Optical amplification of a multimode evanescentlyactive planar optical waveguide

William Y. Liu and Oscar M. Stafsudd

The electric field distribution and complex propagation constants of a multimode dielectric optical waveguidewith complex index of refraction were calculated. A waveguide with a similar structure was fabricated by ionexchange into a glass substrate. The complex index of refraction region was provided by flowing a dyesolution over the waveguide surface. Optical gain as high as 40%/cm was obtained. An experimentaldetermination of the gain of various propagating modes was made and found to be in good agreement withtheoretically calculated values.

I. Introduction

Active dielectric waveguides, i.e., guides that havegain, are of considerable interest. It is difficult tofabricate a waveguide in most active media. However,passive guides are very easy to fabricate in ordinaryglasses by ion diffusion. The low cost and high perfec-tion of glass substrates make them very attractive forintegrated optic devices. A passive dielectric opticalwaveguide can be made into an optical amplifier oractive guide by coupling some of the mode energythrough its evanescent field to an active medium out-side the guide. The scheme was first suggested anddemonstrated by Hill et al.' and Watanabe et al. 2 Atypical evanescently coupled active waveguide has across section as shown in Fig. 1. The structure has ahigher index of refraction region sandwiched betweentwo lower index regions. The fundamental mode fielddistribution is also shown in Fig. 1. If region 1 has gainor loss, indicated here by the complex index of refrac-tion nj, the evanescent field amplitude in this regionwill be amplified or attenuated, respectively. For thecontinued satisfaction of the boundary conditions, theentire mode will exhibit gain or loss. This has beendemonstrated experimentally by Ippen and Shank,3and Jansen.5 By the measurement of gain in opticalfibers surrounded by dye solutions, the cylindrical ge-ometry of the fibers made it very difficult to measureindependently the gain of each propagating mode.

The authors are with University of California, Los Angeles, De-partment of Electrical Engineering, Los Angeles, California 90024-1594.

Received 9 June 1989.0003-6935/901213114-04$02.00/0.© 1990 Optical Society of America.

Planar geometry, on the other hand, allows separationof energy of each propagating mode and, therefore,independent measurement of gain or loss. In thispaper, we report the properties of planar active wave-guides.

11. Theory

A 1-D slab waveguide was analyzed using the ap-proach taken by Miller.6 The analysis was begun byforming Maxwell's equations for TE modes propagat-ing in the waveguide choosing a coordinate system asshown in Fig. 2. TE modes rather than the TM modeswere chosen because of their relatively large evanes-cent field and, therefore, stronger evanescent waveinteraction. This assumption is strictly true only inthe case of isotropic media. However, anisotropy pro-vided by the ion exchange was in fact not observed inour samples. The index variation was taken as arbi-trary in the X-direction, and the electromagnetic wavewas assumed propagate in the Z-direction:

Ex = O, E = - =0,ay

aH. MY~ MYW aH .OHy atx = d~ e~x)at az ax ' (1)

aH, aEy

Equation (1) summarizes the TE mode preconditionsin such a guiding structure. No further assumptionswere made in the subsequent derivation. The result-ing eigenequations, Eqs. (2a) and (2b), associate withthe complex permittivity e, where E' and E" are thereal and imaginary parts of the electric field, ko is thefree space wavenumber, and e',e" are the real and imag-inary parts of the permittivity:

3114 APPLIED OPTICS / Vol. 29, No. 21 / 20 July 1990

nl+ie, active region

n2, waveguide region

k

n3, substrate region (Fig. 1. Optically active waveguide.

1.525

n'(x)1.53 1

1.525

1 ~~~Xt 5pm

n"(x) I

10-5 lII

0 I x

5pm

Fig. 3. Real and imaginary index of refraction as a function of X.

n+ie, active regionn. 2-l * ,. aveuide rg io f.. I

p.F.,,w1, ........................ ° < > < 1 i ;

x y

-z v

Fig. 2. Dielectric slab waveguide.

1a2 E(x) + [e(x)- i]Eoy(x)k2 ax2 ~["X l'9'X

- [(x) - =ii]E; = 0;

k1 ax 2 + [er(x) - I6ir]E;y(x)

(2a)

+ [e,(x) - ii]Ey = 0. (2b)

The argument x indicates the variability of e in thatdirection. The permittivity e as a function of x de-scribes the slab waveguide completely. Finally, f3ii andAir are the real and imaginary components of the eigen-values determined from the known index profile of thewaveguide. The permittivity function and eigenva-lues are related to the index profile and mode indices,respectively, as shown in Eq. (3). The 1-D Helmholtzequation is a special case for the complex eigenequa-tion with the zero imaginary part of e:

e'(x) = n'2() - n'2 (X), e (x) = 2n'(x)n"(x), (3)

where n' = real index and n" = imaginary index.Equations (2a) and (2b) must be solved simulta-

neously because of the interdependency of the electricfield distribution function and the eigenvalues. Thereal part of the eigenvalue is the mode index, and theimaginary part of the eigenvalue is the gain or losscoefficient of the propagating modes in the waveguide.The fundamental and higher-order TE modes werecalculated numerically. The problem is not unlike aquantum mechanical finite potential well problem,and the standard Taylor method7 was used. Because

Table I.

Mode 1 2 3 4 5 6

Analytical 2.2400 2.2196 2.2059 2.1961 2.1889 2.1838

Numerical 2.2431 2.2219 2.2077 2.1973 2.1899 2.1844

Difference (%) 0.14 0.11 0.082 0.055 0.045 0.027

Table II.

Mode 1 2 3

TheoreticalReal index 1.525 1.528 1.530Gain (%) 3.11 15.45 50.81

ExperimentalGain (%) 3.6 14 43Power carried 73.8 23.1 3.1

the imaginary part of the permittivity was many ordersof magnitudes smaller than the real part, an initialsolution was found by ignoring the imaginary eigenval-ue in Eq. (2a). After obtaining this trial solution, Eq.(2b) was used to obtain the imaginary part of theeigenvalue. The imaginary eigenvalue was then usedin Eq. (2a) again to obtain an adjusted real eigenvalue.This iterative process was repeated until the desiredaccuracy was reached. It shall be noted that this pro-cess converges rapidly, requiring only one or two itera-tions in any practical example.

To verify the validity of these numerical solutions,the real part of the eigenvalues was compared to themode indices obtained by Conwell,8 assuming an expo-nential index profile. The results are shown in TableI. It should be noted that the mode indices are in veryclose agreement. The actual index profile of the dif-fused soda-lime glass waveguide used in this study wasdetermined by using the m-line spectroscopic tech-nique.9 The profile was found to be almost an abruptstep as shown in Fig. 3. Figure 3 shows that real andimaginary index profiles which were used to model theactual active waveguide structure. The permittivity e

used in Eqs. (2a) and (2b) was simply the square of thereal and imaginary index profiles. The eigenvalues

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n3 , substrate region

:. I.

.7

calculated from these index profile parameters arecompared to the experimental results and are summa-rized in Table II.

Ill. Experiment

Ion diffused strip dielectric waveguides were madefrom an ordinary soda-lime glass (microscope slide)using an evaporated aluminum mask, conventionalphotolithography, and ion exchange processes. Thechannel waveguide mask openings were 50 ,um wide.Rubidium ions were diffused into the glass by immers-ing the masked substrate in a RbNO3 melt at 4000 C for3 h.10 This produces a diffused guide with a width of50 Arm and depth of 5 ,um, i.e., a 10:1 aspect ratio.Figure 4 shows the typical experimental setup used todetermine the gain or loss for each mode of the activeguide. A He-Ne laser at 632.8 nm was used as a probeand launched many waveguide modes. The energy ofeach mode was sampled by means of a prism couplerand photomultiplier in a manner similar to m-linespectroscopy. Gain was provided by a flowing dyesolution over the substrate's surface. The dye solventwas a mixture of methanol and benzyl alcohol. Theratio of methanol to benzyl alcohol was 1:9 to bring thereal part of index of refraction of the solution to 1.526,a close match to the substrate index. The rhodamine6G concentration of the dye solution was 10-3 M. Thelargest reported small signal gain of such a solution is230 dB/cm at line center."1 The changes of outputintensity of each mode as a function of optical pumpapplied to the dye solution indicated the amplificationby the active waveguide of each mode. The opticalpump was a nitrogen laser at 337 nm. Using thisexperimental arrangement, the gain and relative pow-er carried by each mode were measured.

IV. Results

The index profile used in calculations is shown inFig. 3. The waveguide was measured to have approxi-mately a step index profile and was 5 Aim deep. Thesubstrate index was 1.515, the waveguide index was1.531, and the dye index was 1.526. Since the pumpsource was a focused nitrogen laser beam, the dyesolution was assumed to be uniformly pumped over thetotal volume of the evanescent field of the probe beam.This assumption implied that the imaginary index ofrefraction for the dye solution was a constant indepen-dent of X and Z for all practical considerations. Be-cause the value for the imaginary index of the dye

UV

Fig. 4. Gain vs mode measurement.

solution was difficult to determine experimentally, itwas used as a single parameter to fit simultaneouslythe experimentally determined gains of the variousmodes.

The results of computer calculation and experi-ments are listed in Table II. Notice that, although achange in imaginary index of the gain medium willchange the theoretical gain, the ratio between gainsassociated with various modes will not be affected.The ratio between gains of each mode is rooted in thestructure of the waveguide. The evanescent field ofhigher-order modes penetrates deeper into the dyesolution than the lower-order modes; therefore, high-er-order modes are amplified more than lower-ordermodes. Since the amount of evanescent field penetra-tion is determined by the waveguide structure, the gainratios between various modes are not changed bychanging the very small imaginary index of the dyesolution. The best of the imaginary index of the dyesolution was determined to be 10-5.

The agreement between the experimental value andthe computer's prediction was remarkable, consider-ing that the 10% experimental error and the modelused was only a 1-D approximation to the 2-D realwaveguide structure. The He-Ne probe beam wave-length, 632.8 nm, is quite far from the line center, 590nm, of rhodamine 6G. Therefore, the small signal gainof the dye at this wavelength is a factor of -10 belowthe line center value. The imaginary index of refrac-tion of saturation pumped rhodamine 6G at line centeris -5 X 10-4. This is calculated from the gain value'quoted in Ref. 11.

V. Conclusion

The real and imaginary parts of the mode indices fora dielectric optical waveguide were found numerically.The real part of the indices was determined by both thenumerical calculation of a step index waveguide andthe m-line experimental data. The results from thetwo methods were in very good agreement. The imagi-nary part of the mode indices was verified by theexperimental results. It should be noted that the per-turbation method, such as discussed by Yeh andYariv,12 will yield similar results for only stepped indexguides. However, the numerical approach can be usedto solve the waveguide with arbitrary real and imagi-nary index profiles. It is a very powerful tool for thestudy of real active waveguide structures.

References1. K. 0. Hill, A. Watanabe, and J. G. Chambers, "Evanescent-

Wave Interactions in an Optical Wave-Guiding Structure,"Appl. Opt. 11, 1952-1959 (1972).

2. A. Watanabe, K. 0. Hill, and R. I. MacDonald, "Amplification ofLight in an Optical Wave-Guiding Structure with Evanescent-Wave Pumping," Can. J. Phys. 51, 761-771 (1973).

3. E. P. Ippen and C. V. Shank, "Evanescent-Field-Pumped DyeLaser," Appl. Phys. Lett. 21, (1 Oct. 1972).

4. H. Injeyan, "Light Amplification by Evanescent Wave Couplingin a Multimode Fiber," UCLA Electrical Engineering Ph.D.Dissertation (1981).

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5. M. Jansen, "Active Optical Fiber Structures," UCLA Electrical 9. R. Th. Kersten, "The Prism-Film Coupler as a Precision Instru-Engineering Ph.D. Dissertation (1984). ment. Part I. Accuracy and Capabilities of Prism Coupler as

Instruments," Opt. Acta 22, 503-513 (1975).6. 5. . Miller, "Integrated Optics: an Introduction,' Bell Syst. 10. V. Neuman, 0. Parriaux, and L. M. Walpita, "Double-Alkali

Tech. J. 48, 2059-2069 (1969). Effect: Influence on Index Profile of Ion-Exchanged Wavegui-

7. D. S. Saxon, Elementary Quantum Mechanics (Holden-Day, des," Electron. Let. 15 (1979).San Francisco, 1969), pp 158-168. 11. F. P. Schafer, "Principle of Dye Laser Operation," Top. Appl.

Phys. 1, 81 (1973).

8. E. M. Conwell, "Modes in Optical Waveguide Formed by Diffu- 12. P. Yeh and A. Yariv, Optical Waves in Crystals (Wiley, Newsion," Appl. Phys. Lett. 23, 328-329 (15 Sept. 1973). York, 1984).

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