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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 6976–6983
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Manipulation of quadratic cascading processes in a locally quasi-periodic χ(2) medium

Wenjie Wang, Yan Sheng, Shaoding Liu, Xiaoying Niu, and Wieslaw Krolikowski  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 6976-6983 (2014)
http://dx.doi.org/10.1364/OE.22.006976


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Abstract

We theoretically and numerically investigate the quadratic cascading effect of third-harmonic (TH) generation in a locally quasi-periodic nonlinear photonic structure. We study the effect of structure parameters on the acceptance bandwidth and conversion efficiency of the cascading process. We demonstrate that the conversion efficiency of the cascading process can be enhanced by using a longer locally quasi-periodic nonlinear photonic crystal, without adversely affecting the acceptance bandwidth of the emitted radiation.

© 2014 Optical Society of America

1. Introduction

Quadratic cascading effect (χ(2) : χ(2)) in nonlinear medium [1

1. J.B. Khurgin, A. Obeidat, S. J. Lee, and Y. J. Ding, “Cascaded optical nonlinearities: microscopic understanding as a collective effect,” J. Opt. Soc. Am. 14, 1977–1983 (1997). [CrossRef]

] as one of the possible frequency conversion techniques has been widely used in a variety of applications, including new laser sources, pulse shaping, all-optical switching, Raman spectroscopy and so on [2

2. S. Zhu, Y. Zhu, and B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997). [CrossRef]

10

10. G. I. Petrov, M. Zhi, and V. V. Yakovlev, “Coherent anti-Stokes Raman spectroscopy utilizing phase mismatched cascaded quadratic optical interactions in nonlinear crystals,” Opt. Express 21, 31960–31965 (2013). [CrossRef]

]. Conversion efficiency and acceptance bandwidth have been the two major factors when designing or improving the performance of these devices. To realize an efficient quadratic cascading process, one has to employ simultaneous phase-matching (PM) conditions for multiple nonlinear processes. For example, both second harmonic generation (SHG, ω + ω = 2ω) and successive sum frequency generation (SFG, ω + 2ω = 3ω) are required to be phase matched for cascaded third harmonic generation (THG) [2

2. S. Zhu, Y. Zhu, and B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997). [CrossRef]

]. The PM conditions can be fulfilled by utilizing the quasi-phase-matching (QPM) technique in the so-called nonlinear photonic crystals with an alternative change of the sign of second-order susceptibility χ(2) [11

11. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998). [CrossRef]

13

13. Y. Sheng, K. Koynov, J. Dou, B. Ma, J. Li, and D. Zhang, “Collinear second harmonic generations in a nonlinear photonic quasicrystal,” Appl. Phys. Lett. 92, 201113 (2008). [CrossRef]

]. Two-dimensional (2D) and quasi-periodic χ(2) modulations are often used for the cascading process as they enable access to more than one independent reciprocal lattice vectors (RLVs) representing the spatial frequency of nonlinearity modulation. However, these nonlinear photonic crystals are efficient only for a particular choice of wavelengths of the interacting waves due to the discrete distribution of RLVs. More importantly, while in these structures the conversion efficiency increases with the length of the crystal, the acceptance bandwidth dramatically decreases. It is actually impossible to manipulate these two features independently in these structures.

Substantial research efforts have been focused on the broadband frequency conversion in novel quadratic structures with the combination of periodic and disordered or chirped modulation of χ(2) [14

14. K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–604 (1994). [CrossRef]

20

20. I. Varon, G. Porat, and A. Arie, “Controlling the disorder properties of quadratic nonlinear photonic crystals,” Opt. Lett. 36, 3978–3980 (2011). [CrossRef] [PubMed]

]. For example, engineered structures with chirped quasi-phase matching gratings for broadband frequency conversion have been demonstrated recently [15

15. A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technology, IEEE J. Lightwave Technol. 26, 343–349, (2008). [CrossRef]

, 16

16. A. Tehranchi, R. Morandotti, and R. Kashyap, “Efficient flattop ultra-wideband wavelength converters based on double-pass cascaded sum and difference frequency generation using engineered chirped gratings,” Opt. Express 19, 22528–22534 (2011). [CrossRef] [PubMed]

]. It has also been proposed to combine both features of chirped and quasi-periodic modulations to enhance the acceptance bandwidth of quadratic proceeses [17

17. J. Yang, X. P. Hu, P. Xu, X. J. Lv, C. Zhang, G. Zhao, H. J. Zhou, and S. N. Zhu, “Chirped-quasi-periodic structure for quasi-phase-matching,” Opt. Express 18, 14717–14723 (2010). [CrossRef] [PubMed]

]. Experimentally broadband cascaded third harmonic generation has been realized in the so-called short-range ordered structure by introducing some degree of structural randomness into otherwise periodic nonlinearity modulation [18

18. Y. Sheng and W. Krolikowski, “Broadband frequency tripling in locally ordered nonlinear photonic crystal,” Opt. Express 21, 4475–4480 (2013). [CrossRef] [PubMed]

]. Moreover, we have recently explored a broadband THG by using fully disordered quadratic medium. However in this case the conversion efficiency was very low [19

19. W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Yu. Kivshar, “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Opt. Express 17, 20117–20123 (2009). [CrossRef] [PubMed]

]. To obtain better trade-off between bandwidth and efficiency, specific designs of disordered χ(2) modulation have been employed that allows one to flexibly control the acceptance bandwidth of a single frequency conversion, e.g. the second harmonic generation [15

15. A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technology, IEEE J. Lightwave Technol. 26, 343–349, (2008). [CrossRef]

, 20

20. I. Varon, G. Porat, and A. Arie, “Controlling the disorder properties of quadratic nonlinear photonic crystals,” Opt. Lett. 36, 3978–3980 (2011). [CrossRef] [PubMed]

]. However, the simultaneous control of the acceptance bandwidth and conversion efficiency of the quadratic cascading process still remains a challenge.

In this work, we propose a novel quadratic nonlinear structure in ferroelectric medium such as lithium niobate crystal, utilizing the co-existence of disorder and quasi-periodicity, which provides an independent control of acceptance bandwidth and conversion efficiency of the cascading process. Our theoretical analysis and numerical simulations show that the bandwidth of the cascading process is determined by the quasi-periodic segment only, while the conversion efficiency depends on the total wave interaction distance, i.e. the length of the whole crystal.

2. Theoretical analysis

The concept of the locally quasi-periodic nonlinear photonic structure is schematically shown in Fig. 1. The structure consists of a set of quasi-periodic segments separated by the extended domains with random length. The quasi-periodic segment contains two basic building blocks A and B of length LA and LB arranged in a quasi-periodic sequence [3

3. K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002). [CrossRef] [PubMed]

, 21

21. K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum. Elect. 35, 1649–1656 (1999). [CrossRef]

]. These building blocks (A and B) represent a pair of 180° antiparallel domains with the corresponding duty cycle DA and DB (defined as the ratio of the width of negative area −χ(2) to the length of basic building block), respectively. Such quasi-periodic structure provides discrete set of reciprocal vectors Gmn = 2π(m + )/D, where D = τLA + LB is the so-called average parameter, τ is an irrational number, and m, n are integers. The total numbers of building blocks A and B in each quasi-periodic segment is the same and denoted as M. Any two neighboring quasi-periodic segments are separated by an extended domain of random length (see Fig. 1).

Fig. 1 The concept of the locally quasi-periodic quadratic structure for broadband second harmonic and cascading third harmonic generation. The locally quasi-periodic structure is constructed by repeating a set of quasi-periodic segments (consisting blocks A and B that are arranged into a quasi-periodic sequence) and separating them with an extended domain of random length. The insets below depict schematically the SHG and THG in the structure.

Considering a monochromatic fundamental wave with frequency ω propagating along the crystal [see Fig. 1] and assuming weak conversion efficiency, stationary cw regime and slowly varying envelope of harmonic waves for each quadratic process, the coupled wave equations governing the cascaded process of THG can be written as
dE2(x)dx=iβ1E12d(x)eiΔk1x,
(1)
dE3(x)dx=iβ2E1E2(x)d(x)eiΔk2x,
(2)
where β1 = 4πω2/cn2, β2 = 8πω3/cn3 is the coefficient for cascaded SHG and SFG process, respectively; d(x) = χ2(x)/2; Δk1 = k2 − 2k1 and Δk2 = k3k2k1 is the phase mismatching factor for the corresponding quadratic process; Ei, ωi = , ni and ki represents the electric envelope, frequency, refractive index and wave vector of the interacting waves, with subscripts i = 1, 2, 3 referring to the fundamental, SH and TH fields, respectively. c is the speed of light in vacuum.

The widths of the extended domains are chosen to be random to randomize the phases of second and third harmonics generated in different quasi-periodic segments. Since the interaction within the extended domains is non-phase-matched we may safely assume that these extended domains predominantly affect only the phases of interacting waves. Denoting this additional phase shift as ϕn and Φn for the SH and TH, respectively, the total electric fields of the SH and TH waves at the end of the crystal consisting of N pairs of quasi-periodic segments and the corresponding random domains can be expressed as
E2=n=1NE2neiϕn,
(5)
E3=n=1NE3neiΦn.
(6)
At this point, we introduce the intensity of harmonic waves defined as Ii(x)=Ei(x)Ei*(x)(i=2,3), which gives the corresponding harmonic intensities I2 and I3 at the end of the crystal as
I2=NI2n+n=1Nm=1,mnNE2n*E2mei(ϕnϕm),
(7)
I3=n=1NI3n+n=1Nm=1,mnNE3n*E3mei(ΦnΦm),
(8)
where I2n, I3n refers to the intensity of SH and TH waves generated in the nth segment, respectively. According to the result in Eq. (3), SH intensity generated in each segment is identical which can be expressed as
I2n=Γ12Δx2sinc2(Δk12Δx).
(9)
The explicit expression for the TH intensity I3n in the nth segment cannot be obtained because of its dependence on the boundary condition of the input SH waves. However, for a large N and because of the random phase introduced by the extended domains, the second term on the right hand sides of Eqs. (7) and (8) can be averaged to zero. The validity of this assumption has been confirmed in direct numerical simulations. Thus the harmonic intensity at the end of the crystal can be simplified to the sum of harmonic intensities generated in each quasi-periodic segment, demonstrating that the presence of extended domains with random lengths introduces disordered phases of harmonic signal which causes signal generated in each quasi-periodic segment to add incoherently during propagation. For the SHG, the overall SH intensity is proportional to the numbers of quasi-periodic segments, exhibiting a linear intensity dependence on the propagation distance; and the overall spectral bandwidth is uniquely determined by the spectral bandwidth of each quasi-periodic segment [as shown in Eq. (9)]. For consistent SFG process, the characteristics of TH intensity and its spectral bandwidth can be further explored based on the behavior of SH waves inside the crystal. Based on results of Eqs. (3) and (7) we will approximate the overall electric field of SH waves in the crystal as follows,
E2(n)=nΓ1Δxsinc(Δk12Δx)eiΔk1(xn1+xn)/2.
(10)
where E2(n) denotes the overall SH field after the nth segment in the crystal. Substituting this expression into Eq. (4) as the boundary condition (nn − 1) the electric field E3n and the corresponding intensity I3n can be presented in the following forms
E3n=(n1+12)Γ1Γ2Δx2sinc(Δk12Δx)sinc(Δk22Δx)eiΔk3(xn1+xn)/2,
(11)
I3n=(n1+12)2Γ12Γ22Δx4sinc2(Δk12Δx)sinc2(Δk22Δx).
(12)
As shown in Eqs. (11) and (12), the TH waves generated in each segment exhibits the same acceptance bandwidth but different intensity. Therefore the overall acceptance bandwidth of the TH waves at the end of the crystal is uniquely determined by the bandwidth of the quasi-periodic segment only while the overall intensity increases with propagation distance. Based on the analysis above, for a large N the overall intensity of harmonic waves at the end of the crystal can be expressed as follows
I2=NΓ12Δx2sinc2(Δk12Δx),
(13)
I3=N22Γ12Γ22Δx4sinc2(Δk12Δx)sinc2(Δk22Δx).
(14)

3. Numerical simulation

To further illustrate the effect of locally quasi-periodic nonlinear modulation on the cascading processes, we resort to numerical simulation by solving the coupled wave equations [Eqs. (1) and (2)] by using fast Fourier transform beam propagation method in one dimensional structure. The quasi-periodic segment is first designed to provide phase matching conditions for the two consistent SHG (e1e1e2) and SFG processes (e1e2e3) at the incident fundamental wavelength λ0 = 1.5μm in lithium niobate crystal [22

22. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quant. Electron. 16, 373–375 (1984). [CrossRef]

]. The two basic building blocks A and B of length LA = 7μm and LB = 5.68μm are ordered in a sequence LBLALBLALBLBLALBLALB... with the duty cycle factors DA = 0.6 and DB = 0 in the corresponding basic building blocks. The basic multiple integers of reciprocal vectors contributed to the fulfillment of quasi-phase matching are m = 1, n = 0, m′ = 1 and n′ = 1, given D = 17.66μm and τ = 1.86. The step size along the propagation direction is chosen as h = 0.05μm, so the length of the crystal is denoted as L = nxh, with nx being a positive integer. To get a perspective on the efficiency of the interaction which was assumed prior to be weak, let us consider for example the amplitude of fundamental wave E1 = 105V/m, M = 40, λ0 = 1.5μm at L = 219h. Then Eqs.(13) and (14) give I2/I1 ≈ 0.25%, I3/I2 ≈ 0.7%, I3/I1 ≈ 0.0018%, respectively.

Fig. 2 Dependence of normalized harmonic intensity on the incident fundamental wavelengths for SHG process (a) and SFG process (b), respectively, in the locally quasi-periodic nonlinear photonic structure with the same number M = 40 of building blocks in each quasi-periodic segment. The spectrum plots of harmonic waves (solid lines) in the figures are sampled at different lengths of the crystal where L = nxh, with nx = 217, 218, 219, which agreeing well with the spectrum lines (dotted lines) calculated from Eqs. (9) and (12), respectively. The intensities have been calculated by averaging over 512 realizations of nonlinear photonic structures.

For further illustration, we investigated the evolutions of the acceptance bandwidth of frequency conversion with different numbers of building blocks (M). In Fig. 3 we depict the normalized SH and TH intensities as a function of the incident fundamental wavelengths in three locally quasi-periodic samples, which have the same lengths but consist of different number of building blocks in the quasi-periodic segment, namely M=30, 40 and 50, respectively. It is seen that in all cases, the acceptance bandwidth of the nonlinear conversion processes broadens with the decrease of the number M of building blocks in the quasi-periodic segment. This result agrees well with the theoretical predictions of Eqs. (13) and (14). It also opens up a way to realize the quadratic cascading process with controllable acceptance bandwidth by tuning the number of building blocks M in the locally quasi-periodic nonlinear photonic structure. The ripples of the signals visible in Figs. 2 and 3 indicate that the whole structure is not completely random and the certain degree of coherence is preserved in harmonic signals.

Fig. 3 Normalized SH intensity (a) and TH intensity (b) as a function of the incident fundamental wavelengths in the locally quasi-periodic domain structures with different values of number M = 30, 40, 50 in each segment, but with the same crystal length L. The corresponding acceptance bandwidth (FWHM) is 56nm, 42nm and 33nm for the second harmonic and 14nm, 10nm and 8nm for the third harmonic, respectively. The intensities have been obtained by averaging over 512 realizations of nonlinear photonic structures.

Finally we investigate the dependence of the intensity of both harmonics on the propagation distance. The results of the intermediate process of SHG and cascaded THG are shown in Fig. 4(a) and 4(b), respectively. It is seen that the second harmonic intensity grows linearly with the interaction distance, in a good agreement with the prediction (dotted lines in Fig. 4(a)) of Eq. (13). Consequently the intensity of the third harmonic grows quadratically with the distance (see solid line in Fig. 4(b)). The dotted lines in Fig. 4(b) represents the analytical result calculated from Eq. (14), which shows a good agreement with the numerical result. With an electric field of fundamental wave E1 = 105 V/m, we numerically calculate the conversion efficiency of the harmonic generations at the end of the crystal, which are I2/I1 = 0.24% and I3/I2 = 0.67% for the SHG and SFG process, respectively. The corresponding conversion efficiency of cascaded THG is I3/I1 = 0.0016%. These results indicate that the conversion efficiency of the quadratic cascading processes can be enhanced by using a longer locally quasi-periodic crystal, without any impact on the acceptance bandwidth of the nonlinear interaction which is determined only by the number of building blocks in the quasi-periodic segment.

Fig. 4 Dependence of SH intensity (a) and TH intensity (b) on the interaction distance in the locally quasi-periodic nonlinear photonic structures with number M = 40 in each segment for different incident fundamental wavelengths λ = 1.496μm, 1.50μm, and 1.504μm. The analytical results derived from Eqs. (13) and (14) are plotted as dotted lines in (a) and (b), respectively, which show a good agreement with the corresponding numerical simulation. The results have been calculated by averaging over 512 realizations of nonlinear photonic structures.

Since the properties of our structure depend on the interplay between quasi-periodicity and randomness one may wonder how they will be affected by variation of domain size presenting always during the fabrication process. It is well known that departure from the exact quasi-phase matching has always detrimental effect on the efficiency of the frequency conversion process the spectral bandwidth actually increases [23

23. M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum. Elect. 28, 2631–2654 (1992). [CrossRef]

]. The analysis shows that in case of purely periodic structures these errors lead to decrease of exactly the same effect that will take place in quasi-periodic segment of our structure. On the other hand, since our structure already exhibits randomness via presence of the extended domains of random length, one may expect that overall our structure will be, in fact, much more tolerant to fabrication inaccuracies.

4. Conclusion

In summary, we have proposed a novel locally quasi-periodic nonlinear photonic structure for the realization of quadratic cascading process with controllable properties of harmonic generation. We show that the acceptance bandwidth of the cascading process is determined solely by the length of the quasi-periodic segment, while the conversion efficiency is dependent on the length of the whole nonlinear photonic crystal. Therefore, this scheme provides an independent control of the acceptance bandwidth and conversion efficiency of the cascading process. The proposed interaction scheme could be realized experimentally in ferroelectric crystal by using electrical poling to fabricate the desired nonlinearity modulation. The same approach can be used for any quadratic cascading effect, such as fourth-harmonic generation in a single crystal and can find applications in nonlinear frequency conversion, such as pulse conversion, reconstruction and monitoring of short pulses.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11204206), Natural Science Foundation of Shanxi Province (Grant No. 2013021017-3), Taiyuan University of Technology Science Foundation (Grant No. 2012L038) and the Australian Research Council.

References and links

1.

J.B. Khurgin, A. Obeidat, S. J. Lee, and Y. J. Ding, “Cascaded optical nonlinearities: microscopic understanding as a collective effect,” J. Opt. Soc. Am. 14, 1977–1983 (1997). [CrossRef]

2.

S. Zhu, Y. Zhu, and B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997). [CrossRef]

3.

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002). [CrossRef] [PubMed]

4.

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001). [CrossRef]

5.

N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, “Group-velocity-mismatch compensation in cascaded third-harmonic generation with two dimensional quasi-phase-matching gratings,” Opt. Lett. 31, 2780–2782 (2006). [CrossRef] [PubMed]

6.

N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, “Cascaded third-harmonic generation of ultrashort optical pulses in two-dimensional quasi-phase-matching gratings,” J. Opt. Soc. Am. B 24, 2394–2405 (2007). [CrossRef]

7.

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. 84, 1055–1057 (2004). [CrossRef]

8.

M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency-generation and difference-frequency-generation processes in periodically poled LiNbO3,” Opt. Lett. 22, 274–276 (1997). [CrossRef] [PubMed]

9.

M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, K. W. DeLong, and R. Trebino, “Ultrafast optical switching by use of fully phase-matched cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Lett. 22, 245–247 (1997). [CrossRef] [PubMed]

10.

G. I. Petrov, M. Zhi, and V. V. Yakovlev, “Coherent anti-Stokes Raman spectroscopy utilizing phase mismatched cascaded quadratic optical interactions in nonlinear crystals,” Opt. Express 21, 31960–31965 (2013). [CrossRef]

11.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998). [CrossRef]

12.

Y. Du, S. N. Zhu, Y. Y. Zhu, P. Xu, C. Zhang, Y. B. Chen, Z. W. Liu, and N. B. Ming, “Parametric and cascaded parametric interactions in a quasiperiodic optical superlattice,” Appl. Phys. Lett. 81, 1573–1575 (2002). [CrossRef]

13.

Y. Sheng, K. Koynov, J. Dou, B. Ma, J. Li, and D. Zhang, “Collinear second harmonic generations in a nonlinear photonic quasicrystal,” Appl. Phys. Lett. 92, 201113 (2008). [CrossRef]

14.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–604 (1994). [CrossRef]

15.

A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technology, IEEE J. Lightwave Technol. 26, 343–349, (2008). [CrossRef]

16.

A. Tehranchi, R. Morandotti, and R. Kashyap, “Efficient flattop ultra-wideband wavelength converters based on double-pass cascaded sum and difference frequency generation using engineered chirped gratings,” Opt. Express 19, 22528–22534 (2011). [CrossRef] [PubMed]

17.

J. Yang, X. P. Hu, P. Xu, X. J. Lv, C. Zhang, G. Zhao, H. J. Zhou, and S. N. Zhu, “Chirped-quasi-periodic structure for quasi-phase-matching,” Opt. Express 18, 14717–14723 (2010). [CrossRef] [PubMed]

18.

Y. Sheng and W. Krolikowski, “Broadband frequency tripling in locally ordered nonlinear photonic crystal,” Opt. Express 21, 4475–4480 (2013). [CrossRef] [PubMed]

19.

W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Yu. Kivshar, “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Opt. Express 17, 20117–20123 (2009). [CrossRef] [PubMed]

20.

I. Varon, G. Porat, and A. Arie, “Controlling the disorder properties of quadratic nonlinear photonic crystals,” Opt. Lett. 36, 3978–3980 (2011). [CrossRef] [PubMed]

21.

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum. Elect. 35, 1649–1656 (1999). [CrossRef]

22.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quant. Electron. 16, 373–375 (1984). [CrossRef]

23.

M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum. Elect. 28, 2631–2654 (1992). [CrossRef]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.7220) Nonlinear optics : Upconversion
(190.4223) Nonlinear optics : Nonlinear wave mixing

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 20, 2013
Revised Manuscript: February 24, 2014
Manuscript Accepted: February 26, 2014
Published: March 18, 2014

Citation
Wenjie Wang, Yan Sheng, Shaoding Liu, Xiaoying Niu, and Wieslaw Krolikowski, "Manipulation of quadratic cascading processes in a locally quasi-periodic χ(2) medium," Opt. Express 22, 6976-6983 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6976


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References

  1. J.B. Khurgin, A. Obeidat, S. J. Lee, Y. J. Ding, “Cascaded optical nonlinearities: microscopic understanding as a collective effect,” J. Opt. Soc. Am. 14, 1977–1983 (1997). [CrossRef]
  2. S. Zhu, Y. Zhu, B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997). [CrossRef]
  3. K. Fradkin-Kashi, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002). [CrossRef] [PubMed]
  4. C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001). [CrossRef]
  5. N. Fujioka, S. Ashihara, H. Ono, T. Shimura, K. Kuroda, “Group-velocity-mismatch compensation in cascaded third-harmonic generation with two dimensional quasi-phase-matching gratings,” Opt. Lett. 31, 2780–2782 (2006). [CrossRef] [PubMed]
  6. N. Fujioka, S. Ashihara, H. Ono, T. Shimura, K. Kuroda, “Cascaded third-harmonic generation of ultrashort optical pulses in two-dimensional quasi-phase-matching gratings,” J. Opt. Soc. Am. B 24, 2394–2405 (2007). [CrossRef]
  7. S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. 84, 1055–1057 (2004). [CrossRef]
  8. M. Asobe, I. Yokohama, H. Itoh, T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency-generation and difference-frequency-generation processes in periodically poled LiNbO3,” Opt. Lett. 22, 274–276 (1997). [CrossRef] [PubMed]
  9. M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, K. W. DeLong, R. Trebino, “Ultrafast optical switching by use of fully phase-matched cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Lett. 22, 245–247 (1997). [CrossRef] [PubMed]
  10. G. I. Petrov, M. Zhi, V. V. Yakovlev, “Coherent anti-Stokes Raman spectroscopy utilizing phase mismatched cascaded quadratic optical interactions in nonlinear crystals,” Opt. Express 21, 31960–31965 (2013). [CrossRef]
  11. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998). [CrossRef]
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