterα, equation (14), is now given by

α = 0.3908C2nLd2d3L

5/30

c2= 0.3908C2

nL2L5/30

c2. (16)

3. Predicted broadening of the Gaussian pulse

To obtain explicit expressions concerning the time domain spreading of a pulse wavepropagating through a random phase screen, let us assume that the input waveform is theGaussian pulse

vi(t) = exp

[− t

2

T 20

](17)

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Optical pulse propagation 555

where we identify the quantityT0 with the input pulse half-width. The Fourier transformof (17) is

Vi(ω) =∫ ∞−∞

exp

[− t

2

T 20

]exp[iωt ] dt = √πT0 exp(− 1

4ω2T 2

0 ) (18)

which has spectral half-width1ω = 2/T0 as defined by the 1/e point. For short pulseson the order of femtoseconds (fs), the bandwidth defined by1f = 1ω/(2π) = 2/(πT0)

is considered ultrawide. Nonetheless, at optical frequencies on the order off0 = c/λ ∼3×1014 Hz, the resulting waveform may still be considered narrowband relative to the carrierfrequency:1f ≪ f0. To preserve the narrowband assumption, therefore, we limit ourdiscussion to pulse widths satisfyingT0 > 20 fs.

The implied temporal broadening of the pulse is deduced from the on-axis temporal meanintensity. The on-axis temporal mean intensity for a Gaussian input pulse, equation (17), isgiven [8] by

〈I (0, L, t)〉 = T 20

4π

∫ ∫ ∞−∞

exp[− 1

2ω2cT

20 − 1

8ω2dT

20

]02(0, ωc + ω0, ωd, L)

× exp[−iωdt ] dωc dωd. (19)

Upon substituting (10) and (11) into (19) and performing the integration, we find that theon-axis mean intensity is given by

〈IFF(0, L, t)〉 =(W 2

0

2Lc

)2(1 +ω2

0T20 )

T0TFFexp

[−2(t − L/c)2

T 2FF

](20)

〈INF(0, L, t)〉 = T0

TNFexp

[−2(t − L/c)2

T 2NF

](21)

whereTNF andTFF are estimates of the received pulse half-width in near and far fields givenby

TNF = TFF = (T 20 + 8α)1/2 (22)

and whereT0 is the initial pulse half-width and the parameterα is given by (16). This expressionfor the received pulse width, equation (22), has the same functional form as the results obtainedfor an extended random medium [5]. In the limit of free space,α = 0, equations (20) and (21)agree with Ziolkowski and Judkins [4].

4. Random slab model for scintillation

We see from equations (22) and (16) that the amount of broadening that an optical pulseexperiences as it propagates through an arbitrary thick slab of random medium depends on thestructure parameter of the slab,C2

n, the thickness of the slab,L2, and the outer scale size of theslab of random medium,L0. C2

n values of the random medium layer can be calculated fromthe scintillation that an optical wave experiences as it propagates through the phase screen.

The scintillation of a continuous wave (k1 = k2) propagating through a slab of randommedium is given [6] by

σ 2I = 2Re [E2 +E3] (23)

whereE2 andE3 are given for a collimated beam in the near and far fields by

E2 = 4π2k2Ld2d3

∫ 1

0

∫ ∞0κ8n(κ) dκ dη

E3 = −4π2k2Ld2d3

∫ 1

0

∫ ∞0κ8n(κ) exp[−κ2βNF/FF] dκ dη

(24)

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whereβ is given by

βNF = iLd3(1 +d2η)

kβFF = iLd3(1 +d2η)[1− d3(1 +d2η)]

k. (25)

Evaluating (23) for the Kolmogorov spectrum, equation (15), with an infinite outer scaleand zero inner scale, yields

σ 2I NF = 1.23C2

nk7/6L

11/63 [(1 +d2)

11/6− 1] (26)

σ 2I FF = 4.75C2

nk7/6L

11/63 Re

{i5/6[(1 +d2)

11/162F1

(− 56,

116 ; 17

6 ; d3(1 +d2))

− 2F1(− 5

6,116 ; 17

6 ; d3)]}

(27)

where2F1 is a generalized hypergeometric function.In the near and far fields, a Gaussian beam propagating through athin random phase

screen, behaves like plane and spherical waves respectively. If we assumed3 � 1, we findthat for the special case of a ‘thin’ phase screen,d2 � 1, equations (26) and (27) agree withthe results of Andrews and Phillips [5].

5. Applications of theoretical results

In optical communication systems we desire high data rates and low error rates. Althoughshorter pulses allow for higher data rates, they also experience more broadening (higher errorrates). Therefore,a priori knowledge of pulse broadening will aid engineers in the design ofoptical communication systems. We have shown in this paper that if a Gaussian space–timeoptical pulse (initial beam radiusW0 and initial half-widthT0) propagates through free spacea distance ofL1, through a random medium a distanceL2, and then through free space adistanceL3 as shown in figure 1, its received pulse width,T , is given by (22) and (16), withc = 3.0× 108 m s−1, L2 is the thickness of the slab in metres,L0 is the outer scale size ofthe random medium andC2

n is the structure parameter of the random medium which can beinferred from the scintillation index,σ 2

I , given by (26) and (27).

5.1. LEO–LEO crosslink

One scenario when a laser beam propagates through a slab of random medium positionedsomewhere between transmitter and receiver is the case of a crosslink between two low Earthorbit (LEO) satellites as shown in figure 2.

We see in (22) and (16) that the amount of broadening that an optical pulse experiencesonly depends on the thickness of the slab of random medium, the structure constant describingthe strength of fluctuations in the index of refraction, and the outer scale size. For propagationthrough the upper atmosphere, we assume outer scale sizes 10 m< L0 < 100 m. Forcollimated beams having diameters 1–10 cm, any LEO–LEO crosslink falls under the far-fieldassumption

2L

kW 20

� 1. (28)

Based on (27), the structure parameter,C2n, can be deduced from knowledge of the

scintillation index

C2n =

σ 2I FF

4.75k7/6L11/63 Re

{i5/6[(

1+L2L3

)11/62F1

(− 56,

116 ; 17

6 ; L2+L3L

)−2F1(− 5

6,116 ; 17

6 ; L3L

)]} .(29)

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Optical pulse propagation 557

Earth

6380km

200-500km

Orbit20km Atmosphere

LEO

Satellite

LEO

Satellite

L2 L3L1

Figure 2. LEO–LEO crosslink.

We see in figure 2 that in the case of the LEO–LEO crosslink,L1 ≈ L3, which means thatthe expected pulse broadening is the same regardless of which satellite is the transmitter andwhich satellite is the receiver.

Let us consider a horizontal crosslink between two LEO satellites in a 300 km orbit aroundthe Earth (of radius 6380 km) and let us assume the midpoint of the propagation through theEarth’s atmosphere,L2, is at an altitude of 19 km. Assuming circular orbits as shown in figure 2,in this example we letL1 = 1420 km,L2 = 226 km, andL3 = 1420 km. Since the structureparameter varies for altitudes between 19 and 20 km, we will assumeC2

n = 4.0× 10−19 [15]and we will assume an outer scale size,L0, of 50 m at those altitudes. Using these values in (16)we see thatα = 2.0×10−28. For an input pulse half-width ofT0 = 40 fs, the expected receivedpulse half-width, equation (22), isT = 56.6 fs, which corresponds to a 40% broadening ofthe optical pulse. In this example, the scintillation index, equation (27), is 0.62 which satisfiesthe weak optical turbulence restriction (σ 2

I < 1).

5.2. UAV–LEO and LEO–UAV crosslinks

Another crosslink scenario is the case of an optical communications link between an unmannedair vehicle (UAV) and a LEO satellite. We will use the term UAV–LEO to denote a crosslinkwhere the pulse is sent from the UAV to the LEO satellite and we will use LEO–UAV torepresent the pulse being sent from the LEO satellite to the UAV. In the UAV–LEO scenario,L1 = 0,L2 is the distance the beam travels from the UAV through atmospheric turbulence andL3 is the distance the beam travels through free space prior to reaching the satellite. For thecase of a LEO–UAV scenario,L1 is the distance the beam travels prior to reaching the Earth’satmosphere,L2 is the distance the beam propagates through the atmosphere prior to reachingthe UAV, andL3 = 0.

Since the amount of expected pulse broadening, equations (22) and (16), is independentof the propagation distances before and after the random medium,L1 andL3, both the UAV–LEO and LEO–UAV links will yield the same expected pulse broadening. These two scenariossatisfy the far-field assumption for beams on the order of 1–10 cm and hence are governedfor scintillation by (27). However, in the LEO–UAV case,L3/L � 1, which reduces (27)to equation (26). Therefore, to deduceC2

n for a UAV–LEO crosslink we useC2n as given by

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equation (29) and for a LEO–UAV crosslink we use

C2n =

σ 2I NF

2.255k7/6L11/62

. (30)

Let us consider the case of a UAV at an altitude of 18 km that transmits short optical pulsesthat are received at a LEO satellite orbiting at an altitude of 300 km. Assuming circular orbitsin this example, we letL1 = 0,L2 = 160 km andL3 = 1760 km. The altitude at the UAV is18 km and when the pulse exits the Earth’s atmosphere it is at an altitude of 20 km. Since thestructure parameter varies with altitude, for this example we will assume that theC2

n at 19 kmrepresents the average structure constant along the propagation path through the atmosphere.Using the Hufnagel–Valley model [15] with a ground-level turbulence of 1.7 × 10−13 andassuming an upper atmospheric wind velocity of 12 m s−1, the C2

n parameter at an altitudeof 19 km is estimated to be 4.0× 10−19. The expected broadening for an optical pulse withwavelength 1.55µm is given in figure 3 for optical pulses with varying half-widths and forthree different outer scale,L0, nominal values and for a structure parameter,C2

n = 4× 10−19.The optical scintillation (27) in figure 3 is 0.08 which constitutesweakoptical turbulence.

40 50 60 70 80 90 100

Initial pulse half-width To (fs)

0

10

20

30

40

50

60

70

80

90

100L 0 = 10mL 0 = 50mL 0= 100m

UAV to LEO Upper Atmosphere Cross-Link

λ = 1.55 µm , Cn2= 4 x 10

-19

% o

f Pu

lse

Bro

ade

nin

g

Figure 3. Expected pulse broadening in a UAV–LEO crosslink. The UAV altitude is 18 km.

We see in figure 3 that the amount of expected broadening is very sensitive to the outerscale size. From (22) and (16) we see that the broadening depends on outer scale, structureconstant, and thickness of the slab of random medium. In figure 4, we see the expected pulsebroadening in a UAV to LEO crosslink for different UAV altitudes. The structure constants,C2n, have been set to 6× 10−19, 4× 10−19, and 2× 10−19 for the 17 km, 18 km, and 19 km

UAV altitudes, respectively. In this case the scintillation, equation (27), is 0.18, 0.08, and 0.03for the respective 17 km, 18 km, and 19 km altitudes.

6. Discussion

In the design of a high data rate, low error rate optical communication system,a prioriknowledge of the expected pulse broadening is required. The only previously derived analytic

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Optical pulse propagation 559

40 50 60 70 80 90 100

Initial pulse half-width To (fs)

0

10

20

30

40

50

60

70

UAV Altitude = 17kmUAV Altitude = 18kmUAV Altitude = 19km

UAV to LEO Upper Atmosphere Cross-Link

λ = 1.55 µm , L0 = 50m

% o

f Pu

lse

Bro

ade

nin

g

Figure 4. Expected pulse broadening in a UAV–LEO crosslink for different UAV altitudes.

models governing the temporal broadening of an optical pulse are for the case of an extendedrandom medium. Most optical communication channels will be satellite to satellite whichis not the case of an extended random medium but rather a slab of random medium locatedbetween slabs of free space. The expressions given in this paper apply to propagation througha slab of random medium (propagation prior to and/or after the slab is assumed to be throughfree space). If we let the thickness of the slab approach the total propagation distance (L2 = L,L1 = L3 = 0) the expressions given in this paper, equations (22) and (16), which are valid inboth the near and far fields, agree with the broadening predicted for an extended medium [3].Examples are given for UAV–LEO crosslinks for different outer scale sizes and UAV altitudes.The expressions obtained in this paper are for collimated Gaussian beams in both the near andfar fields and applicable only to pulses greater than 20 fs propagating in weak optical turbulence(σ 2I � 1). Although general Gaussian beam expressions based on the Rytov method are only

valid in the weak fluctuation regime, we expect that the expressions developed in this paperfor near and far fields are valid into the strong irradiance regime.

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[12] Jakeman E, Klewe R C, Richards P H and Walker J G 1984 Application of non-Gaussian scattering of laser lightto measurements in a propane flameJ. Phys. D: Appl. Phys.171941–52

[13] Booker H G, Ferguson J A and Vats H O 1985 Comparison between extended-medium and the phase-screenscintillation theoriesJ. Atmos. Terrest. Phys.47381–99

[14] Andrews L C, Phillips R L and Weeks A R 1997 Propagation of a Gaussian-beam wave through a random phasescreenWaves Random Media7 229–44

[15] Beland R R 1993 Propagation through atmospheric optical turbulenceThe Infrared and ElectroOptical SystemsHandbookvol 2, ed F G Smith (Bellingham, WA: SPIE Optical Engineering Press) ch 2

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