Heterodyne detection of optical signals propagating through the atmosphere in the presence of scattering and turbulence

Volume 53, number 1

OPTICS COMMUNICATIONS

1 February 1985

HETERODYNE DETECTION OF OPTICAL SIGNALS PROPAGATING THROUGH THE ATMOSPHERE IN THE PRESENCE OF SCATTERING AND TURBULENCE Nobuhiro SAGA and Kazumasa TANAKA Nagasaki University, Faculty of Engineering, Department of Electronics, Nagasaki 852, Japan Received 21 September 1984

The effects of the spatial coherence degradation of optical signals on optical heterodyne detection are investigated. The expression of the signal-to-noise (SNR) is derived, taking into account the receiving optics, the atmosphere turbulence, and the scattering, in a form suitable for numerical computation. This expression is in terms of the mutual intensity function of the homogeneous signal field. It is shown that the effective detector radius is nearly equal to that of the Airy disk, and that the effective radius of the receiving aperture approximately equals the coherence length of the signal on the aperture plane.

1. Introduction Heterodyne detection has been investigated for optical signals whose spatial coherence is degraded by the scattering (e.g. molecules and aerosols), the atmospheric turbulence, and both [1-8]. The effect of degradation of the spatial coherence of the source itself must also be considered, which we do not take into consideration in this paper. We assume that the mutual intensity function (MIF) of the signal is known over the receiving aperture plane. Let the receiving optical system consist of the receiving lens located at the front end and a photodetector. This lens focuses the received light onto a photodetector located at the focus of the lens. For the derivation of the signal-to-noise ratio (SNR), we use the analytic expressions derived by Yura [2] for the heterodyne signal power. This expression, however, involves multiple integrals, and it is difficult to calculate it numerically. Recently, general formulas for calculating the SNR have been given by several authors [3,4,6-8], taking into account the receiving optics, the atmosphere turbulence, and the scattering. The formulas, however, take a rather long time to calculate numerically. The purpose of this paper is ( 1 ) t o derive the analytic expression of the SNR which is particularly well suited for numerical calculations and (2) to determine 0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

the effective heterodyne receiver size in the presence of scattering and atmospheric turbulence. Further, the performance ofheterodyne detection is examined for various parameter values of the medium and the receiving optics.

2. Average heterodyne signal power We assume the receiving optics as illustrated in fig. 1. The receiving lens with aperture radius a 1 and focal length f is located in the aperture plane. The photodetector, whose radius is a2, is located at the focus of the lens. The average heterodyne signal power (S) is given by

[21 Aperture

I

Beamsplitter

~ l e n s /

Detector

~

Signal

z=O

z=f

Fig. 1. Geometry of heterodyne detection.

u:('2) d2rzl2) '

(1)

where ( ) denotes an ensemble average and U~ is the local oscillator field. The integration is carried out over the areaA 2 of the photodetector, and

ce = ½(r~eglhv)2R,

(2)

where r/is the quantum efficiency of the photodetector,g is the gain associated with the current amplifier, and R is the resistance of the load. Moreover e, h, v are the electronic charge, Planck's constant, and optical frequency, respectively. In eq. (1), Ud can be expressed by the HuygensFresnel formula as follows Ud(r2) = (ik/2n)

1 February 1985

OPTICS COMMUNICATIONS

Volume 53, number 1

fUs(r l) t(r I ,r 2) d2rl ,

2+ik~lri'r2)(7)

X f d 2 r 2 U : ( r 2 ) e x p ( - i k2fr2

A2

X f d 2 r 2 UiT(r2) exp i ~ r 2 A2 where the area A ~ is a circle with a radius of unity. Let the local oscillator field UQ be a gaussian beam, which is given by g~(r 2) =

%lye)expl--lKQ(l' 2 + i~)r~

-- ik(S- z~) + i tan-1 ~ ] ,

(8)

where (3)

A1 where k is the wavenumber, Us is the incident signal, and G(r 1 ,r2) is given by

G ( r l , r 2 ) = f e x p ( _ i k f + i k ~ r l -2i


k l r l 2f - r 2 l -]. ~ (4)

Here r l , r 2 are vector coordinates over the aperture plane A 1 and the detector plane A 2, respectively. Substituting eq. (3) into eq. (1), we obtain

~Q = 2(f - z~)/kw~,

~Q = vr2/w~(1 + ~2)1/2.

(9)

This beam has the smallest spot size w e at z = ze. Thus, the integral over the area of the detector in eq. (7) is expressed as • k

S A2

2

kal

"

(10)

= 2v~a2K~ exp[ik(f- z~) - i tan -1 ~ ] f(rl), where

f

(S):c~(kl2n) 2

d2r2 f d2r~ U:(r2) U~(r2)

A2

: ~ [~ - i(K2~ - klf)],

x = kala2if,

A2 1

x

f d2rl f A1

(11)

f(rl) : f exp(-~a2r2)Jo(Xrlr2) r 2 dr 2, 0

d2r'l(Us(rl)~(r'l ))

At

and eq. (8) is used. Then, using eq. (10), we obtain

X G(r I ,r2) G *(p; ,r2),

(5)

where * denotes the complex conjugate. Assume isotropy and homogeneity of the statistics. Then, the mutual intensity function (MIF), M(p), is written as [1]

M(R) = (Us(rl) Us (rl)),

(12)

P = [r 1 - r 1 [.

(6)

Substituting eqs. (4) and (6) into eq. (5), we obtain

X f d2r'l M(alo)f(rl)f*(r't).

(13)

At In eq. (13), the integral with respect to r~ is regarded as the convolution of the product of truncated functions. Thus the properties of the convolution allow us

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OPTICS COMMUNICATIONS

to expand M(alP ) in a Fourier-Bessel series in the interval [0,3] as follows

_1

_ ~ [j2(.) +j~(a)],

0 ~< p ~< 3,

(14)

(S)=.4rm2(ka2a2/f) 2

where Jv is the vth order Bessel function of the first kind and sk is the positive kth root of the equation J0(3s) = 0. The expansion coefficients A k may be written as [9] 2 a- [ M ( a--l P ) -J 0 ( S- k P ) p d p ' 9 [J1 (3Sk)] 2 0 2

Ak -

(- =/3). (19)

The re fore sub stit uting eqs. (17)--(19) into eq. (13), we obtain the average heterodyne signal power IS) expressed as

00

M(alP) = k~l= Akdo(skP ),

1 February 1985

oo

×

1

12

f

0

k = 1 , 2 ..... 3. Heterodyne efficiency

t

p = Ir 1 - r l l .

(15)

Introducing the root mean squared signal power

Using eq. (14) and the formula [9]

Ps = : (IUd(r2)12) d2r2 ,

Jo(skP) = Jo(Sk Ir 1 - r'1 I) 0o

= ~

Jn(Skrl)Jn(Skr'l ) exp[in(01 - 0]) l, (16)

(21)

we can write the SNR of the shot-noise limited heterodyne detection in the form

n=_oo

SNR = (Ps/hVB)7, we get

where B is the detector noise bandwidth and

f d2rl f d2,'lM(alP)f(rl)f*(rtl ) Ai

~A k

k=l

(S)/a

3' =

Ai

=4rr 2

(22)

f°*(I Ud(r2 )12) d2r2

1

f f(rl)Jo(Serl)rldr 1 0

[2 (17)

(23)

fA2 [ U~(r2)12d2r2

The quantity 7 is the heterodyne efficiency. The integrals in eq. (23) are carried out as follows o0

In eq. (17), the integral is written by using eq. (2) as follows 1

f

(I Ud(r 2) [2)d2r2

= rra~M(O)

(24)

and

f f(rl)Jo(skrl)r 1 dr 1 0 1 -- exp(--~'alr~) h(Sk,Xr 2) r 2 dr 2 ,

f

f

(18)

0

1

h(.,D =

G

(r2)[2d2r2 = 1 - e x p ( - 2 1 ,

w2

"2 =72, (25)

where w 2 is the spot size of the local oscillator on the photodetector. Substituting eqs. (20), (24), and (25) into eq. (23), we obtain

where

f 0

A2

= h(.,/3), Odl (") Jo (/3) -/3Jo (") J1 (/3)

.2 _/32

(. 4:/3),

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OPTICS COMMUNICATIONS

8X2 3' =

'a2

[1

1 February 1985

1.0

A~I0 L = 10 3 ,

e x p ( - 2 / a 2 ) ] 34(0)

-

....

** X

1

2

0.8

2

?lAk 2 k,Xr2) dr 2 = 0f r2e-~a2r2h(s

10 5

L=IO

S - z =0

(26) Ca)

0,6

This result coincides with that of ref. [5]. Since this expression involves only simple integrals and a sum, it is particularly well suited for numerical calculations. Let the overall atmospheric MIF with both scattering and turbulence be given by [6]

M(p) = exp[-(p/ap)2Sa z] exp[-(p/l)5/3], P <,ap, = exp(-SaZ ) exp[-(p/l) 5/3 ],

p > ap,

(27) where ap is the radius of the scattering particle, S a is the scattering coefficient, z is the length of the propagation path of the signal, and l is the e x p ( - 1 ) coherence length. In this case, introducing A = a 1/ap and L = l/ap, eq. (15) becomes

Ak =

{2/9

0.5

0.4

0.2

it

~

-'- . . . . . . . . . . .

-o-U

''~- ....... I 2

0

I 4

1.0

X

I 6

'°I

I 8

Lz: -

IO'Z

[Jl(3Sk)] 2)

I 10

5

(b)

-I X

{/

p exp[--(pA)2SaZ --(pA/L)

5/3]

Jo(SkP)dp

/ ,,'.

I0 -3

................

2

+ exp(~5'aZ )

f P exp[-(pA/L) 5/3 ] Jo(skP)dP }" A-1 (28)

and M(0)

=

1.

(29)

From eqs. (11) and (26), it follows that the imaginary part o f ~"must be equal to zero to maximize the heterodyne efficiency ?. That is,

K~ ~ = k/.t'.

(3o)

Since f is equal to the radius of the curvature of the average phase front of the signal on the detector, this expression shows that the phase front of the local oscillator field must coincide with that of the incident signal. Then, when calculating the heterodyne efficiency, the beam parameters (z~ ,w~) of the local oscillator are chosen to be identical with those of the optimum local oscillator in the absence of both scattering and turbulence (see eq. (19) in ref. [10]).

/;/,'/

---

//,[ I 0 ''+ ["I' 0

.

.

.

Lo,o+, ,o' .

.

I

I

2

4

x

I

I

6

8

i I0

Fig. 2. Heterodyne efficiency .~ as a function of x (= 3.83 a2/ wA) for A (=al/ap)= 10 and for various values of L (= I/ap) and Saz. The parameters are;a 2 : detector radius, WA: radius of the Airy disk,a I : aperture radius,ap: radius of scattering particle, h coherence length, Sa: scattering coefficient, z: path length of the signal.

In figs. 2 and 3, the heterodyne efficiency 3' as a function o f × (= 3.83 a2/wA) for A (= al/ap) = 10 and 103 , respectively is shown, where w A is the radius of the Airy disk. From these figures, we can See that the heterodyne efficiency 7 rapidly increases as X increases and further increase of X lets 7 approach some constant determined by the values of Saz and L (= l/ap). The

Volume 53, number 1

1.0

OPTICS COMMUNICATIONS

A=I0 a

_ _

L=I0 s

.....

L=I0 3

1 February 1985 Sa z = 0

10-'4

0.5 1.0

(a)

(c)

0.8

A=10a

10 "a 0.6

L =10

r

3

0.5 0.4

I I,"

,.o

/

0.2

/

. . . .

#

/""

I

~--

i

0

2

.

.

.

4

I0" ~-

7"

~ .

I 0 -r

- . . .5. . . . . . . . . .

.

X

.

.

.

.

.

.

.

.

.

6

.

.

9

.

8

10

0

2

4

X

6

8

10

&.lOs

(b)

IO2~

5

,/;//

io t-/1'

r

.I'I,"

9

,o

/'// 10-6 0

2

4

X

6

8

10

constant value of 3' occurs at X ~ 3.8. This denotes that the detector size is approximately equal to the Airy disk. Furthermore, we see that the effects of coherence length on the heterodyne efficiency decrease as the scattering attenuation SaZ increases. The case Saz = 0 corresponds to the state o f non-scattering and coincides with that o f ref. [5 ]. Fig. 4 shows the dependence of the heterodyne efficiency on the coherence length for A = 10 and X = 2.0, 3.5. As can be seen in fig. 4, there is a clear " k n e e " in each curve at L = 103 , i.e. at a 1 = l. When the aper-

Fig. 3. Heterodyne efficiency 3' as a function of × (= 3.83 a2/ WA) for A (=al[ap) = 103 and for various values ofL (=l/ap) and Saz. The other parameters are the same as fig. 2.

ture radius a 1 is larger than the coherence length l, the heterodyne efficiency decreases almost linearly with the coherence length. The heterodyne efficiency is hardly improved as the aperture radius a 1 is less than the coherence length l. Thus l is the maximum collection aperture radius. The decrease o f X which is related to the detector size leads to that o f the heterodyne efficiency. Fig. 5 shows the comparison between the heterodyne efficiency for the optimum local oscillator in the presence o f b o t h scattering and turbulence, and that in the

Volume 53, number 1

OPTICS COMMUNICATIONS So Z = 0

,,,

_

~o 5 ~ - _ - _ . - - - - 2 2 51o.5 ° ~ ~ - . ". . . . . . . I~ o

IO-I

I II

10 -2 ii

Y

z II

5

. ~-- ......

I III Ill I

I ll/

10-5

¢i I

7

I0 -4

II

III iiI

1/

10-a

i O-e

//

I0

~l

I

I

I

I0 2

10 5

10 4

10 5

L

Fig. 4. Heterodyne efficiency "r as a function of L (= l/ap) for A (=al/ap) = 103 and x (= 3.83 a2]wA) = 2.0, 3.5 and for several values of Saz. The other parameters are the same as fig. 2.

1 February 1985

4. Conclusion The SNR in heterodyne detection of optical signals propagating through the atmosphere is investigated in the presence of scatterign and turbulence, taking into account the spatial coherence of the signal. We have derived the expression of the SNR in a form well suited for numerical calculations. If the effective detector size is chosen to be almost equal to the size of the Airy disk, the maximum value of the heterodyne efficiency for each coherence length and each scattering attenuation can be obtained approximately. There is a limitation in the achievable heterodyne efficiency no matter how large the collection apearture is. The maximum aperture radius is nearly equal to the coherence length of the signal, regardless of the detector size and the scattering attenuation. And if we use the local oscillator beam whose parameters are chosen to maximize the SNR for a coherent signal field, the SNR in heterodyne detection of the signal field with degraded spatial coherence is practically maximized.

Acknowledgement absence of them. In this example ofSaz = 0.5, we see that the spot size of the local oscillator field is insensitive to the heterodyne efficiency.

The authors wish to thank Prof. O. Fukumitsu and Dr. T. Takenaka of Kyushu University for their encouragement.

0.3 References [1] [2] [3] [4] [5 ]

0.2 A=L=I0

a

[6] .....

0.1

moximum heterodyne

[7] [8] [9]

efficiency curve

in F i g . 3 ( o )

[10] ,

I

I

2

4

X

I

I

I

6

8

10

Fig. 5. Comparison between the heterodyne efficiency in fig. 3a for A (=al]ap) = L (=flap) = 103 and Saz = 0.5 (solid line) and the maximum heterodyne efficiency (dashed line).

D.L. Fried, IEEE Proc. 55 (1967) 57. H.T. Yura, Appl. Optics 13 (1974) 150. B.J. Rye, Appl. Optics 18 (1979) 1390. D. McGuire, Optics Lett. 5 (1980) 73. T. Takenaka, N. Saga and O. Fukumitsu, Trans. IECE Japan J64-C (1981) 553. L.G. Kazovsky and N.S. Kopeika, Appl. Optics 22 (1983) 706. J. Salzman and A. Katzir, Appl. Optics 22 (1983) 888. J. Salzman and A. Katzir, Appl. Optics 23 (1984) 1066. N.N. Lebedev, Special functions and their applications (Dover, New York, 1972). T. Takenaka, N. Saga and O. Fukumitsu, Trans. IECE Japan J63-C (1980) 423.