Three-Dimensional Vector Diagrams as an Aid inAnalyzing Optical Devices and Systems

E. 0. Ammann

The usefulness of three-dimensional vector diagrams in analyzing optical systems is described. The dia-

grams involved are three-dimensional plots of the positive frequency spectrum of the signal. By using

two diagrams to represent a given signal, one for each of two orthogonal linearly polarized components,information on signal polarization is also conveyed. Devices or systems are analyzed by notinghow the diagrams of the input signal are successively altered by passage of the signal through the

system. The techniques described are most useful with systems containing birefringent and dichroicelements, and with signals containing one or a few discrete spectral components. The diagrams can be

used to determine both the output (optical) signal and the (rf) signal resulting from its detection by

a square-law photodevice. Examplessideband modulator, and a balanced

1. Introduction

The purpose of this paper is to show the utility ofthree-dimensional vector diagrams in understandingand analyzing many optical devices and systems.These vector diagrams are particularly useful with sys-tems such as laser systems, where the optical signal con-sists of one or a few discrete frequencies. The effects ofpassing such a signal through various birefringent anddichroic optical components (retardation plates, mod-ulators, polarizers, etc.) can be found using the tech-niques of this paper. Vector diagrams are simple andfurthermore give insight into the physical aspects of theproblem, and hence are useful as a supplement to, or insome cases an alternate to, formal mathematical anal-ysis.

Two-dimensional phasor diagrams have been widelyemployed for many years in the analysis of networks.'Phasors represent single-frequency, sinusoidally vary-ing functions of time such as voltage and current in acircuit. Three-dimensional vector diagrams have alsobeen employed previously. Bracewell,2 for example,has used them to represent multifrequency signals andtheir spectra. Thus the concept of such diagrams isby no means new. We believe, however, that the use-fulness of these diagrams in analyzing optical systemsand devices is not generally appreciated. In this paper,

The author is with the Electro-Optics Organization, SylvaniaElectronic Systems-Western Division, Mountain View, Cali-fornia 94040.

Received 18 September 1969.

considered include two types of amplitude modulator, a single-optical detection system.

the techniques for working with three-dimensionalvector diagrams are described and several examples aregiven.

11. The Vector Diagrams

The vector diagram of this paper is a three-dimen-sional plot of the positive-frequency spectrum of thesignal represented. The diagram contains amplitude,frequency, and phase information for each spectral com-ponent present. More concisely stated, a vector dia-gram is the fourier transform of the analytic signalrepresentation' of the signal. Formidable as this maysound, the diagrams of a variety of commonly en-countered signals are quite easy to construct.

A. Single-Frequency Signal

Suppose that we have a monochromatic optical signalwhose electric field is given by A coscot. The complexrepresentation of this signal, which is also its analyticsignal, is Aeiwt. The vector diagram of this signal isshown in Fig. 1(a), with a simplified version of thediagram shown in Fig. 1(b).

The amplitude of the signal is denoted by the lengthof the vector. This length will be written alongside thehead of the vector as in Fig. 1(b). The frequency ofthe signal determines the vector's position along the waxis. Since we are dealing with optical signals, andtherefore with co's of approximately 1014, the origin (w= 0) will be impractically far removed from the vector,and thus will often not appear on a diagram drawn toscale. Hence for convenience the u and v coordinateaxes will usually be shown intercepting the w axis atsome arbitrary point as in Fig. 1 (b), rather than at w =0. The exact location is unimportant as long as it is

July 1970 / Vol. 9, No. 7 / APPLIED OPTICS 1683

U

(FREQUENCY}(a} (b

Fig. 1. Complete (a) and simplified (b) vector diagram of mono-chromatic signal.

labeled or is otherwise apparent from the problemunder consideration. When amplitude- or frequency-modulated signals are involved, a logical choice is toplace the axes at the carrier frequency.

Progression of the phase of the signal is shown byrotation of the vector about the w axis. The vector'sprojection upon the u-v plane is a complex plane plot ofAeiwt, with u the real axis and v the imaginary axis.The vector diagram is a snapshot of the signal at aparticular instant of time, and thus shows the phase ofthe signal at that instant. In Fig. 1 we have chosen toview the signal at a time such that eiwt = 1.

Let us now consider how the vector diagram of ourmonochromatic signal varies with time for two im-portant cases. First suppose that we station ourselvesat a particular point in space and observe the opticalsignal as it propagates past. The vector diagram ofFig. 1(b) applies, wvith the vector rotating about the waxis at the circular frequency . We will call the direc-tion of rotation shown in Fig. 1(b) positive, and theopposite rotation direction, negative. A wave travel-ing in the opposite direction will result in a vector ro-tating in the negative direction.

For the second case, suppose that we move with thesignal as it propagates. Since we are traveling with thesignal, its phase does not appear to change. Hencethe diagram of Fig. 1(b) applies, but with the vectorfixed in position rather than rotating. Finally, if wetravel in the direction of propagation but with a velocityv, which is less than the signal velocity, the vector willrotate at the reduced circular frequency v/c, where cis the signal velocity in the medium.

B. Multifrequency Signals

Now consider a signal E(t) containing several dis-crete frequencies, such as

E(t) = Aeiwlt + Beiw2t + Ceiw3t,

vector C appears to have a positive rotation frequency°3- W2) while vector A appears to have a negative ro-

tation frequency ( - W.2) The effect is thus thatvector C gains on vector B, while vector A falls behindB.

If we travel with the signal rather than remainingstationary, all the vectors of Fig. 2 will be fixed in posi-tion.

Thus far we have said nothing of how signal polariza-tion enters into the discussion. The polarization of asignal may be described by stating the complex ampli-tudes (i.e., amplitudes and phases) of two orthogonallinearly polarized components. Hence we convey polar-ization information by using two vector diagrams todescribe a signal, rather than one, with each diagramdescribing one linear polarization component.

Additional characteristics of the vector diagrams areperhaps best described by means of examples. There-fore, we will begin with a simple illustration and proceedto more complicated examples.

111. Examples

The first two examples illustrate how vector diagramscan be used to analyze optical devices. The quantity ofinterest in these examples is the spectrum of the opticalsignal transmitted by the device. In later examples,we will see how vector diagrams can also be used todetermine the rf or microwave output which resultsupon detection of the optical signal by a square-lawphotodetector.

A. Calculation of Output Optical Spectrum

For any problem of this type, the basic procedure isessentially the same. We begin with an incomingsignal, draw the vector diagrams corresponding toorthogonal linear polarization components, and thennote how each element of the optical device or systemalters the diagrams. The two output vector diagramsdescribe the amplitude, frequency, phase, and polariza-tion of each spectral component.

1. Electrooptic Amplitude Modulator

Consider first a conventional electrooptic amplitude

(1)

where co3 > C2 > wi. Each new frequency component isrepresented by a vector appropriately positioned on thew axis. At t = 0, the vector diagram corresponding toEq. (1) is shown in Fig. 2. If we again assume a sta-tionary observer, each vector will rotate about the waxis with its corresponding frequency. Note that therotation frequency of vector C relative to B is (3 - C2).

That is, if we sit upon the head of vector B as it spins, Fig. 2. Vector diagram of multifrequency signal.

1684 APPLIED OPTICS / Vol. 9, No. 7 / July1970

(a)

\ X\XI MODULATING

SIGNAL

Fig. 3. Conventional AM electrooptic modulator.

modulator of the type described by Kaminow.4 Thenormal modulator configuration, shown in Fig. 3, con-sists of a polarizer, electrooptic crystal, quarter-waveplate, and polarizer. Typically the crystal is so orientedthat upon application of a longitudinal electric field,principal axes (shown dotted in Fig. 3) are induced inthe x' and y' directions. The index of refraction is in-creased in one of these directions and decreased in theother by the applied electric field. Assume that theapplied field (which is the modulating signal) is asimple sinusoid of frequency cin.

The incoming optical signal is assumed to be mono-chromatic with frequency coc. Passage of this signalthrough the first polarizer results in the linearly polar-ized signal of Fig. 4(a). We have shown a separatediagram for the (zero) y component to emphasize thatseparate diagrams are needed for the orthogonallypolarized components.

The signal next travels from the polarizer to theelectrooptic crystal. During this time, its vector dia-grams do not change since our reference point travelswith the signal. Having reached the crystal, we shouldredraw the vector diagrams of Fig. 4(a) using x'-y' co-ordinates since the crystal's principal axes lie along thesedirections. The x-y and x'-y' coordinate systems arerelated by

EV = (ED/A/2) + (Ex/V2),

lb)

(c)

iJ(m)

(d)

Ey

J,() i(m)

J(m)

(e)

(2a) 'J,Im)(f)and

Ex, = - (E,,/V/2) + (Ex//2). (2b)

Applying Eqs. (2) to the vectors of Fig. 4(a), we obtainthe vectors shown in Fig. 4(b) which describe the x' andy' signal components at the input to the crystal.

Consider for a moment the Ey, component. Duringits passage through the crystal it is phase modulated bythe refractive index variations that occur. The result-ing signal is of the form

ExJ,(m) J,(m)

J1(m)(go " -

Ey = (1/A/2) exp i(wct + m sincmt),

which when expanded gives

Ey, = (1/V2)[Jo(m) exp iwct + Ji(m) exp i(c, + cm)t

-J 1(m) exp i(wc - wm)t + higher order sidebands]. (3b)

(hI

Fig. 4. Vector diagrams for AM electrooptic modulator.

July 1970 / Vol. 9, No. 7 / APPLIED OPTICS 1685

(3a)

E ,

Ey

If the modulation index n is small, only the first upperand lower sidebands will have significant amplitudes,and hence the higher order sidebands can be neglected.The vector diagram of EU' at the output of the crystalwill then be as shown in Fig. 4(c).

Next consider the effect upon Ex, of passage throughthe crystal. We stated earlier that changes in the indexof refraction along x' were opposite in sign to those alongy'. Thus if the variations in the y' index go as sincott,those in the x' index will go as-sinclet. E,,,, as it leavesthe crystal, will therefore be given by

E. = (1/ V2) exp i(wjt - sinw,,t). (4a)

Expanding this expression gives

E. = (1//2)[Jo(n) exp iwt - J10() exp i(w, + Win)t

+ J(m) exp i(w, - w)t + higher order sidebands], (4b)

which, when higher order sidebands are neglected, givesthe vector diagram of Fig. 4(c). Thus E' and Ex, areboth phase modulated by passage through the crystal,but their modulations differ by 180.

EU' and Ex, next reach the quarter-wave plate. Thefast (F) and slow () axes of the quarter-wave plate liealong the x' and y' axes and hence it is not necessary tochange coordinate systems. Since a quarter-wave platechanges by 900 the relative phases of signals polarizedalong its F and S axes, the vector diagrams of Es, andEU' are as shown in Fig. 4(d) at the output of thequarter-wave plate. Note that during passage throughthe quarter-wave plate, we have chosen to use Ex as ourreference wave (that is, to travel with Ex,). The resultis that Ey, appears to have lagged 900 in phase. Thisoccurs because E.' travels slightly faster than E,, in thequarter-wave plate. Thus from our vantage point onEz,, E, appears to be going in the opposite direction,and according to the convention which we have chosen,its vectors rotate in a negative direction. We couldhave equally well used E, as our reference and it wouldthen appear that Ex (which is traveling faster than EU,,our reference wave) had advanced 900 in phase.

The final element through which the light passes is apolarizer whose transmission axis is in the x direction.Let us therefore convert the vector diagram of Fig. 4(d)back into an x-y coordinate system. Using the rela-tions,

El = (Ems/VI \)- (E // 2),

E = (EU'/,/2) + (E'//V2),

While this diagram is somewhat clearer than the pre-vious one, its interpretation can be made still morestraightforward by the following means. Let us againchange our viewing position on the wave, but this timewe will let the signal slip ahead many optical wave-lengths. All three vectors make many revolutions attheir respective frequencies during this slipping, but theupper sideband is catching up with the carrier while thelower sideband is falling behind. If we establish ournew vantage point having let sufficient signal go by tomake the upper sideband and carrier parallel, and if weare further careful to choose our position on the properfraction of the wavelength to cause the carrier to bevertical, we obtain the more familiar AM spectrum ofFig. 4(h).

Thus we see that small shifts (a fraction of an opticalwavelength) in reference position will rotate all vectorsby the same amount, while large shifts will allow thevectors to rotate with respect to each other. It willusually be advantageous to manipulate the final vectordiagram(s) in this fashion to make its interpretation asstraightforward as possible.

2. Sinle-Sideband llodulatorFor the second example, let us consider the optical

single-sideband suppressed-carrier modulator of Buhreret al.5 The modulator has the configuration shown inFig. 5, consisting of input polarizer, quarter-wave plate,two electrooptic crystals, quarter-wave plate, and out-put polarizer. The first birefringent crystal has its in-duced birefringent axes in the x and y directions, whilethe second has its induced axes in the x' and y' directions.The signal applied to the second crystal is shifted 900in phase with respect to that applied to the first.

We again assume that a monochromatic signal of fre-quency c enters the modulator. After passing throughthe input polarizer, the signal is represented by thevector diagrams shown in Fig. 6(a). The next element,a quarter-wave plate, has its fast and slow axes in the x'and y' directions, and hence we use Eqs. (2) on thediagrams of Fig. 6(a) to give the corresponding diagramsin x'-y' coordinates. The results are shown in Fig.6(b).

Passage of the signal through the quarter-wave plate

(5a)

(5b)

we obtain the diagrams of Fig. 4(e).When the signal of Fig. 4(e) passes through the out-

put polarizer, we are left with the output of Fig. 4(f), anamplitude-modulated (AM) signal which is linearlypolarized. It is probably not obvious that the vectordiagram of Fig. 4(f) indeed represents an AM signal, solet us perform two manipulations which will make thismore apparent. Figure 4(f) represents the diagramwhich we see from our reference position on the outputsignal. Suppose that we change our position by of anoptical wavelength (at co). If we let the wave slipahead by that amount, the new diagram is shown in Fig.4(g).

Y

"y I

Ix.

Fig. 5. Single-sideband suppressed-carrier modulator.

1686 APPLIED OPTICS / Vol. 9, No. 7 / July 1970

Sin umt

,I

la)

EU,l T

1W

E'.

I2 I

Ex,

E , __

(c}

E, I

11

d}

J~I(M

U -'I

MM)m Ey)

J i7 _ J ,M:~~~' V-2

(I}

E,,

Qc

i imi E,:7 _2 J~(lmi

: J7imi

V5JOimJ(mi

EVy

ilimi _ = im

Ji2 Jm i

i(Mi lm

JIm

Ex.

M , JO(m)Jm I(miJLP imi

Ex.

E,

(gi

J"(M, Ex.,

7 TI 4lim)

(hi

(I}Eii

I :5~i- ~ml

(J

Fig. 6. Vector diagrams for single-sideband suppressed-carriermodulator.

(e}

advances Ev by 900 with respect to EI, and gives asthe output from the quarter-wave plate the diagramsof Fig. 6(c). Using Eqs. (5) on these diagrams, weobtain the x-y diagrams of Fig. 6(d) for the wave-plate output. The signal now passes through the firstcrystal and, as explained earlier, x and y signal com-ponents are phase modulated, with their modulationsbeing 1800 out of phase. Figure 6(e) shows the vectordiagrams for the signal leaving the first crystal. Wehave again assumed that the modulation index m issmall, and therefore that only the first upper and lowersidebands have significant amplitudes.

Converting back to an x'-y' coordinate system, weobtain the diagrams of Fig. 6(f). The signal now passesthrough the second crystal, where each spectral com-ponent is phase modulated. The 900 phase shift of themodulating signal must be taken into consideration,along with the fact that EU' and E.' are modulated 1800out of phase. We have previously seen how the lattereffect is handled.

The 900 phase shift in modulating signal means we aredealing with an exp im coscmt term rather than an expim sinwt term. Expansion of exp im cosW t gives theresult that each upper sideband is rotated an additional900 in a positive direction with respect to its carrier,and each lower sideband an additional 900 in a negativedirection. If we assume that the modulation index forthe second crystal is the same as for the first, the outputfrom the second crystal is given by Fig. 6(g). In Fig.6(g) the sidebands from the second modulation areshown dotted, while those from the previous modulationare shown solid. Simplifying Fig. 6(g) by adding vec-tors wherever possible (and making all vectors solid),we obtain Fig. 6(h). At this point, the first lower side-band has been cancelled in both EU' and E.', but thecarrier remains.

Passing the signal through the second quarter-waveplate, we obtain the diagram of Fig. 6(i). Convertingnext to x-y coordinates, the signal is shown in Fig. 6(j).Finally, passage through the output polarizer selectsE. of Fig. 6(j) as the output signal. E contains onlya single frequency which has been shifted by om fromthe input frequency. The carrier and other sidebandshave been suppressed.

Since during the analysis only first upper and lowersidebands were included for each phase modulation ofthe signal, the results are, of course, approximate.The error caused by this approximation depends uponthe value of modulation index employed. The approxi-mation greatly reduces the difficulty involved in ma-nipulating the vector diagrams and should be used when-ever possible. Thus when phase or frequency modula-tion is involved, vector diagrams will be most manage-able, and hence most useful, when only a few frequenciesare needed to represent the modulated signal.

B. Detection of the Output Optical Signalby a Square-Law Photodetector

In many optical systems, the ultimate quantity ofinterest is not the output optical signal per se, but ratherthe signal that results from its detection by a square-

July 1970 / Vol. 9, No. 7 / APPLIED OPTICS 1687

E,

<Z::::�_

E ,

, _L

-112

In a similar fashion, the beat amplitude at cw - W3

arising from mixing between vectors A and C is propor-tional to

Fig. 7. Vector diagram forsignal given by Eq. (6).

i J3

law photodetector. It is possible to determine thissignal, using the simple techniques to be described fromthe vector diagrams of the output optical signal.

Consider a linearly polarized, multifrequency signalEx(t) whose analytic representation is Eq. (6) and whosevector diagram is given by Fig. 7. Let this signal

Ez(t) = A exp i(wilt + 'PI) + B exp i(W2 t + '02)

+ C exp i(W 3t + 'P3) (6)

be incident upon a square-law photodetector. It iswell known that the photodetector output current willbe proportional to E,(t)E*(t), where the asterisk de-notes complex conjugate.

Using Eq. (6), we obtain

Ez(t )E2(t) = (A2 + B2+ C

2)

+ AB coS[(W - 2)t + P - P2)]

+ BC COS[(W2 - W3)t + (P2 - '03)1+ AC COS[(W - 3)t + (P - 3)1- (7)

The first term is a dc current term, while other termsrepresent different frequencies generated by photo-mixing between the various optical frequencies. Con-sider the term AB cos [(w - c02)t + (1 - 2) ] generatedby mixing between vectors A and B of Fig. 7. Thisterm has an amplitude proportional to the product ofthe amplitudes of the vectors A and B, a frequency equalto the difference between the frequencies of vectors Aand B, and a relative phase equal to the difference be-tween the phases of vectors A and B. A moment's re-flection will show that this has to be the case, since theterm arose from the products [A exp i(wlt + s0D)] [Bexp - i(CO2 t + IP2)} and [A exp - i(wit + sol)] [B expi(CO2t + IP2)].

None of the information is lost if we write AB cos1- 2)t + (y - ~02)] in its exponential form, AB

exp i[(w - 2)t + ( - 2)]. But the exponentialform can be generated by the product [A exp i(woit +X01) ] [B exp -i(CO2 t + 2) ], and hence we have a usefulsimple rule. The complex amplitude of the beat producedby mixing vectors A and B is proportional to the productof the complex amplitude of vector A and the conjugate ofthe complex amplitude of vector B. In a simplified nota-tion, the complex amplitude is given by

A/sl B/-s02 = AB/oi - 02, (8)

which contains all the information present in AB cos[(X1 - W2)t + ( - 'P2)]-

A/<l C/-S3 = ACoi_-3 (9)

If other optical frequencies spaced W - co3 apart werepresent, their contribution would also have to be addedin to obtain the total beat amplitude at co - w3.

Two points should be emphasized. In using therule of multiplying vector A by the conjugate of vectorB, vector A should have the higher frequency of thetwo. The second point is that, as can be seen fromFig. 1, taking the conjugate of a vector simply changesthe sign of its phase p, and hence the notation used inEqs. (8) and (9) is especially handy. We will nowconsider two additional examples which make use of

the techniques just described.

1. Optical MI/Iodulation by Light Bunching

A modified version of the electrooptic AM modulatorof Fig. 3 has been proposed and demonstrated byBuhrer.6 Buhrer's device, shown in Fig. 8, differsfrom the conventional modulator of Fig. 3 in that theoutput polarizer is replaced by a naturally birefringentcrystal. This allows one to utilize all the light leav-ing the quarter-wave plate, instead of absorbing halfat an output polarizer.

Since the devices of Figs. 3 and 8 are identical upthrough the quarter-wave plate, we may begin ouranalysis by stating that the vector diagrams of thesignal leaving the quarter-wave plate are identical tothose of Fig. 4(e). These diagrams are repeated inFig. 9(a). If we change our reference position by 8

wavelength we obtain the diagrams of Fig. 9(b).The signal next enters the long birefringent crystal of

Fig. S. The term long is used to denote that the effec-tive path length within the crystal is greatly differentfor x and y components. We will again choose E asour reference; therefore its vector diagram is unchangedby passage through the crystal. E, however, ispolarized along the crystal's fast axis and appears totravel faster than our reference wave. Thus each of thevectors of E in Fig. 9(b) rotates in a positive direction,making many revolutions due to the length of the

NTURALLY

BiREFRINGENTW ~ ~~~~F CRYSTAL

MODULATINGSIGNAL

Fig. 8. Modified AM electrooptic modulator.

1688 APPLIED OPTICS / Vol. 9, No. 7 / July 1970

JI(m) JJm) EM

I JamJ W=±I,

____

i,(m)VT

tJi(m)Jo(m)/-9o

+ 1J(m)J(m)/-9o + J(m)J(m)/-9o+ JO(m)JI(m)/-90,

which gives the final result,

Jj(m)J0(m)/-9o°.

If the long birefringent crystal were not present, theE. and EU contributions would cancel instead of adding,and no output at m would occur.

(bW

Ji(m) E --- _

6 v2

j,(m) Ex 2 M

J, (m)_V 2

(c)

Fig. 9. Vector diagrams for modified AM electrooptic modulator.

crystal. As discussed earlier, the upper sideband gainson the carrier while the lower sideband falls behindduring these rotations. The crystal length is chosenso the upper sideband gains 1800 on the carrier, whilethe lower sideband lags by a like amount. The result-ing diagram for EU at the output of the crystal is shownin Fig. 9(c).

The important result to be noted in Fig. 9(c) is thatthe E. and EU diagrams represent amplitude modula-tions which are in phase with each other, whereas theE, and EU amplitude modulations of Fig. 9(b) are 1800out of phase. It does not matter that the phase of thecarrier of EU in Fig. 9(c) is 0 (which can have any value).All that matters is that the sidebands and carrier ofEU have the proper relative phases.

The unimportance of a can be demonstrated by con-sidering the demodulation of the signals of Fig. 9(c) byphotodetection. When two, rather than one, linearpolarizations are present, the output photocurrent isproportional to Ex(t)E*(t) + EU(t)E*(t). We areinterested in the output component at frequency cin,since this is the modulation which we imposed upon thelight. E and EU will each produce om components dueto the upper and lower sidebands mixing with the carrier.Using the simplified notation described in Sec. III.B,we find that the complex amplitude of the m beat isproportional to

I {[J (m)/ 9 0O][Jo(m) /O0 ] J )t2 -V -/ IL - I-- + [0(m)/ IO

[J(m) /-90o] + [j, /-90° ± 0][Jo(m) 7-]

+ [J L)Ž][JA)/9O .0]}

where the first two products are due to E! and the lasttwo due to E,. This becomes

2. Balanced Optical Detection

For our final example, consider the scheme of Fig. 10for combining and photomixing two optical beams.The two beams might be, for example, the informationcarrying beam (signal) and local oscillator beam (LO)in an optical heterodyne receiver. Vector diagrams willbe used to analyze the performance of such a system.

Suppose that the signal is a single-sideband modu-lated beam whose vector diagram is shown in Fig. 11 (a).Suppose also that the LO is a monochromatic beamwhose frequency differs from that of the signal carrierby WA. Figure 11(b) shows the vector diagram of theLO. Both the signal and LO are assumed to be linearlypolarized in the x direction. There are no opticalelements in the system which will change the polariza-tion of the signal or LO, and hence x subscripts are un-necessary and will not be shown.

The half-silvered mirror divides each incident beaminto reflected and transmitted components. Thesecomponents are equal in amplitude, but differ (as canbe shown from energy considerations) in phase by 900.We are not concerned with the absolute phase changeswhich take place, only with the relative phase changesbetween the transmitted and reflected components.

The reflected component of the signal and trans-mitted component of the LO proceed toward detector 1of Fig. 10. This signal, which we call E1 , is shown atan appropriately chosen reference plane by the vectordiagram of Fig. 11(c). Likewise E2, which consists ofreflected LO and transmitted signal, has the diagram ofFig. 11(d) at an equivalent reference point.

Using the techniques of Sec. III.B, the photocurrentii from detector 1 is found to be

Fig. 10. Balanced opticaldetector. (x direction is out

of page.)

PHOTODETECTOR 2

July 1970 / Vol. 9, No. 7 / APPLIED OPTICS 1689

(a)

J(m J(m)

E,(m)

-'5

E ,;

AmA

(a)

E

(C)

b)

A

(d)

Fig. 11. Vector diagrams for balanced optical detector.

i = (k/2) [AB/-90 cos(wA + w,.)t + AB/-90' coswAt

+ nzA2/0 coscjmt], (a)

where k is a constant of proportionality. Similarly i isfound to be

i2 = (k/2) [mAB/90' cos(wA + w )t + AB/900 coswlt

+ mA 2/00 coscowt]. (lOb)

The output is obtained by subtracting i2 from i, whichgives

i- i2 = (k/2) [(mAB/-90 -mAB/90") cos(coA + w)t

+ (AB/-90 - AB/90") cos caAt

+ (mA 2/0" - mA2/00) coswmt]

= k [mAB/-900 cos(cuA + cu.)t + AB/-90. coswAt]. (11)

The output consists of a carrier of frequency A anda single (upper) sideband of frequency A, + w.. Thusthe signal spectrum has effectively been shifted down infrequency from the optical portion of the spectrum tothe rf portion. The important characteristic of the

system of Fig. 10 is that output occurs only when thereis mixing between a signal frequency and an LO fre-quency. lixing of two signal frequencies (or two LOfrequencies) does not produce an output.

IV. Conclusions

The use of three-dimensional vector diagrams inanalyzing optical devices and systems has been de-scribed. The graphical techniques of this paper arepotentially useful as supplements or alternates to formalmathematical analysis. The diagrams are most usefulin connection with signals containing one or a few dis-crete frequencies. Passage of such a signal throughbirefringent and dichroic elements results in certainsimple changes in the signal's vector diagrams. Inaddition to providing information on the optical outputsignal itself, vector diagrams also can be used to deter-mine the rf (or microwave) output occurring upondetection of the signal by a square-law photodevice.

This work was supported by the Sylvania IndependentResearch Program.

References

1. See, for example, H. H. Skilling, Electrical Engineering Circuits(Wiley, New York, 1957), Chap. 3.

2. R. N. Bracewell, The Fourier Transform and Its Applications(McGraw-Hill Book Co., Inc., New York, 1965).

3. Suppose that we have a real signal f(t). The analytic signalrepresentation for f(t) is found by (a) taking the signal anddeleting the negative frequencies; and (b) doubling thepositive-frequency amplitudes. A discussion of analyticsignals may be found in Ref. 2.

4. I. P. Kaminow, Phys. Rev. Lett. 6, 528 (1961).5. C. F. Buhrer, V. J. Fowler, and L. R. Bloom, Proc. IRE 50,

1827 (1962).6. C. F. Buhrer, Proc. IEEE 51, 1151 (1963).

Neal R. Hinkle of Aerospace Controls Corpora-tion of Los Angeles.

1690 APPLIED OPTICS / VcI. 9, No. 7 / July 1970