Nonlinear Raman-Nath second harmonic generation with structured fundamental wave

Transcription

1 Vol. 4, No. 4 Jul 6 OPTICS EXPRESS 5666 Nonlinear Raan-Nath second haronic generation with structured fundaental wave HAIGANG LIU,, JUN LI,3 XIAOHUI ZHAO,, YUANLIN ZHENG,,,4 AND XIANFENG CHEN,,5 State Key Laboratory of Advanced Optical Counication Systes and Networks, Departent of Physics and Astronoy, Shanghai Jiao Tong University, Shanghai 4, China Key Laboratory for Laser plasa (Ministry of Education), Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 4, China 3 Science and Technology College, Jiangxi Noral University, Jiangxi 337, China 4 ylzheng@sjtu.edu.cn 5 xfchen@sjtu.edu.cn Abstract: We proposed and experientally deonstrated that nonlinear Raan-Nath second haronic can be achieved in real tie when a fundaental wave with the phase periodically odulated, tered as structured fundaental wave, incident in a hoogeneous nonlinear ediu. The diffraction of second haronic originates fro the structured fundaental wave, rather than the grating of a nonlinear photonic crystal. Nonlinear second haronic generation, in fors of both one- and two-diensional, was investigated in our experient. This ethod circuvents the liitation of nonlinear photonic crystals in soe extend and has potential applications in nonlinear freuency conversion, optical signal processing and bea shaping, etc. 6 Optical Society of Aerica OCIS codes: (9.9) Nonlinear optics; (9.76) Coputer holography; (5.97) Diffractive optics; (9.6) Haronic generation and ixing. References and links A. Arie and N. Voloch, Periodic, uasi-periodic, and rando uadratic nonlinear photonic crystals, Laser Photonics Rev. 4(3), (). J. A. Arstrong, N. Bloebergen, J. Ducuing, and P. S. Pershan, Interactions between light waves in a nonlinear dielectric, Phys. Rev. 7(6), (96). S. N. Zhu, Y. Y. Zhu, and N. B. Ming, Quasi-phase-atched third-haronic generation in a uasi-periodic optical superlattice, Science 78(5339), (997). K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenan, Multiple nonlinear optical interactions with arbitrary wave vector differences, Phys. Rev. Lett. 88, 393 (). T. Ellenbogen, N. V. Bloch, A. G. Padowicz, and A. Arie, Nonlinear generation and anipulation of Airy beas, Nat. Photonics 3(7), (9). Y. Q. Qin, C. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in doain engineered structures, Phys. Rev. Lett. (6), 639 (8). Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Nonlinear Cerenkov radiation in nonlinear photonic crystal waveguides, Phys. Rev. Lett. (6), 6394 (8). A. R. Tunyagi, M. Ulex, and K. Betzler, Noncollinear optical freuency doubling in strontiu bariu niobate, Phys. Rev. Lett. 9(4), 439 (3). S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, Multiorder nonlinear diffraction in freuency doubling processes, Opt. Lett. 34(6), (9). K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Ibrock, C. Denz, and W. Krolikowski, Enhanced Čerenkov second-haronic eission in nonlinear photonic structures, Opt. Lett. 37(), (). H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, Nonlinear Cherenkov radiation in an anoalous dispersive ediu, Phys. Rev. Lett. 8(), 39 (). H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, Enhanced nonlinear Cherenkov radiation on the crystal boundary, Opt. Lett. 38(), (3). V. Roppo, K. Kalinowski, Y. Sheng, W. Krolikowski, C. Cojocaru, and J. Trull, Unified approach to Čerenkov second haronic generation, Opt. Express (), (3). S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, Generation of secondharonic conical waves via nonlinear bragg diffraction, Phys. Rev. Lett. (), 39 (8). A. Shapira and A. Arie, Phase-atched nonlinear diffraction, Opt. Lett. 36(), (). #6774 Journal 6 Received 7 May 6; revised 4 Jun 6; accepted 4 Jun 6; published 3 Jun 6

2 Vol. 4, No. 4 Jul 6 OPTICS EXPRESS S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, Nonlinear diffraction fro a virtual bea, Phys. Rev. Lett. 4(8), 839 (). 7. S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, Generation of secondharonic bessel beas by transverse phase-atching in annular periodically poled structures, Jpn. J. Appl. Phys. 47(8), 6777 (8). 8. W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, Multiple Čerenkov second-haronic waves in a two-diensional nonlinear photonic structure, Opt. Lett. 35(), (). 9. P. Molina, M. O. Rairez, B. J. Garcia, and L. E. Bausa, Directional dependence of the second haronic response in two-diensional nonlinear photonic crystals, Appl. Phys. Lett. 96(6), 6 ().. Y. Sheng, D. L. Ma, and A. Krolikowski, Randoized nonlinear photonic crystal for broadband optical freuency conversion, J. Phys. At. Mol. Opt. Phys. 46(), 54 (3).. Y. Sheng, W. J. Wang, R. Shiloh, V. Roppo, Y. F. Kong, A. Arie, and W. Krolikowski, Čerenkov thirdharonic generation in χ, Appl. Phys. Lett. 98(4), 44 ().. X. H. Hong, B. Yang, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, Nonlinear volue holography for wave-front engineering, Phys. Rev. Lett. 3(6), 639 (4). 3. N. V. Bloch, K. Sheer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, Twisting light by nonlinear photonic crystals, Phys. Rev. Lett. 8(3), 339 (). 4. A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, Two-diensional nonlinear bea shaping, Opt. Lett. 37(), (). 5. T. Wulle and S. Heringhaus, Nonlinear optics of Bessel beas, Phys. Rev. Lett. 7(), 4 44 (993). 6. A. Libster-Hershko, S. Trajtenberg-Mills, and A. Arie, Dynaic control of light beas in second haronic generation, Opt. Lett. 4(9), (5).. Introduction Nonlinear photonic crystals (NPCs) [,], in which the second-order susceptibility χ is spatially odulated, open a door in the field of nonlinear wave ixing, like second-haronic generation (SHG), su-freuency generation, difference-freuency generation and so on. The phase atching of collinear and noncollinear wave ixing can be achieved by the copensation of an reciprocal vector of NPCs without wavelength liitation. Both onediensional (D) and two-diensional (D) structures have widely been fabricated to realize optical paraetric processes in uasi-phase atching (QPM) anners [3 6]. More recently, transverse schees (perpendicular to traditional QPM direction) of noncollinear wave ixing processes have attracted uch attention. Various nonlinear processes, such as nonlinear Cherenkov radiation [7 3], nonlinear Raan-Nath and nonlinear Bragg diffraction [9,4 6] have been intensively studied in NPCs. And various patterns of D nonlinear photonic structures have also been deonstrated, including suare, hexagonal lattices, annular periodical, or even rando structures [7 ]. Especially, with holography concept introduced to nonlinear optics, arbitrary two-diensional nonlinear bea shaping, such as vortex and airy beas, could be realized [ 4]. However, all these ethods, which anipulate the structure of the NPCs, have several drawbacks, including coplex fabrication and unchangeable second haronic (SH) pattern. For desired spatial shaping of the SH bea, an alternative approach has been used to control the incident FW [5,6]. The odulation of χ structures periodically flips the phase of second-order nonlinear polarization excited by the fundaental wave (FW). Considering a degenerate SHG process, the nonlinear polarization P ω excited by the FW in a nonlinear ediu can be expressed as P r t = r E r t () ω(,) εχ () ω(,), where ε is the vacuu perittivity, χ is the second-order susceptibility and E ω is the electric field of FW. In NPCs, the constructive interference of the radiation of polarization P ω is achieved by periodically odulating the sign (or value) of the coefficient χ in space. While, the physics behind E. () hints that periodically odulating the phase (or aplitude) of FW can also achieve the phenoenon in NPCs. In this Letter, we study the process of SHG using structured FW as the input in a hoogenous χ ediu. The wavefront of the FW is periodically odulated to provide a

3 Vol. 4, No. 4 Jul 6 OPTICS EXPRESS 5668 transverse structure in the process of SHG. In D geoetry, due to the structure of the FW, the SH exhibits as ultiple discrete spatial spots. Extending this concept to D situation, SH rings and suares can be attained. Under the condition of nondiffraction in a sufficient short ediu and G k., the generated SH appears in nonlinear Raan-Nath diffraction direction. The SHG process is assisted by the structure of the FW itself, rather than any nonlinear grating or structure in the nonlinear ediu.. The coupled wave euation The incident FW is defined in the for of E = A exp[ i ( k r ωt )], where A and k are the aplitude and wavevector of the FW, respectively. Suppose that the FW propagates along the y-axis of the crystal. When the wavefront is periodically odulated, the FW can be ore intuitively written as an expansion of Fourier series: E = A C exp( ig r)exp[ i( k y r ωt)] where C and G =π Λ are the Fourier coefficients and reciprocal vectors of the FW, Λ is the period of the FW odulation. The relation between G and k y are G + ky = k. In order to obtain SH in nonlinear Raan-Nath diffraction directions, soe restrictions need to be done. Under the condition of G k. (for larger values of, C are sall enough to be negligible), the diffraction of FW is negligible, which holds in a sufficient short ediu (the critical length L kλ G Λ). The crystal is a sufficient short ediu when its thickness L c is shorter than twice the critical length L. For siplicity, under this assuption, the odulation function can be written as: E = A exp[ i( k r ωt)] C exp( ig r) (3) We define the expression of SH wave E = Aexp[ i( kt r+ ky y ωt)]. Under the assuing that the FW is undepleted, the evolution of the SH wave is directly given by [] da = κ A b exp[ i( ky k) y]exp[ i( kt G) r], (4) dy + n= where κ is the nonlinear coupling coefficient. b = C n, Cn is the Fourier coefficient of the nonlinear polarization and the suation runs fro to +. In E. (4), the first exponential ter stands for the longitudinal phase isatch. The second exponential ter is the nonlinear Raan-Nath diffraction phase isatch, in which the structure function originates fro the FW. Thus, we can get SH in nonlinear Raan-Nath diffraction directions under the assuption entioned above. If the phase of incident FW period is sharply odulated fro to φ and the duty cycle D is.5. The Fourier coefficients can be derived as iφ C = + e i[cos( π ) ]. =, ±, ±, (5) ( iφ C = e ) π Note that the Fourier coefficients C are non-zero only for odd values of (except C ). For a certain order of, the SH intensity is proportional to b and can be expressed as χ I b I, where I and I are the FW intensity and the SH intensity, respectively.

4 Vol. 4, No. 4 Jul 6 OPTICS EXPRESS Experient results and discussion In our proof-of-principle experient, the FW at a wavelength of 64 n (Nd:YAG nanosecond laser) was phase odulated by a spatial light odulator (SLM). The SLM had a resolution of 5 5 pixels, each with a rectangular area of μ. The light was then iaged by a 4-f syste (agnification of.5) to iprint the odulated wavefront pattern to the onset of a χ crystal. The bea waist was reduced to. After the crystal, a shortpass filter was used to filter out the FW. Finally, the generated SH bea was projected on a screen in the far-field and recorded by a caera. For siplicity and without loss of 3 generality, a 5ol% MgO:LiNbO 3 bulk crystal (.3 in x y z diensions) was used in our experient. The FW was kept as o-polarized and propagated along the y-axis of the crystal at roo teperature. In this situation, should saller than 9. In the following experient Λ μ was used ( L.4), hence the thickness of the ediu in our experient ( L =.3< Lin ) could be treated as a sufficient short ediu. In order to illustrate the role of such structure of the FW in the process of SHG, we did a coparison experient using a unifor and a wavefront odulated FW input, as shown in Fig.. For siplicity, the hologra loaded on the SLM is used to represent the wavefront profile. Firstly, due to the near birefringent phase atching (BPM) condition, there is only one collinear phase-isatched SH spot observed in the case of FW without any phase structure (shown in Fig. (a)-(c)). The type of SH interaction is oo-e and the phase-atching geoetry is shown in Fig. (b). Secondly, the phase of the FW is periodically sharply odulated fro to π in D structure. The hologra loaded on SLM is illustrated in Fig. (d), which exhibits : duty cycle (D). The phase-atching geoetry is shown in Fig. (e). The output is a set of syetrically distributed SH spots, in which, ±, ± and ± 3 orders were experientally observed, shown in Fig. (f). Fro the Fourier coefficient of E. (5) and + n= the relation of b = C n, Cn, in this case, the Fourier coefficients b are non-zero for each value of and this is in consistence with the experient results. Figure (g) show the experiental iage of the FW fro the output end of the nonlinear crystal, as can be seen the diffraction of FW was not noticeable. Fig.. (a)(d) Hologras loaded on SLM representing the phase structure of the FW. (b)(e) Phase-atching geoetries of the nonlinear Raan-Nath diffraction SHG process, corresponding to (a) and (d) under the assuption of nondiffraction of FW, respectively. (c)(f) Observed SH diffraction patterns of (a) and (d), respectively. (g) The experiental iage of the FW fro the output end of the nonlinear crystal. For precise deterination of the angle of the SH eission, we used various odulation periods ranging fro 5 to 5 μ. The angle of the th diffraction order nonlinear Raan- Nath diffraction SH spot is defined by the transverse phase-atching condition:

5 Vol. 4, No. 4 Jul 6 OPTICS EXPRESS 567 sin α =,,,, G k = ± ±. According to the Snell s law n sinα = sin β, the e external radiation angles are β = arcsin( λ Λ ), =, ±, ±,, where n e and λ are the refractive index and wavelength of the SH. The external angles of the ±, ± and ± 3 order nonlinear SH spots are shown in Fig. (a). The SH diffraction angles are increasing while the odulation period of the FW decreases. The theoretical prediction and experiental results are also in well agreeent with each other. Apart fro the angular inforation, the influence of paraeters of FW such as φ and the duty cycle D on the SH pattern are also investigated. In case of D =.5, fro the Fourier coefficient of E. (5), the value of φ deterines the existence of C and the relative values of C (the value of C is if is even). According to the relation of b + n= = C C, n, n the value of φ only influence the relative value of b. However, it is interesting when the paraeter φ = π, the Fourier coefficients b becoes non-zero only for even s. ' The corresponding experiental result is described in Fig. (b), in which only and ± orders are present. When the duty cycle D.5 ( C = D, C = sin( πd) π ), each order of b becoes non-zero again. The issing odd orders of SH spots reappear as observed in the experient. It is worth noticing that the excited nonlinear polarization wave experiences a π phase in two adjacent opposite doains, which is caused by the sharp χ odulation fro + to. And it only relates to once Fourier transfor of the structure of the crystal. While, in our case, the nonlinear polarization wave involves the product of two Fourier expansion series of the FW [see E. ()]. Fig.. (a) The external angles of +, +, and + 3 order SH as a function of the FW odulation period. Theoretical prediction (solid curves) and easured ones (signs) are in well agreeent with each other. (b) Observed SH pattern with D =.5 and φ = π in D structure. In addition to D geoetry, we also extended such structured FW to D situations. The phase of FW period was sharply odulated fro to π in D patterns and the period of FW corresponding to 4 μ with the duty cycle D =.5. For the annular structure of FW (Fig. 3(a)), the phase-atching geoetry with radial reciprocal vectors is shown in Fig. 3(b). The SH fors ultiple rings structure in Fig. 3(c). The -, -, 3- and 4-order SH rings were observed in experient. The SH of different order rings are circularly syetric with unifor intensity distribution and lower orders of the SH rings are brighter due to larger Fourier coefficients b. Moreover, a suare patterned FW phase was applied (Fig. 3(d)) with the phase-atching geoetry shown in Fig. 3(e). The reciprocal vectors of the FW decreases with increase of the FW period around the aziuthal direction (perpendicular to the bea propagation direction) of the FW to for a suare structure. Different orders of SH for ultiple suares structure (Fig. 3(f)). However, unlike annular structure of the FW, the SH intensity distribution is not unifor with the corner direction of SH suare pattern exhibiting higher intensity. This is caused by the syetric property of this structure, in

6 Vol. 4, No. 4 Jul 6 OPTICS EXPRESS 567 which ost of the FW only feel the structure along horizontal and vertical directions. When the phase of FW period was sharply odulated fro to π, the odd orders of the SH rings, dots and suares were not presented. This is siilar to those in D cases. The Fourier coefficients b = ake odd orders of the SH rings, dots and suares disappear. Fig. 3. (a)(d) Hologras loaded on SLM representing the phase structure of the FW. (b)(e) The phase-atching geoetries of the nonlinear Raan-Nath diffraction SHG process with different structures of the FW, corresponding to (a) and (d) under the assuption of nondiffraction of FW, respectively. (c)(f) SH pattern observed in experient, corresponding to (a) and (d), respectively. The phase odulation of the FW itself for nonlinear wave ixing features any advantages. Actually, copared to χ odulation, it is ore flexible to odulate the FW itself [6]. Firstly, the proposed ethod does not reuire any special fabrication. Secondly, the FW can be controlled by an SLM in real tie, whereas SHG in NPCs is fundaentally restricted by their predesigned structures. Moreover, the odulated FW can propagate along any direction of bulk nonlinear ediu. This is iportant for efficient SHG by utilizing larger coponents of the χ tensor. Unlike the work in [6], which dealt with on-axis shaping, the present paper deals with off-axis nonlinear Raan-Nath diffraction. The phase of FW can be continually changed in our ethod while the values of odulated χ are only + and in NPCs. This ethod will largely enrich the SHG process, but not liited to. The concept can be adopted in other schees as well, for exaple the investigation of nonlinear Cerenkov radiation in bulk edia. Other potential applications ay lie in nonlinear freuency conversion, optical signal processing, bea shaping and so on. 4. Conclusion In suary, under the condition of nondiffraction of FW, nonlinear Raan-Nath SH can be achieved when a structured FW incident in a hoogeneous nonlinear ediu. The change of pattern can be done in real tie. In the D case, the SH exhibits as ultiple discrete spatial spots. And in D cases, SH rings and suares were also observed in experient. The schee siplifies the procedure where NPCs are needed, and provides uch ore flexibility in generating coplex SH patterns. The concept can be adopted in other schees and has potential applications in nonlinear freuency conversion, optical signal processing and bea shaping, etc. Acknowledgent This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 6553, 6359 and 464, the Foundation for Developent of Science and Technology of Shanghai under Grant No. 3JC483.