Cascading of Second-Order Nonlinearities - ACS Symposium Series

Chapter 37

Cascading of Second-Order Nonlinearities Concepts, Materials, and Devices 1

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William E. Torruellas , Dug Y. Kim , Matthias Jaeger , Gijs Krijnen , Roland Schiek , George I. Stegeman, Petar Vidakovic , and Joseph Zyss Downloaded by NORTH CAROLINA STATE UNIV on October 14, 2012 | http://pubs.acs.org Publication Date: August 11, 1995 | doi: 10.1021/bk-1995-0601.ch037

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Center for Research and Education in Optics and Lasers, University of Central Florida, 12424 Research Parkway, Orlando, FL 32826 France Telecom, Centre National d'Etudes des Télécommunications, Paris B, BP 107, 196 Avenue Henri Ravera, 92225 Bagneux, France 2

The concept of utilizing second order nonlinear optical processes to mimic third order nonlinearities, in particular the optical Kerr effect, is discussed. Nonlinear organic materials with their large second order nonlinearities offer definite advantages for the implementation of such concepts. As a demonstration, we present both experimental and numerical results in the two important telecommunication windows on a DAN single crystal corefiberoriginally designed for efficient second harmonic emission in the Čerenkov geometry at 820 nm.

When third order nonlinear phenomena are investigated in non-centrosymmetric materials, two successive (cascaded) second order processes can contribute to the intrinsic third order response for some effects. (1) As early as 1967, Ostrowskij predicted that cascading could play a prime role in self-action processes or any of the geometries involving the mixing of four degenerate waves.(2) The two contributions which lead to intensity-dependent effects for a single input beam are shown schematically in Figure 1. The presence of the so-called cascaded nonlinearities can interfere with and even mask the direct or intrinsic third order effects. In the early days of nonlinear optics such interference effects were reported in studies of nonlinear phenomena in GaAs.(3,4) Cascading has also been used to calibrate the third order nonlinear susceptibility of reference materials such as α-quartz whose nonlinearities are in turn are used as references for Third Harmonic Generation evaluations of nonlinearities in fused silica etc.(5) In general cascading in bulk materials has not led to a large intensity-dependent refractive mdex.(6-9) However, recent advances have produced both new organic second order materials with large nonlinearities as well as new methods of phase-matching existing materials in waveguides utilizing the large nonlinear coefficients normally not available for phase-matching bulk crystais.(10-]3) This has led to new experiments in bulk organic materials as well as in waveguides/5,14-16) The low powers expected in waveguides have led to a number 0097-6156/95/0601-O509$12.00/0 © 1995 American Chemical Society In Polymers for Second-Order Nonlinear Optics; Lindsay, G., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

Downloaded by NORTH CAROLINA STATE UNIV on October 14, 2012 | http://pubs.acs.org Publication Date: August 11, 1995 | doi: 10.1021/bk-1995-0601.ch037

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POLYMERS FOR SECOND-ORDER NONLINEAR OPTICS

of predicted applications to signal processing, spatial solitons etc. (77-22,) In fact, a novel form of all-optical switching has already been demonstrated.(23) Experimental evidence for surprisingly large nonlinearities which were subsequently identified to be due to cascading was available already in 1990.(24) Yamashita and coworkers reported self-phase modulation in DAN crystal core fibers which corresponded to record high, instantaneous, intensity dependent refractive indices/2^ They used the effect to pulse compress the output of a femtosecond CPM dye laser.(25) Thesefiberswere initially fabricated for efficient Cerenkov doubling of near infrared light into the b\ue.(26) By operating farfromany molecular resonance, we have now clearly identified both experimentally and numerically the origin of Yamashita's SPM result as being due to cascading via the presence of a Cerenkov SHG in thosefibers.(75,27jAdditional confirmation of cascading was obtainedfromZ-scan measurements in a bulk single crystal of the same material used in the crystal core fibers.(75; The latter results open, in our opinion, a very interesting area of research. Although SHG in organic crystal core fibers may be considered to be an attractive approach to direct doubling of diode lasers into die blue region of the spectrum, this approach has been slowed down because of photo-induced damage in the blue as well as the recent improvements in short wavelength diode lasers/25,29) Phase matchable organic materials are, however, still extremely attractive because of their large figure of merit for SHG in devices using cascading as the source of nonlinear phase distortion, in particular within the two telecommunication windows demonstrated here. We show that such an effect gives rise to effective nonlinear coefficients of the same order of magnitude as the largest intrinsic optical Kerr coefficients reported at 1.3 and 1.6 \im.(30,31) Extrapolating these results to other materials and geometries, the possibility of exceeding these latter values by several orders of magnitude makes this approach in organic polymers and crystals probably the only viable one for reducing the switching power of a lossless all-optical switch to below the magical 1 Watt level. Concepts: As previously mentioned, cascaded nonlinearities play an important role in noncentrosymmetric materials with phase-matchable nonlinearities. In the case of SHG and for plane waves in the CW regime we can describe this effect by invoking a simple coupled mode analysis. This leads to the following two coupled partial differential equations describing the propagation of the twofieldsat the fundamental and second harmonic frequencies(7#j: BE*

2

a

iLkz

±KE »E 'e '

dz'

iK(Jg? ) e- ' w

2

iAJrz

dz'

Here K defines a coupling coefficient which is proportional to the effective second order

In Polymers for Second-Order Nonlinear Optics; Lindsay, G., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

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Cascading ofSecond-Order Nonlinearities511

TORRUELLAS ET AL.

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nonlinear susceptibility d ^ ) for the particular SHG interaction (propagation direction, light polarizations etc.), and Akz' is the linear phase mismatch between the twofields.Integrating these equations in the low depletion approximation is instructive for the understanding of the physics involved in the cascading process: dx" _

dz'

7

sintAJcz ) A* K 1 - cos (AJcz )

2

a

2

2

e f f

2


"

=

"

7

A J T

KMC vac

w

Here I refers to the fundamental light intensity, and n (henceforth written as just nj) is the effective cascaded nonlinearity. K is proportional to [d ^ ^] /n which has to be as large as possible. Hence the samefigureof merit occurs for cascading and SHG. An additional prerequisite for large nonlinearities is that the SHG interaction should be almost phase-matched. Tofirstapproximation, one can see that the "space averaged" nonlinearity will be proportional to 1/Ak, for large AkL. Finally a consequence of the previous derivation is that the sign of the effective intensity dependent nonlinearity changes when the phase mismatch changes sign, a clear signature of the cascading process being present and dominant. Note that the above analysis is only valid very far from phase matching. Because cascading is a consequence of a strong coupling between the twofieldspresent, the low depletion approximation is primarily useful for pedagogical reasons and to gain some insight into the problem. Examining the above equations, some analogy with a two photon absorber is apparent. The phase-matching condition has in fact created an electromagnetic resonance for the fundamental field. However, when solved more completely, the above equations show significant differences relative to two-photon absorption. (18,32) However the most important one is that in the cascading case no photonic energy is lost to the material system as would be the case in two photon absorption. The energy is only transferred to other electromagnetic frequencies (second harmonic in this case). Therefore it is possible to recover the energy in the fundamental field. From this perspective the cascading approach to enhancing the effective Kerr nonlinearity is lossless in nature and hence does not induce thermo-optic effects if linear absorption is absent. If no second-harmonic is desired it is possible to design the "electromagnetic resonance" in order to induce large nonlinear phase shifts and no or small amounts of fundamental depletion as demonstrated in the following discussion. 2

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e

Materials: Table I clearly indicates that organic materials are far more promising than existing inorganics for cascading. For operation within one of the telecommunication windows at 1300nm or 1550nm these materials do not sufferfromthe detrimental absorption and damage associated with blue-green generation in organics.^^ Table I also shows that


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this approach could lead to an enhancement of several orders of magnitude over the existing non-resonant third order nonlinearities of semiconductors (~ 10' cm /W) and ID rc-electron conjugated polymers(~4xl0" ca?Nf).(30,31,33) As a result we believe thatfroma materials perspective, cascading in organics is extremely promising. 13

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Table I: k refers to the estimated cut-off wavelength for linear and nonlinear absorption, d^ and dy are respectively the largest diagonal and off-diagonal second order nonlinear coefficients. n is the intensity dependent refractive index coefficient. P is the estimated switching power for a 2 cm long device with a 5 pm channel waveguide area producing a 2n nonlinear phase shift at 1500 nm. It is approximately equal to 4xl0" / n (Watt).


c

2

2

s

12

2

K

dii

nm

AlGaAs 31

33

p T S

KTP

7

LiNb03 DAN NPP

14

1 5

10

MMONS Poled Polymer DAST

40

38

(pm^V)

(pm/V)

n

e

f

f

n

2> cn?AV

Watt

1450 nm

lxlO

-13

37.5

1300 nm

2xl0

-12

2

700 nm

6

18

lxl0"

13

700 nm

6

25

2xl0"

13

-12

40 25 3.75

900 nm

50

lxlO

900 nm

98

5xl0"

12

0.8

4xl0"

12

1

900 nm

85

-11

900 nm

200 (DANS)

lxlO

1400 nm

600

lxlO"

39

10

0.4 0.04

However at this time inorganic materials are more generally available and phasematching techniques are much better developed in inorganics than for organic materials.(12) Hence one can expect a time lag in the implementation of organic cascading devices. Nevertheless we believe that because of their flexibility and integrability, organic polymers will ultimately be the material class of choice when alloptical devices and architectures arefinallyimplemented. One should also note the extremely high d coefficient of DAST single crystals, in which the implementation of phase-matching involving other than a birefringence technique, quasi-phase matching or Cerenkov phase matching for example, could result in a very large enhancement. (11) 3 3


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Nonlinearity Measurements: Single crystalfibersof organic molecules of the paranitroaniline family have previously been grown for efficient SHG in the blue-green spectral region.(26,34,35) The single mode fiber geometry for DAN crystal core is shown in Figure 2, where the angle between the Z axis and thefiberaxis has been determined to be 36 degrees/35,) Such fibers are designed to be phase matched under the Cerenkov condition for blue light generation/26^ As shown in Figure 3, while the fundamental wavelength at 1320 nm can be guided in the case of an SF1 cladding, the second harmonic polarized along the Y axis has a refractive index in the core smaller than that of the cladding. Efficient SHG for a 1320 nm (and previously 820 nm) input was observed. At the Cerenkov radiation angle in the cladding, the projection of the SHG wavevector along the fiber axis is twice that of the guided wave at the fundamentalfrequency.Figure 4 shows this condition. Although the radiated SHG is not strongly confined in the core of the fiber, efficient interaction between the guided mode at the fundamentalfrequencyand the portion of the SHG trapped in the core can be realized. Such an interaction gives rise, in fact, to a phase matching condition similar to that observed in the more classical case of copropagating fundamental and harmonic. We have studied the second harmonic and nonlinear phase shift processes numerically by using a standard BPM (Beam Propagation Method)/3# Shown in Figure 5 is the variation in nonlinear phase shift with the difference between the refractive index of the cladding at the second harmonicfrequencyand the effective refractive index of the guided fundamental. Similar to the bulk case, a change in sign occurs when going through such a phase matching resonance, allowing both negative and positive nonlinearities to be obtained in a Cerenkov geometry. Ourfiberwas designed with a change in refractive index, abscissa of Figure 5, close to 8xl0" , implying as noted in the previous section that a phase shift of approximately -50° is reached with no considerable depletion being observed, less than 5%. Additional modifications of the phase matching conditions could improve in principle such a loss figure. Note that even though Figure 5 resembles the solution of the standard plane wave, infinite medium model, in our case the abscissa is an arbitrary parameter that we decided to vary in our model and is not the phase mismatch between the guided fundamental mode and the radiated Cerenkov wave. A number of experiments were performed on single crystal DAN, both in fiber and bulk crystal form at 1064, 1320 and 1550 nm.(J5,27) The most sophisticated utilized the DANfiberin one arm of a Mach-Zehnder interferometer and propagation in air for the reference arm.(15,37) A typical fringe pattern observed when the length of the reference arm is changed (scanned) is shown in Figure 6. As the intensity of the radiation at 1320 nm in thefiberis increased, there is an additional phase shift due to cascading and the interferometerfringepattern shifts relative to its low power case, as shown in Figure 6. Purposely moving one of the mirrors of the interferometer allowed us to calibrate the interferometer for a positive or negative nonlinear phase shift. The case shown in Figure 6 corresponds to a reduction of optical path length equivalent to a phase shift of -rc/4. A value for the effective n of -8xl0" cm /W was obtained for a D A N core fiber with a SF1 glass cladding, in reasonable agreement with the calculated value. 3

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Figure 1. Schematics of the cascaded (small depletion limit) and direct Kenprocesses. Both lead to effective nonlinearities proportional to IE" | . 2

Z

X

Figure 2. Single crystal DANfibergeometry and the polarizations for the input fundamental and second harmonic generated by the Cerenkov process. 1.95

r-

1.90

i-

1 gQ I I I I I I I 1 1 I I I I I I 1 I I I I I I I 4 III1IIII1II 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

I I

III11III

Wavelength ( X m )

Figure 3. Refractive index dispersion with wavelength of the SF1 glass and the two normal mode polarizations of the DAN fiber. At 1320 nm n ^ ^ A N ) > riclad showing that the fundamental is guided, where-as n j j(660nm) > n (DAN,1320nm) indicating the possibility of a Cerenkov phase matching condition. c a(

xz


I


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Figure 4. The Cerenkov phase matching condition with SHG generation at an angle 6 into the cladding. Note that the SH field in the core is reflected at the core-cladding interface. For small angles 6 the reflection coefficient of the SH field approaches unity, overlaps well the fundamental and can copropagate with it for fairly long distances. N is the effective index of the fundamental and n is the cladding index at the harmonic frequency with cos8 = N / n 2 . The schematic Cerenkov phase matching condition is also represented. C

C

w

2(l)

c

w

w

150

50 0)

u W

0)

in n) JZ

Q_

0

-50

-100 -150

-20

-10

0

10

20

1.10 1.00 0.90 0.80 c

0.70

G- 0 . B 0 3

0.50

Q_° 0 . 4 0 0.3C 0.20 0.10 0.00

-20

-10

0

Cn (2u)-/V c

ef f

10

(u) ) / 1 0 "

20 3

Figure 5. Results of a numerical calculation for the fundamental power and nonlinear phase shift at the output of a single mode DAN fiber under the Cerenkov condition. Here n ^ u ) and N g(o)) are the cladding refractive index and guided wave effective index respectively. e



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Figure 6. Low and high intensity scans of the fringe patternsfromthe MachZehnder interferometer measured at 1320 nm corresponding to a negative intensity dependent refractive index.


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Broadening of the frequency spectrum of a short pulse due to Self-Phase Modulation (SPM) was also used to estimate the effective nonlinearity at 1550 nm. Modulation of the carrierfrequencycould be observed when femtosecond pulses were launched into the DAN fiber. In this case the self-phase modulation signal (Figure 7) corresponds to an intensity dependent refractive index | n | of 4x10'' cm AV. Although only the magnitude of the effective nonlinearity can be deduced with this technique, reasonable agreement is obtained with the more accurate interferometric measurement at 1320 nm. (In fact the same value is not expected since the Cerenkov cross-section will change with wavelength.) Both experiments had an important feature in common. In both cases the observation of an effective Kerr nonlinearity was correlated with the presence of a strong SHG field in the core. When different cladding glasses were used, the SHG intensity in the core was considerably reduced making the observation of a phase shift or a broadening due to SPM impossible at similar fundamental input powers. The correlation is that second harmonic conversion is necessary for large nonlinear phase shifts. In thefibersthe orientation of the crystal relative to thefiberaxis isfixedby the growth technique used and the dependence of the nonlinearity on phase mismatch cannot be investigated conveniently. Therefore in order to verify the change in the sign of the nonlinearity at phase-matching, the nonlinear phase shift and effective nonlinearity n were measured using Z-scan in a DAN bulk crystal at 1064 nm near the phase matching condition. The crystal was generously donated by the IBM group headed by Dr. G. Bjorklund. Figures 8 and 9 show the results for the fundamental and second harmonic powers, and the effective nonlinearity versus incidence angle near phase-matching. Integrating Eq.l for a gaussian pulse in time and space under the 1 3

2


2

2

1569

1573

1577 Wavelength

1581

1585

1589

[ nm ]

Figure 7. High power spectral broadening of 0.5 psec pulses outputfroma D A N crystal core fiber relative to the low input power case. Self-phasemodulation occurs due to cascading at 1550 nm, showing the ultrafast response of the nonlinearity.



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-25

-15

-5

5

15

25

P h a s e - M i s m a t c h Ak.L

Figure 8. Phase matching resonance for SHG in a DAN single crystal. Note the large depletion of the fundamental beam (open circles) and the strong conversion to second harmonic (solid circles). AkL is the cumulative phase mismatch in the crystal L long.

_3

5

li

"-25

11111111

11111111111111111111111111111111

-15

-5

5

15

11111111

25

P h a s e - M i s m a t c h Ak.L

Figure 9. Effective intensity dependent refractive index coefficient measured at 1064nm with the Z-scan technique near the phase matching resonance versus cumulative phase mismatch AkL. The solid line represents the results of a numerical model which integrates Eq.l for no walk-off and negligible diffraction.


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assumptions of no diffraction and small spatial walk-off gives good agreement with the experimental results. The change in sign in n clearly occurs at phase-match. Therefore we conclude that cascading of two second order processes is clearly the dominant mechanism for self action in D A N crystals when operating near a phase matching resonance. In conclusion we have shown that within both telecommunication windows, cascaded nonlinearities in organic single crystal core fibers can mimic very large third order effects. Comparison with other organic crystals and polymers shows that this approach could lead to effects several orders of magnitude larger than those observed in the currently most effective third order materials. The research at CREOL, was supported by a grant from the AFOSR. While at C N E T , W T was a NSF/NATO postdoctoral fellow. The authors are grateful to the I B M group, for allowing them access to a D A N single crystal.


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11. 12. 13. 14.

see for example Flytzanis,C., in Quantum Electronics; ed. Rabin, H . ; Tang, C.L. (Academic, N Y , 1975), Vol 1, Part A . Ostrowskij, L . A . JETP Lett., 10, 281 (1967). Yablonovitch, E.; Flytzanis, C.; Bloembergen, N . Phys. Rev. Lett., 29, 865 (1972). Gustafson, T.K.; Taran, J.-P.E.; Kelley, P.L.; Chiao, R . Y . Opt. Comm., 2, 17 (1970). Meredith, G.R. Phys. Rev. B, 24, 5522 (1981); Buchalter, B.; Meredith, G.R. Appl. Opt., 21, 3221 (1982). Belashenkov, N . R.; Gagarskii, S. V . ; Inochkin, M . V . Opt. Spectrosc., 66, 1383-6 (1989). DeSalvo, R.; Hagan, D.J.; Sheik-Bahae, M . ; Stegeman,G.I.; Vanherzeele, H.; Van Stryland, E.W. Opt. Let., 17, 28 (1992). Danielius,D.; Di Trapani, P.; Dubietis, A.; Piskaskas, A.; Podenas, D.; Banfi, G.P. Opt. Lett., 18, (1993) 574. Nitti, S.; Tan, H . M . ; Banfi, G.P. and Degiorgio, V . Opt. Comm., 106, 263 (1993). Zyss, J.; Chemla, D.S., "Quadratic Nonlinear optics and Optimization of the Second-Order Nonlinear Optical Response of Molecular Crystals", in Nonlinear and Optical Properties of Organic Molecules and Crystals; eds. Chemla, D.S.; Zyss, J., Vol 1 (Academic Press, 1987) pp23-192 Marder,S.R.; Perry, J.W.; Schaefer, W.P., Science 245, 626 (1989). Fejer, M . M.; Magel, G. A.; Jundt, D . H . ; Byer, R. L., IEEE J. Quantum Electron. 28, 2631 (1992). van der Poel, C. J.; Bierlein, J. D.; Brown, J. B.; Colak, S., Appl. Phys. Lett. 57, 2074 (1990). Sundheimer,M.L.;Bosshard, Ch.; VanStryland, E.V.; Stegeman, G.I.; Bierlein, J.D., Opt. Lett., 18, 1397 (1993).


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Kim, D.Y.; Torruellas, W.E.; Kang, J.; Bosshard, C.; Vidakovic, P.; Zyss, J.; Moerner, W.; Twieg, R.; Bjorklund, G., Opt. Lett., 19, 868 (1994). Schiek, R.; Sundheimer, M . L . ; Kim, D.Y.; Baek, Y.; Stegeman, G.I.; Suche, H.; Sohler, W., "Direct Measurement of Cascaded Nonlinearity in Lithium Niobate Channel Waveguides", Opt Lett., in press Schiek, R., J. Opt. Quant. Electron., 26, 415 (1994). Stegeman, G.I.; Sheik-Bahae, M . ; VanStryland, E.; Assanto, G., Opt. Lett., 18, 13 (1993). Assanto, G . ; Stegeman, G.I.; Sheik-Bahae, M . ; VanStryland, E., Appl. Phys. Lett., 62, 1323 (1993); Assanto, G.; Stegeman, G.I.; Sheik-Bahae, M . ; VanStryland, E . , "Coherent Interactions for AllOptical Signal Processing via Quadratic Nonlinearities", J. Quant. Electron., submitted St. Jean Russell, P., Electron. Lett., 29, 1228 (1993). Werner, M.J.; Drummond, P.D., Opt.Lett., 10, (1993) 2390. Tomer, L.; Menyuk, C.R.; Stegeman, G.I., "Excitation of Soliton-like Waves with Cascaded Nonlinearities", Opt. Lett., in press Hagan, D.J.; Sheik-Bahae, M . ; Wang, Z.; Stegeman, G.I.; VanStryland, E.W., "Phase Controlled Transistor Action by Cascading of SecondOrder Nonlinearities in KTP", Opt. Lett., in press Yamashita, M . ; Torizuka, K.; Uemiya, T., Appl. Phys. Lett., 57, (1990) 1301. Yamashita, M.; Torizuka, K.; Uemiya, T.; Shimada, J., Appl. Phys. Lett., 58, 2727 (1991) see for example, Chikuma, K.; Umegaki, S., J. Opt. Soc. Am. B, 9, 1083 (1992). Torruellas, W.E.; Schiek, R.; Kim, D.Y.; Krijnen, G.; Stegeman, G.I.; Vidakovic, P.; Zyss, J., "Cascading Nonlinearities in an Organic Single Crystal Core Fiber: The Čerenkov Regime", Opt. Comm, in press Heard, D.; Yano, K.; Inoue, Y.; Kijima, T.; Ide, T.; Arai, H . ; Yano, S., Technical digest of CLEO'94 (Opt. Soc. Am., Washington, 1994), paper CTuK13 p90. for example, Ishihara, T.; Brunthaler, G.; Walecki, W.; Hagerot, M . ; Nurmiko, A.V.; Samarth, N.; Luo, H . ; Furdyna, J., Appl. Phys. Lett., 62, 2460 (1992). Kim, D.Y.; Lawrence, B,L.; Torruellas, W.E.; Baker, G.; Meth, J., "Assessment of Single Crystal PTS as an All-optical Switching Material at 1.3 μm", Appl. Phys. Lett., in press Lawrence, B.; Cha, M . ; Kang, J.U.; Torruellas, W.; Stegeman, G.I.; Baker, G.; Meth, J.; Etemad, S., Electron. Lett., 30, 447 (1994). Eckardt, R. C.; Reintjes, J. , IEEE J. Quant. Electron., QE-20, 1178 (1984). Villeneuve, A . ; Yang, C.C.; Stegeman, G.I.; Lin, C.-H.; Lin, H.-H., Appl. Phys. Lett., 62, 2465 (1993). Vidakovic, P.V.; Coquillay, M . ; Salin, F., J. Opt. Soc. Am., B, 4, 998 (1987).


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Kerkoc, P.; Zgonik, M . ; Sutter, K.; Bosshard, Ch.; Gunter, P., J. Opt. Soc. Am. B, 7, (1990) Hoekstra,H.J.W.M.; Krijnen, G.J.M.; Lambeck, P.V., Opt. Comm., 97, 301 (1993). Kim, D . Y . ; Sundheimer, M . ; Otomo, A.; Stegeman, G.I.; Horsthuis, W.G.H.; Mohlmann, G.R., Appl. Phys. Lett., 63, 290 (1993). Bierlein, J . D . ; Cheng, L . K . ; Wang, Y . ; Tam, W., Appl. Phys. Lett., 56, 423 (1990). Otomo, A.; Stegeman, G.I.; Horsthuis, W.H.G.; Mohlmann, G.R., Appl. Phys. Lett., 65, 2389 (1994). Lawrence, B.L., M.S. Dissertation, M.I.T., May 1992.

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Cascading of Second-Order Nonlinearities - ACS Symposium Series

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