Heterodyne efficiency for a partially coherentoptical signal

Toshiyuki Tanaka, Mitsuo Taguchi, and Kazumasa Tanaka

Heterodyne efficiency is discussed for a partially coherent signal and a coherent local oscillatorbeam. Both fileds are assumed to have Gaussian amplitude distributions. An input aperture is used toreduce the background noise. As the coherence of the signal decreases, the efficiency also decreases.However, there is a simple relation between the beam parameters and the detector dimensions tomaintain optimum efficiency. The effect of the offset of the signal from the detector axis is alsodiscussed, assuming the Gaussian probability of the deviation. In this case, the optimum parametersthat give the maximum efficiency change with the average deviation.

Key words: Diffraction, heterodyne detection.

IntroductionOptical heterodyning is one of the most promisingmethods for detecting weak signals, and it has cometo be used in practical measurement.' One of thedifficulties of the method is finding a match betweensignal and local oscillator (LO) fields. If there aresome misalignments, the signal-to-noise ratio (SNR)in the output will be reduced.2 The ratio of a realSNR to an ideal one is referred to as mixing orheterodyne efficiency.3 4 In addition, the SNR is alsoreduced by the coherence of the signal.5

In this paper the effects of the degree of coherenceare examined for the signal field whose amplitudedistribution is Gaussian. An aperture is used atthe input plane to reduce the background radiation.This aperture affects the incident signal field, andthe effect of diffraction must be taken into consider-ation.

The LO field is assumed to be large enough com-pared with the signal and background fields that theLO shot noise is the dominant noise in the system.

Expression of Heterodyne Efficiency in Terms of SignalCoherenceFollowing the notations in Refs. 4 and 5, we find thatwhen the signal field is partially coherent, the expres-

The authors are with the Department of Electrical Engineeringand Computer Science, Nagasaki University, Nagasaki 852, Japan.

Received 16 July 1991.0003-6935/92/255391-04$05.00/0.o 1992 Optical Society of America.

sion for the heterodyne efficiency is given by

(f IIs UU*dS2 )(1)

I us 12 dS 2 fS2 I U, 12 dS 2

where ( ) denotes an ensemble average, U, and U, arethe signal and the LO fields, respectively, and theasterisk represents the complex conjugate. The inte-gral is carried out over the detector surface S 2 located atz = z2. The numerator is proportional to the averagepower of the intermediate frequency signal Pif.

Let the signal be the diffraction field from an inputaperture S, at z = z1. If the detector is in the Fresnelregion, the diffraction field on the detector is obtainedby using the Kirchhoff-Huygens diffraction formula.For a partially coherent signal, (Pif) is given by5

(Pif) = (k/2,T)2£S2 £S2 IU1*(X2 , Y2, Z2)

x U1(x2 ', Y2', Z2')]dS2dS2' fs1 fs1

< U*(x1, y1) U(x1tX y1') > G(x1,yi; 2, Y2)

x G*(xi', y'; x2 ', Y2')dSdS,',

where

1G(xi, y,; x2 , Y2) = - z) x-jk(z2 - Z,)

- Xl) 2 + (Y2 - y)2-jk1 (Z2 - ZO) J

(2)

(3)

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The average (Us*(xlyi)U8 (xlylt)) is related to thedegree of coherence Yc of the input signal on the inputaperture and the intensity I by

(U8*(xl, yl) U8(Xl' y1')) = yjI(X1 yl)I(xI', y1')]112.

(4)

Let the intensity of the signal on the input aperturebe Gaussian whose 1/e radius is w80 as follows:

coherence may take some specified form, dependingon the situation. The most familiar and useful oneis the Gaussian form,

(x1 - x 1')2 + (Y- yl)2]l Z(xy; xi,,Y) = exp - 12

(6)

where is the coherence length of the signal field.

2(x1 - )2 + 2 yi2

The parameter is the offset of the signalthe optical axis of the detector system. Tb

0. 6

a[/a2

0. 5

0. 4

0. 3

0. 23 .

0. 6

a/a2

t0. 5

0. 4

0.3

0.23. 0

0 3. 5 4. 0 4. 5 5. 0 5. 5 6. 0 6. 5 7. 0

- 2 / WA(a)

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

> a2/WA(b)

2. Note that form of Eq. (6) for Yc applies only to theincident beam that has its waist at the position of theinput aperture.

Substituting these forms into Eq. (1), we calculate1 field from the heterodyne efficiency. The geometry of the detec-.e degree of tion system is assumed to be square. The input

0. 6

ai/a2 Ft

0.5 -

0.3

0.2 K3. 0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

- a2/WA(c)

Fig. 1. Contours of the constant heterodyne efficiency for acoherent signal: (a) w8 /wl = 0.8; (b) wSIwl = 1.0; (c) ws/wl = 1.2.

5392 APPLIED OPTICS / Vol. 31, No. 25 / 1 September 1992

I(x1 , Y1) = exp[-

aperture and the detector surface are square with thesides 2a1 and 2a 2, respectively.

Numerical CalculationsThe incident signal beam and the LO beam areassumed to have their waists at the input aperture(beam-waist coincidence). Figure 1 shows the con-tours of constant efficiency for a coherent signal(a11/ = 0) and no offset. The ratio w81wl is variedfrom 0.8 to 1.2, where ws and w1 are the smallest 1/eradii of the incident signal and LO beams, respec-tively. From this figure, we can see that the parame-ters a1/a 2 and a2/wl are almost inversely proportionalfor some value of the efficiency for a given ratio w8/w l.Therefore, to obtain high efficiency we must choosethe optimum value, such as (a1 /a2 ) x (a21wd) = 1.6,1.8, and 1.8, for w5Iwl = 0.8, 1.0, and 1.2, respectively.These values correspond to al1w, = 2.0, 1.8, and 1.5,respectively. We can also see that the optimumvalue of the ratio w8Iw is 1.0, which gives a maximumefficiency of 99.7%. As we expected, this resultcoincides with that obtained previously in the litera-ture.2

As an example for a partially coherent signal field,we show the contours of the constant efficiency forthe case when the ratio al/1 = 1.0. The resultingcontours are shown in Fig. 2. This figure also showsthe inverse proportionality between the parametersaj/a2 and a2 /w1 for a constant ratio of wllws = 1.0.The value of (ai/a2 ) x (a 2/wl), which gives a highefficiency, is 2.6. The value of al/w, that corre-sponds to it is also 2.6.

Next we examine the effects of the degree ofcoherence and the offset of the signal field. From theresults just obtained, we choose w81wl = 1.0. Be-cause the ratios ai/a 2 and a 2 /wl are approximately

0. 6

ai/a2

0. 5

0. 4

0. 3

0.2 L5. 0 5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9. 0

> 2/WL

Fig. 2. Contours of the constant heterodyne efficiency for apartially coherent signal. The coherence length is equal to theside of the input aperture (a, = 1).

1.0

' 0.8

a0/A =0.0

r0.61-

0.4-

0.2p-

1.0

2.0

I I IIlI I I I I I

0.0 4.0 6.0 8.0 10.0 12.0> 2/WA

I I I

1.0 2.0 3.0 4.0 5.0> a /Ws

Fig. 3. Heterodyne efficiency as a function of a2Iwl or al/w, forvarious values of coherence length.

inversely proportional for a given value of wSIwl, wecan choose one of them arbitrarily while keeping theproduct a constant. In the following calculations,ai/a 2 = 0.4. The value alw, is obtained from therelation a/w = (a1 /a2 ) x (ailw) x (wi/w) =

0.4(a2 /w1 ).Figure 3 shows the heterodyne efficiency as a

function of al/w, for given values of al/1. From thisfigure we can see that the efficiency decreases as thecoherence of the signal is reduced. The optimumaperture size is 1.8 times the smallest 1/e radius ofthe coherent signal beam. For a partially coherentsignal field, the optimum values of al/w, are 2.6, 3.2,and 3.7 for al/l = 1.0, 2.0, and 3.0, respectively.

Figure 4 shows the effects of the offset of the signalfield for a partially coherent signal. From this figure

0.8 F

0.6 -

0.4 -

0.2 _

I I I I I I I I I I I

0.0 4.0 6.0 8.0 10.0- a2

I I I - _ l I

12.0/ WA

1.0 2.0 3.0 4.0 5.0->a /Ws

Fig. 4. Heterodyne efficiency for a partially coherent signal withor without axial offset: solid curves, 8/al = 0.0; dashed curves,8/a, = 0.1.

1 September 1992 / Vol. 31, No. 25 / APPLIED OPTICS 5393

. . .. . . . . . . .

1.0 _-

I.Or

0.8 1

0.6

0.4 -

0.2

ai/R =0.0

1.0

3.0

1 1 11111 I I I I 11111

10.0 100.0> aG/O

Fig. 5. Average heterodyne efficiency for a partially coherentsignal with an offset that has a Gaussian probability distribution.

we see that the optimum values of al/w8 are smallerthan for the aligned case. Nevertheless, as the coher-ence decreases, the difference becomes negligible andwe can choose the parameter allw, as the parameterfor the zero-offset case.

In practical detection systems, however, the propa-gation axis of the signal beam will fluctuate. Conse-quently, we introduce a Gaussian probability distribu-tion for the offset as

P(8) = exp(- 2 /u 2 )

and calculate the average efficiency. The results ofthis calculation are shown in Fig. 5. The figure

shows that by making the value al/a larger than 3.0,we can obtain an efficiency of at least 70% of theinfinite-aperture efficiency for any degree of coher-ence.

ConclusionsThe effect of the coherence of the signal field isdiscussed. The results show that, for a given ratio ofthe smallest 1/e radius of the signal beam to that ofthe LO beam, an approximately inverse proportionalrelation holds between the ratio of the dimension ofthe input aperture to that of the detector and theratio of the dimension of the detector to the smallest1/e radius of the LO beam. Therefore, we canchoose one of the ratios arbitrarily, depending on thepractical situations. The other ratio is determinedfrom the inverse proportional relation.

As for the effects of the offset of the signal, aheterodyne efficiency larger than 70% of the infinite-aperture efficiency is obtained by making the inputaperture at least three times the standard deviationof the offset distribution of the signal.

References1. A. L. Migdall, B. Roop, Y. C. Zheng, J. E. Hardis, and Gu Jun

Xia, "Use of heterodyne detection to measure optical transmit-tance over a wide range," Appl. Opt. 29,5136-5144 (1990).

2. K. Tanaka and N. Ohta, "Effects of tilt and offset of signal fieldon heterodyne efficiency," Appl. Opt. 26, 627-632 (1987).

3. R. H. Kingston, Detection of Optical and Infrared Radiation(Springer-Verlag, Berlin, 1978), chap. 3.

4. D. Fink, "Coherent detection signal-to-noise ratio," Appl. Opt.14, 689-690 (1975).

5. T. Takenaka, N. Saga, and 0. Fukumitsu, "Heterodyne detec-tion of partially coherent optical signal," Trans. Inst. Electron.Inform. Commun. Eng. Jpn. J64-C, 553-559 (1981).

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