Optical limiting in a periodic materials with relaxational nonlinearity

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1 Optical limiting in a periodic materials with relaxational nonlinearity Xue Liu, 1, Joseph W. Haus 1,* and M. S. Shahriar, 1 Electro-Optics Program, University o Dayton, 300 College Park, Dayton, OH , USA Electrical Engineering and Computer Science, Northwestern University, 145 N. Sheridan Road, Evanston, IL 6008, USA *Corresponding author: Joseph.Haus@notes.udayton.edu Astract: We numerically investigate counter-propagating eams in a one-dimensionally, periodic structure with non-instantaneous Kerr nonlinearity or the design o eicient optical limiters. The perormance o the Photonic Band Gap optical limiter with dierent response times is compared with the instantaneous case. Dynamic range and the cuto intensity can e improved over a range o relaxation times. 009 Optical Society o America OCIS codes: Instailities and chaos; Nonlinear optics, transverse eects in; Sel-action eects. Reerences and links 1. T. Xia, D. J. Hagan, A. Dogariu, A. A. Said, and E. W. Van Stryland, "Optimization o optical limiting devices ased on excited-state asorption," Appl. Opt. 36, (1997).. J. S. Shirk, R. G. S. Pong, F. J. Bartoli, and A. W. Snow, Optical limiter using a lead phthalocyanine, Appl. Phys. Lett. 63, (1993). 3. J. Shirk, R. Pong, S. Flom, F. Bartoli, M. Boyle, and A. Snow, Lead phthalocyanine reverse saturale asorption optical limiters, Pure Appl. Opt. 5, (1996). 4. P. Tran, All-optical switching with a nonlinear chiral photonic andgap structure, J. Opt. Soc. Am. B 16, (1999). 5. J. P. Dowling, M. Scalora, M. J. Bloemer and C. M. Bowden, The photonic and edge laser: A new approach to gain enhancement, J. Appl. Phys. 75, (1994). 6. M. Scalora, J. P. Dowling, M. J. Bloemer and C. M. Bowden, The photonic and edge optical diode, J. Appl. Phys. 76, (1994). 7. M. Scalora, J.P. Dowling, C.M. Bowden and M. J. Bloemer, Optical limiting and switching o ultrashort pulses in nonlinear photonic and gap materials, Phys. Rev. Lett. 73, (1994). 8. M. Scalora, N. Mattiucci, G. D'Aguanno, M. C. Larciprete, and M. J. Bloemer, Nonlinear pulse propagation in one-dimensional metal-dielectric multilayer stacks: Ultrawide andwidth optical limiting, Phys. Rev. E 73, (006). 9. B. Y. Soon and J. W. Haus, One-dimensional photonic crystal optical limiter, Opt. Express 11, (003). 10. B. J. Eggleton, C. M. desterke, R. E. Slusher and J. E. Sipe, Distriuted eedack pulse generator ased on nonlinear ier grating, Electron. Lett. 3, (1996). 11. J. W. Haus, B. Y. Soon, M. Scalora, C. Siilia, I. Mel nikov, Coupled-mode equations or Kerr media with periodically modulated linear and nonlinear coeicients, J. Opt. Soc. Am. B 19, 8-91 (00). 1. J. W. Haus, B. Y. Soon, M. Scalora, M. Bloemer, C. Bowden, C. Siilia, and A. Zheltikov, Spatiotemporal instailities or counter-propagating waves in periodic media, Opt. Express 10, (00). 13. M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, "Sel-trapping o partially spatially incoherent light," Phys. Rev. Lett. 77, (1996). 14. X. Liu, J. W. Haus and S.M. Shahriar, Modulation instaility or a relaxational Kerr medium, Opt. Commun. 81, (008). 15. M.-F. Shih, C.-C. Jeng, F.-W. Sheu, and C.-Y. Lin, Spatiotemporal Optical Modulation Instaility o Coherent Light in Noninstantaneous Nonlinear Media, Phys. Rev. Lett. 88, (00). (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 696

2 1. Introduction One o the key challenges in the development o optical limiters is the search or appropriate materials that rapidly respond to the applied laser intensity, have good nonlinear asorption characteristics over a road range o wavelengths and have a high damage threshold. To protect the eyes the transmitted luence through the device should e clamped under 500 nj/cm over a range o visile wavelengths. Dangerous IR radiation could e eliminated y a ixed high pass ilter. Organic materials have shown great promise or optical limiting applications. They possess reverse saturale asorer action where molecules are designed to possess larger excited-state asorption than ground-state asorption. The principal eature is their enhanced nonlinear asorption at higher intensities, ut they also have nonlinear reraction eects [1]. The spectral range o materials can e engineered to cover ands. Lead phthalocyanines have asorption ands that can cover the visile and near- IR regimes [,3]. One dimensional Photonic Band Gap (PBG) structures have een proposed as a arication strategy to improve the optical limiting perormance o materials. The unusual spatial and temporal dispersion properties o PBGs make them excellent candidates or manipulating the dispersion characteristics o the composite material. This has given rise to a wide range o proposed photonic devices [4-8], among which nonlinear PBG optical limiters have emerged as showing potential due to their compact structure and straightorward arication process. Earlier work on PBG optical limiters was ased on exploiting transverse and longitudinal modulation instailities using Kerr media [9]. Compared with traditional homogeneous nonlinear optical limiters, this design has shown promising perormance or optical limiting, i.e., high transmission or normal light and low or intense eams, eective in protecting optical elements and sensors against damage y exposure to malicious or unexpected high-intensity light. The PBG can provide access to the nonlinearity o materials, such as metals, that are not normally considered as candidates or optical limiting materials. Scalora et al. recently developed a chirped PBG device with copper layers to induce roadand optical limiting [8]. The chirped dielectric layer thicknesses are designed to improve the linear transmittance and concentrate large ields inside the copper to access the large nonlinearity. An added advantage is that the stack ilters the rest o the spectrum outside the visile regime. In this paper, the counter-propagating waves in one dimensional PBG structure with relaxational nonlinearity are numerically investigated. The role o the spatial and temporal modulation instailities on reshaping the orward and ackward ields as a unction o the relaxation time is elucidated. The inite response time o the Kerr eect is taken into consideration in the model. Based on the nonlinear, deocusing eect o the modulation instailities, an optical limiter is exploited y placing an aperture in the ar ield ehind the PBG structure. The role o nonlinear response time on the perormance o optical limiter is examined.. Mode-coupled equations and computational approach The dynamical equations or counter-propagating waves in a non-instantaneous nonlinear PBG structure can e derived y applying the multiple scales method to the classical nonlinear wave equation [10]. The dynamics o the medium is descried y a simple relaxational model with the medium s response time. (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 697

3 1 E E i = + E + iδ E + iκ E + iη N E v t z F 1 E E i = + E + iδ E + iκ E + iη N E v t z F N t N t 1 ( N E E ) = + + τ 1 ( N E E ) = + + τ, (1-a), (1-), (1-c), (1-d) where the ields E = E ( x, z, t) and E = E ( x, z, t) descrie the electric ield envelope o the orward and ackward waves in space and time. We denote y x the transverse coordinate, since we will treat only a single transverse dimension in our PBG structure; v denotes group velocity, F is the Fresnel parameter, δ = k k0 stands or the detuning o the initial pulse π wave numer rom the center o the stop and, and k0 = whereλ is the lattice constant. κ denotes the coupling coeicient o the grating, andη is Λ the nonlinear Kerr coeicient, which is assumed homogeneous throughout the medium. In our numerical model, the index change o the two materials is assumed to e weakly periodic. N and N represent the excitation densities o the nonlinear medium or the orward and ackward waves, respectively. The inite response time o the Kerr eect o the PBG structure is denoted yτ. From Eqs. (1c) and (1d) it is clear that the dynamics o N is not only related to the local ield intensities o oth waves, ut also the response time o the medium. We consider the case that the signal is only launched into one side o the PBG structure. Thereore, the oundary conditions we applied to the model are E ( x,0, t) = E ( x, t) (-a) E (,, ) 0 x L t = (-) In our previous research [11], a novel split-step method has een introduced to solve Eq. (1). By separating the propagation into x matrices, we express Eq. (1) in the symolic matrix orm where the state vector is and the operators are du = ( L ɶ + N ɶ ) U dt, (3) U = ( E, E ) T, (4) i + + iδ 0 z F 0 iκ Lɶ = + = Vɶ + Kɶ, (5-a) i iκ iδ z F i N 0 Nɶ η =. (5-) 0 iη N Matrices L and N, representing the linear and nonlinear parts o the partial-dierential (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 698

4 equation, are treated dierently in separate steps. For the linear step, ecause the operator L can e decomposed into a diagonal and an o-diagonal matrix, we can urther simpliy the calculations i the step size is small enough. i.e. dt Vdt Kdt U L ( t+ ) = exp( )exp( ) U ( t) (6) where the suscript L stands or the linear propagation contriution. Equation (6) can e readily solved in the Fourier space. In our calculation, 048 and 104 points are used or Fast Fourier Transorm in the longitudinal and transverse directions, respectively. For the nonlinear step, we irst ind the analytical solutions or Eqs. (1c) and (1d) 1 N x z t e E x z t E x z t dt t ( t t ') τ (,, ) = ( (,, ') (,, ') ) ' τ + (7-a) 1 N ( x, z, t) e ( E ( x, z, t ') E ( x, z, t ') ) dt ' (7-) t ( t t ') τ = + τ By assuming that N and N stay constant or a small step size, the solution to the nonlinear step can e written as dt U N ( t+ ) = exp( Ndt) U ( t) (8). where the suscript N reers to the nonlinear contriution to the propagation. Comining the two separated steps together, the second-order solution or U at the incremented time t+ dt is Vdt Kdt Vdt Kdt U ( t+ dt) = exp( )exp( )exp( Ndt)exp( )exp( ) U ( t) (9). We monitor the accuracy o the numerical method y examining the law o energy conversation at each time step. 3. Results 3.1. Propagation The orward propagating pulse is placed outside the PBG structure at certain distance z0 so that the initial ield in the PBG structure vanishes. The intensity distriution o the pulse is in a Gaussian shape transversely and longitudinally, i.e. E x z = A z z σ x σ (10) (,,0) exp( ( 0 ) / z )exp( / x ) The initial condition or the ackward propagating pulse is E (,,0) 0 x z = (11) To simpliy the calculations, the parameters in Eq. (1) are scaled as ollows: v= 1, κ = 1, η = 1. We tune the incident laser pulse central angular requency away rom the center o the PBG stop and y changing the value oδ. The selection o δ is closely related to the spatio-temporal modulation instaility o the nonlinear PBG structure, which is essential in inluencing the propagation o the pulse. When the irst transmission peak o the PBG or positive angular requency, also called the and-edge, overlaps with the center o the pulse spectrum the reak-up o the pulse is maximized y the spatio-temporal modulation instaility (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 699

5 [1]. The transmission curve or the PBG structure used in our simulation with a length L=6.1 is plotted in Fig. 1. The oset rom the center o the pulse spectrum to the and-edge o the PBG isδ = 1.1, Fig. 1. Transmission curve o the PBG structure and the spectrum o the input pulse. Previous papers show that the threshold o initial amplitude or modulation instaility to occur is around A=0.3. In our simulations, we set the amplitude at A=0.75, to ensure a high modulation instaility gain. The dierent nonlinear response time chosen or the PBG structure areτ 1 = 0.1, τ = 1 andτ 3 = 4. The case or an instantaneous PBG structure is also simulated or comparison. Fig. (a). A snapshot o the pulse evolution through an instantaneous nonlinear medium placed in a PBG. Top rame is the initial pulse, which is launched outside the medium. The central rame and ottom rame are the orward propagating and ackward propagating pulse, respectively. The initial amplitude o the wave is A=0.75. (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 700

6 Fig. (). A snapshot o the orward- and ackward waves with a non-instantaneous nonlinear medium emedded in the PBG. The relaxation time is τ 1 = 0.1 and the amplitude is the same as in Fig. (a). Fig. (c). A snapshot o the orward- and ackward waves with a non-instantaneous nonlinear medium emedded in the PBG. The relaxation time is τ = 1 and the amplitude is the same as in Fig. (a). Fig. (d). A snapshot o the orward- and ackward waves with a non-instantaneous nonlinear medium emedded in the PBG. The relaxation time is τ 3 = 4 and the amplitude is the same as in Fig. (a). (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 701

7 Snapshot views o the pulse orward-ackward pulse intensities or dierent relaxation times are shown in Figs. (a)-(d).the position o the PBG medium is denoted y a pair o green ars placed etween the range z=(40,50). The asolute amplitude o the ield at each pixel is represented y a color with the amplitude shown in the vertical scale on the right side. The vertical axis displays the transverse coordinate and the horizontal axis is the propagation direction. The top igure is the input Gaussian ield; its pulse width is shorter than the medium so that it samples a large portion o the dispersive spectrum in the medium. The middle igure is the continuation o the orward-propagating pulse as it passes into and through the PBG sample. The ottom igure shows the propagation o the ackward propagating pulse in the 7 medium. In our computations F is chosen to e extremely large ( F = 10 ) so that the transverse coupling is weak and does not noticealy aect the pulse tuning at a selected requency. Comparison among Figs. (a) through (d) leads to several oservations aout the eect o the non-instantaneous nonlinear PBG structure on pulse propagation. In all our cases, the orward propagating pulse with initial Gaussian proile undergoes a strong distortion inside the medium. For the instantaneous case, the modulation instaility leads to a rapid anomalous spread o the pulse in oth transverse and longitudinal directions, which demonstrates that phase changes o the eam in the transverse direction are suicient to modulate the amplitude y the end o the medium. For a short response time ( τ = 0.1 ), not only does the proile o the orward propagating pulse undergo a major distortion into an arch-like shape, ut also the energy redistriution in oth transverse and longitudinal directions ecomes discontinuous. The intensity peaks appear at discrete spots, which can e readily seen in the 3-D plot o Fig. 3. A close look at the orward propagating pulse shows that more high requency components exist than in the case o an instantaneous nonlinear PBG structure. This phenomenon can e explained y interpreting the Modulation Instaility as a degenerated orm o Four Wave Mixing in the Kerr medium, the perturation eing the proe and the CW pump eam [13]. The energy o the two photons rom the pump produces two new conjugate requency photons. In the instantaneous case, to generate MI, phase-match condition is required rom the pump and the proe, thus the generated requency is strictly limited. But when the medium response time is taken into account, a Raman-like nonlinearity is developed or which sel-phase-match is achieved, leading to a gain region outside the original sideand. Previous studies show that the inite response time o the Kerr eect alters the MI y lowering the gain and removing the gain cuto requency [14,15]. For a short response time, the overall gain or small perturations is only slightly decreased, ut the whole gain region is widely extended, leading to a stronger reak-up o the pulse. In the su rame or the ackward propagating wave, we also oserve an arch-like ut weaker distortion than the orward propagating wave due to the limited passage in the nonlinear PBG structure. A ri structure is displayed which is also a characteristic o the longitudinal modulation instaility. As seen rom Fig. (c), even or τ = 1, the reshaping o oth the orward and ackward propagating waves into an arch pattern can still e oserved. The distortion and ri structures are less ovious due to the weaker modulation instaility. (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 70

8 Fig D plot o the orward propagating pulse. As the response time largely increases as in Fig. (d) ( τ = 4 ), the pulse shape is only moderately distorted. The reason is that the overall gain that resulted rom the modulation instaility is greatly lowered y the long response time. The eect o the nonlinearity is aated. Also or the ackward propagating pulse, the ri structure o the ackward wave almost disappears, which is also a sign o the weak modulation instaility. 3. Optical limiting The aility o the non-instantaneous nonlinear PBG structure to reak up a pulse makes itsel a promising candidate or optical limiting. Since the energy o the incident pulse is spread longitudinally and transversely y the modulation instaility, we place an aperture in the Fraunhoer regime ater the PBG structure to urther enhance the optical limiting as shown in Fig. 4. The size o the aperture is set to e the FWHM o the initial pulse, such that when the intensity is low, most part o the optical pulse is passed, and when the intensity gets undesiraly high, a large portion o the spread optical eam is clipped. A proe is positioned ehind the aperture to detect the transmitted energy. (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 703

9 Fig. 4. Aperture placed in the Fraunhoer regime to lock out energy low ( A= 0.75, τ = 1 ). 7 The Fresnel numer is F = 10. In order to evaluate the perormance o the optical limiter,certain criteria need to e set up or comparison. Two igures o merit are proposed to categorize the perormance o optical limiters. First, we deine the transmission dynamic range (TDR) to e TDR T max = ψ = (1) T min Note that i T min is zero, TDR is ininite or any nonzero T max. Generally, TDR is more sensitive to the change in T max. We deine a second igure o merit related to the transmission cuto (TCO) response and; it is deined y the intensity dierence etween the normalized 0% and 80% transmission points, i.e. T -T TCO T +T 80% 0% = (13) 80% 0% TCO shows how ast the transmission drops at the cut-o point. The smaller TCO is, the closer it approaches the ideal optical limiter. Generally, an ideal optical limiter perormance is approached as TDR and TCO=0. (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 704

10 Fig. 5. Transmission curves or PBG structures with dierent response time. Tale 1, The igures o merit or instantaneous and non-instantaneous nonlinear media emedded in a PBG optical limiter. Relaxation time TDR TCO τ = 0 (Instantaneous) τ = τ = τ = N/A Figure 5 summarizes the transmission characteristics or the non-instantaneous medium distriuted in a PBG. We note that or a small response time ( τ = 0.1 ) the wave has a higher T max, lower T min and decays aster than the instantaneous PBG. In terms o igures o merit, the optical limiting perormance is improved in all aspects: higher TDR and lower TCO as shown in Tale 1. Even or relatively long response timeτ = 1, the optical limiting properties are comparale to the instantaneous case. But or a PBG structure with a long response time ( τ = 4 ), due to the weak modulation instaility gain, the reak-up o the pulse in the ar ield is dramatically suppressed, which reduces the optical limiting perormance. 4. Conclusion In this paper, we presented a numerical model or counter-propagating waves in the one dimensional non-instantaneous nonlinear PBG structure. The case that the pulse is incident rom one side o the PBG structure is especially stressed. The and-edge o the photonic crystal is manipulated to overlap with the center o the spectrum o the incident pulse to maximize the modulation instaility. Our study shows that a small inite response time o the Kerr eect ( τ = 0.1 ) o the PBG can strengthen the spatio-temporal modulation instaility o the medium, leading to a rapid spread o the pulse in the transverse direction and stronger temporal reak-up o the pulse aove threshold intensity. For relaxation times o the order unity the perormance o our optical limiter is not degraded over the instantaneous case. High requency components are oserved due to the extended gain region o the modulation instaility, which leads to additional perormance improvements or optical limiting applications. However, or long relaxation times, the nonlinear eect is weakened y lower (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 705

11 gain o the modulation instaility. In Section 3 we applied the spatio-temporal modulation instaility in non-instantaneous nonlinear PBG to design an optical limiter. An aperture is placed in ar-ield to ilter high-intensity light eore entering the detector. The perormance o our design is evaluated in terms o the dynamic range (TDO) and the transmission cuto (TCO). It is ound that oth igures o merit are improved in the case where the response time is short ( τ = 0.1 ) compared to the pulse duration, which shows promising optical limiting characteristics. It is also ound that material with long Kerr relaxational response times is not suitale or our design. The optical limiter can e urther improved y optimizing the parameters in the system. The other actors that we can modiy to urther improve the perormance o the optical limiter include the aperture size, the length o the PBG structure, the complex third-order nonlinearity o the medium and the diraction parameter F, eam ocusing in the medium, and the eam shape. Based on earlier pulications, longer pulses should experience etter optical limiting. (C) 009 OSA 16 Feruary 009 / Vol. 17, No. 4 / OPTICS EXPRESS 706