Enhanced optooptical light deflection using cavity resonance

William H. Steier, Gregory T. Kavounas, Richard T. Sahara, and Jayant Kumar

The efficiency of optooptical light deflection by nondegenerate four-wave mixing can be increased significant-

ly by placing the grating in a resonant cavity. Theory and experiment are presented for linear and ringcavities. Efficiency improvement by an order of magnitude is predicted, and improvements by a factor of 5.6

have been demonstrated.

1. Introduction

Dynamic optooptical beam steering has been report-ed in several photorefractive materials. 1 2 In theseswitches, a grating is written by two optical controlbeams to steer or deflect an optical signal beam. Ar-rays of these switches are essentially real-time holo-grams, and they have potential use in dynamic opticalinterconnects for optical computing and in optical net-work switching. The efficiency v (v = the ratio of thedeflected to the input beam) of these switches is typi-cally only a few percent for materials with reasonablyfast response times. One approach to increasing theefficiency is to go to very long grating wavelengths andvery high electric fields. Herriau et al. 3 reported 95%efficiency in BGO at 20-,gm grating wavelength and 14-kV/cm applied field. We present here another tech-nique that can potentially enhance the efficiency by anorder of magnitude or more by using the grating tocouple into a resonant cavity. This is similar in con-cept to the resonated holograms discussed by Collins.4

11. Linear Cavity

Figure 1 shows the linear cavity configuration. Thecontrol beams at X, write a grating in the nonlinearmaterial. The input beam (X) is at Bragg angle for thegrating. A portion of the input is deflected into thecavity which is resonant at X. The light coupled

The authors are with University of Southern California, Depart-ment of Electrical Engineering, Center for Photonic Technology,Los Angeles, California 90089-0483.

Received 13 August 1987.0003-6935/88/081603-04$02.00/0.© 1988 Optical Society of America.

through one of the mirrors is the output of the switch.Blocking a control beam will erase the grating and turnoff the switch. The surfaces of the nonlinear materialare set at Brewster angle to minimize reflection lossesin the resonator.

The device can be analyzed by following the ampli-tude and phase of the electric field in the cavity andthen equating the field to itself after one round trip.All fields are assumed scalar, and a plane wave analysisis used. Figure 2 shows the location of the electricfields that are related in the following expressions:

(1)

(2)

(3)

(4)

(5)

Eout = CT El,

E2 = -C E1 ,

£3 = E2 exp(ik.l)

E 4 = -- E 3 - i E,

E, = -V1 - A E4 exp(2ik,1 2),

E6 = 1-v

El = E 6 exp(ik,11),

Er = E-iF E3.

E, = -i.,F E5.

(6)

(7)

(8)

(9)

In these expressionsR = intensity reflection coefficient of the output

mirror,T = 1 - R = intensity transmission coefficient of

the output mirror,v = I(Bragg scattered)/I(incident) = deflection

efficiency of the grating,A = round-trip cavity loss except that due to the

grating and mirror coupling, andk = 27r/Xs

It is assumed for simplicity in the analysis that allthe cavity loss A is located at the rear mirror andreduces the reflection from this mirror. The cavity

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1603

Output 2

R

Flat Mirror onPiezo elotri mountR=1.0

Mirror

Scope

R = 100%

Input

Control

Fig. 1. Enhanced optooptical light deflection using a linear cavity.The dashed lines are the signal beam; the solid lines are control

beams.

H.-N. Lasr

Fig. 3. Experimental layout.

E/Er

E3

Fig. 2. Linear cavity showing the location of the various E fieldsused in the analysis.

lengths 11 and 12 are shown in Fig. 2. The optical pathlength of the grating is assumed to be included in 11 and12. The 1800 phase shift in the reflected beam fromeach mirror is included. A -900 phase shift betweenthe Bragg scattered beam from the grating and inci-dent beam 5 has also been included in Eqs. (4), (8), and(9).

Using Eqs. (1)-(7), the field E can be set equal toitself after one round trip:

E= i v( - v)(I - A) E exp[ik,(212 + 11)] (1)

-(1 - ) R(1 - A) exp[2ik,(11 + 12)]At resonance k(11 + 12) = mIl; here m is an integer.The output intensity Io in terms of the incident inten-sity Ii can be calculated from Eq. (1) at resonance:

Io Tv(1 - v)(1- A)I, [I1- (1-vl- A)Rj2

To maximize this ratio, the output mirror reflectivityshould be

R(optimum) = (1 - A)(1 - v)2. (12)

From Eq. (8), the transmitted intensity It can be calcu-lated:

It (1-)[1- R(1 A) 2 (13)

From Eq. (9) the reflected intensity can be calculated:

Ii [1 - (1-v) R(1-_ A)] 2(14)

The balance of the power is absorbed in cavity losses.

The efficiency of the combined grating and cavitycan be more than an order of magnitude larger than v ifthe cavity losses can be kept low. The rise and decaytimes of the switch will be increased by the cavity risetime; but for modest grating writing speeds, thisshould not be a factor. The bandwidth of the channelbeing switched will be reduced to the width of thecavity resonance. The greatest practical difficulty iskeeping the cavity resonant at X,.

To demonstrate the concept, we have completed theexperiment shown in Fig. 3. The gratings were writtenin LiNbO3:Fe using the 4880-A line of the Ar-ion laser.The signal wavelength was 6328-A. The cavity con-sisted of a 0-cm concave mirror with 90% reflectivityand a flat mirror with high reflectivity. The mirrorswere spaced 12.5 cm apart. The LiNbO 3 :Fe was 5 X 5X 2 mm3 With the c axis perpendicular to one of the 5 X2-mm 2 faces. The c axis was oriented for the maxi-mum photorefraction. The LiNbO3 plate was placedat Brewster angle for the 6328-A beam to minimizereflection loss within the cavity.

The photorefraction in LiNbO 3 is slow, so this mate-rial was used to demonstrate the cavity enhancementand not otential switching speed. Gratings writtenat 4880 A would last for over an hour after the writingbeams were turned off. One mirror of the cavity wasmounted on a piezoelectric drive so that the cavitycould be swept through resonance at by applying asawtooth voltage waveform.

Using a 0.36-,vm wavelength grating with a measured1.5% diffraction efficiency, the data shown in Fig. 4were taken. Figure 4 is a tracing of a photograph of anoscilloscope trace. The input beam was modematched to the cavity, and the cavity was carefullyaligned to minimize higher-order transverse mode ex-citation. The lower trace is the sawtooth drive voltagein the piezoelectric sweep while the upper trace is the6328-A power coupled through the cavity output mir-ror. Two longitudinal modes of the cavity can be seen.The diffraction efficiency of the grating-cavity combi-nation was 7.2%, a factor of 4.8 enhancement over thegrating diffraction efficiency.

1604 APPLIED OPTICS / Vol. 27, No. 8 / 15 April 1988

Fig. 4. Oscilloscope traces showing the experimental results for thelinear cavity: upper trace, cavity output showing two cavity longitu-dinal modes; lower trace, sawtooth voltage applied to the piezoelec-

tric mirror mount.

OUTP

OUTPUT 1

INPUTkc

Xc -

Fig. 6. Ring cavity showing the location of the various E fields usedin the analysis.

E, = E2 exp(ikh1),

E,= J-- E,- i,_Ej,

E5= - 1 -A E4 exp(ik,12),

El= E5 exp(ik.13),

E, -ir-E 3 + 1-v E.

=100%

Fig. 5. Ring cavity resonator. The dashed lines are the signalbeam paths; the solid lines are the control beam paths.

From the observed finesse of the cavity F, the round-trip cavity loss can be calculated. If the round-triplosses are not too large, the finesse can be approximat-ed by 6

F211

6 = fractional round-trip loss

= A+2v+T.

From the observed finesse of 23 and the knownvalues of T and v, the cavity loss A can be calculated.Using these values in Eq. (11), the calculated efficiencyof the combined grating cavity is 6.9% compared withthe experimentally observed 7.2%.

Il. Ring Cavity

The ring cavity, shown in Fig. 5, has advantages overthe linear cavity since no power is reflected back at theinput and the circulating mode sees the loss and scat-tering of the LiNbO3 only once per round trip. Thedisadvantage is the added scattering and absorptionloss of the third mirror.

Figure 6 shows the fields of the ring cavity which arerelated by

E-,,.. = ~~~~_T E,, ~(15)

(17)

(18)

(19)

(20)

(21)

As in the linear cavity, the cavity round-trip loss A isassumed to be located at one mirror. The other quan-tities are the same as for the linear cavity. The field isset equal to itself after one round trip and the outputintensity at resonance is calculated as

IO Tv(1 - A)

Ij [1 - _(1-v)(-A)R] 2

The transmitted intensity is

I, [1- _1-v)(1-_A)R 2

(22)

(23)

The reflected power is only due to misalignment of theBrewster angle and is very small.

The ring cavity consisted of the same 50-cm radiusmirror and the flat mirror but used at -20 fromnormal. The third mirror was a flat with high normalincidence reflectivity at 6328 A used at ,-.51' fromnormal. None of the mirrors used had coatings de-signed for off normal use and hence introduced moreloss than necessary.

Figure 7 is a tracing of an oscilloscope photographand shows the results for the ring cavity. The inputbeam was mode matched, and the cavity was carefullyaligned to minimize higher-order transverse modes.In this case, the grating had a measured 2.3% diffrac-tion efficiency at 6328 A. The cavity-grating combi-nation had a measured diffraction efficiency of 12.8%for a factor of 5.6 enhancement.

From the observed cavity finesse, the round-trip lossA can be measured. The output mirror transmissionwas measured at ,,-20O from normal and still found tobe 0.1. For the ring cavity,

F2116

6= A + v + T.

(16) The measured finesse was 25.5 which gave A = 0.10.

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1605

E2 = -rR El,

Fig. 7. Oscilloscope traces showing the experimental results for thering cavity: upper trace, cavity output showing two cavity longitu-dinal modes; lower trace, sawtooth voltage applied to the piezoelec-

tric mirror mount.

Tranrmdted

Fig. 8. Confocal resonator for enhanced efficiency beam steering.The mirror radii and mirror spacing are equal. The dashed lines arethe beam path of the signal beam. The fixed gratings maintain the

Bragg condition as X, is varied as per Ref. 10.

Equation (22) predicts the diffraction efficiency of thecavity-grating combination to be 13.5%, very close tothe observed 12.8%.

IV. Comments and Conclusions

These experiments demonstrate the validity of theconcept and of the derived equations. The practical-ity of this approach will be determined by one's abilityto keep the cavity losses low and the cavity on reso-nance. The experimental results reported here werelimited by unnecessarily high mirror and LiNbO3 sur-face scattering; losses which can be reduced in practice.

With regard to cavity losses the following commentsare pertinent. Because of optical activity, use of theBrewster angle is not possible when using BSO orBGO; antireflection coatings would be required.When using the semiconductors such as GaAs or InP,the material absorption and hence the cavity losses canbe kept low only for signal wavelengths at least 1.3 Atmlong.

A practical stable resonator will likely require fabri-cation from a single piece of material and may requirethe high temperature stability of an oven. The geome-

try used by Henshaw7 which brings the control beamsin from the side of the crystal may be a more practicalway of getting all the beams simultaneously into thephotorefractive material.

Wide angular scanning of the diffracted beam byvarying the wavelength of the writing beams has beenreported.8 9 Angular deviations of up to 30° have beenreported using BSO.10 However, the diffraction effi-ciency was only a few percent. By using the confocalresonator shown in Fig. 8 the overall diffraction effi-ciency can be increased. The output mirror is highlyreflecting over one-half of its surface and partiallyreflecting over the remainder. The fixed gratings areused to maintain the Bragg condition as X, is varied asdescribed in Ref. 10. As X, is varied the angle of thebeam coupled into the cavity varies and the beam pathwithin the cavity changes. The confocal nature of thecavity makes the beam path repeat after two roundtrips and causes the output beam from the cavity tosweep in angle. For small angular deviations one canshow that the resonant frequency for the on-axis andoff-axis beam paths are the same so that the cavitylength can be held fixed. Hence diffraction efficiencyenhancements of the same order of magnitude as pre-dicted and demonstrated here could be achieved in thebeam steering devices.

The authors wish to acknowledge the support of theAir Force Office of Scientific Research under grantAFOSR-84-0207 and the Joint Services ElectronicsProgram under contract F49620-85-C-0071.

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9. G. Pauliat, J. P. Herriau, A. Delboulbe, G. Roosen, and J. P.Huignard, "Dynamic Beam Deflection Using PhotorefractiveGratings in Bi12SiO 2 0 Crystals," J. Opt. Soc. Am B 3,306 (1986).

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