Giant enhancement of second harmonic generation in nonlinear photonic crystals with distributed Bragg reflector mirrors

doi:10.1364/OE.17.014502

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Giant enhancement of second harmonic generation in nonlinear photonic crystals with distributed Bragg reflector mirrors

Ming-Liang Ren and Zhi-Yuan Li »View Author Affiliations

Optics Express, Vol. 17, Issue 17, pp. 14502-14510 (2009)
http://dx.doi.org/10.1364/OE.17.014502

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Abstract

We theoretically investigate second harmonic generation (SHG) in one-dimensional multilayer nonlinear photonic crystal (NPC) structures with distributed Bragg reflector (DBR) as mirrors. The NPC structures have periodic modulation on both the linear and second-order susceptibility. Three major physical mechanisms, quasi-phase matching (QPM) effect, slow light effect at photonic band gap edges, and cavity effect induced by DBR mirrors can be harnessed to enhance SHG. Selection of appropriate structural parameters can facilitate coexistence of these mechanisms to act collectively and constructively to create very high SHG conversion efficiency with an enhancement by up to seven orders of magnitude compared with the ordinary NPC where only QPM works.

The quasi-phase-matching (QPM) technique is known as an attractive way to obtain good phase matching and increase the efficiency of second-harmonic generation (SHG) and other nonlinear optical frequency conversion processes [1

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]

–6

S. Kawai, T. Ogawa, H. S. Lee Robert, C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr_0.6Ba_0.4Nd₂O₆ single crystals,” Appl. Phys. Lett. 73(6), 768–770 (1998). [CrossRef]

]. The QPM technique relies on the periodic modulation of the second-order nonlinear susceptibility

χ^{(2)}

to compensate for the mismatch

Δ k = k_{2} - 2 k_{1}

between the wave vectors of the second-harmonic and fundamental waves. Such an optical structure is now generally called nonlinear photonic crystal (NPC) [2

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998). [CrossRef]

]. The NPC can provide a particular reciprocal lattice vector

G_{i}

to satisfy QPM by making

Δ k = G_{i}

Other approaches have been explored to improve the conversion efficiency of SHG. One way is to utilize the slow light effect at photonic band gap edges (PBGEs) of photonic crystals made from nonlinear optical materials. Enhancement by several orders of magnitude of second-order nonlinear interactions has been reported by several groups [7

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Masselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24(12), 793–795 (1999). [CrossRef]

–11

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, C. Me’riadec, and A. Levenson, “χ⁽²⁾ semiconductor photonic crystals,” J. Opt. Soc. Am. B 19(9), 2094–2101 (2002). [CrossRef]

]. Another way is to introduce resonant cavities to achieve better field confinement and enhancement [12

T. M. Liu, Ch. T. Yu, and Ch. K. Sun, “2 GHz repetition-rate femtosecond blue sources by second harmonic generation in a resonantly enhanced cavity,” Appl. Phys. Lett. 86(6), 061112 (2005). [CrossRef]

–16

H. Cao, D. B. Hall, J. M. Torkelson, and C. Q. Cao, “Large enhancement of second harmonic generation in polymer films by microcavities,” Appl. Phys. Lett. 76(5), 538 (2000). [CrossRef]

]. Alternatively, one can make use of defect modes in nonlinear photonic crystals to greatly enhance local fields [17

B. Shi, Z. M. Jiang, and X. Wang, “Defective photonic crystals with greatly enhanced second-harmonic generation,” Opt. Lett. 26(15), 1194–1196 (2001). [CrossRef]

–20

L. M. Zhao and B. Y. Gu, “Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material,” Appl. Phys. Lett. 88(12), 122904 (2006). [CrossRef]

]. For instance, when the fundamental wave (FW) and the second harmonic wave (SHW) are both tuned at the defect states, giant enhancement of SHG will be obtained [18

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70(24), 245109 (2004). [CrossRef]

]. In our previous works [21

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007). [CrossRef] [PubMed]

,22

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007). [CrossRef]

], we have found that combination of QPM and PBGE effects in one-dimensional (1D) NPCs can significantly enhance the SHG conversion efficiency by three to four orders of magnitude.

In this work, we will show that it is possible to incorporate three of different physical mechanisms as mentioned in the above, namely, the QPM, the slow light effect, and cavity effect into a single NPC structure. By selection of appropriate structural parameters, the three different mechanisms can act collectively and constructively to realize multiplication of individual enhancement to SHG by each mechanism. The result is that the conversion efficiency of SHG in the designed NPC structure can be enlarged by a factor of up to seven orders of magnitude compared with that in a traditional NPC where only QPM works.

The basic geometry of the designed NPC structure is depicted in Fig. 1(a) . The structure is a NPC with two distributed Bragg reflector (DBR) mirrors at its two ends. The central NPC has periodic modulation on both the linear and second-order susceptibility, which is similar to that discussed in Ref. [22

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007). [CrossRef]

]. The unit cell of the NPC is consisted of the poled LiNbO₃ and air, as is schematically shown in Fig. 1(b). Layers A1 and A2, which are the poled LiNbO₃ crystal, possess the same refractive index but inverse signs of

χ^{(2)}

, and Layer B is an air layer, having a different refractive index from that of layer A1 and A2. As a result, a series of photonic band gaps can be produced in the 1D NPC, whose numbers and spectral positions can be flexibly tuned. In this NPC, QPM and PBGE effect each can occur and work to improve the SHG, but special caution must be taken in order to make them act collectively and constructively to create a significantly large enhancement of SHG [21

,22

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007). [CrossRef]

Fig. 1 (a) Schematic diagram of a 1D NPC structure with DFR mirrors at the two ends. N stands for the period number of the NPC, while

D_{1}

and

D_{2}

represents the period number of the left and right DBR mirror. (b) Schematic diagram of the central NPC with both periodic modulation on the refractive index and second-order nonlinear susceptibility. Layers A1 and A2 represent the poled LiNbO₃ crystal, and Layer B is an air layer. The arrows inside the crystal indicate the polarization direction.

The whole structure in Fig. 1(a) is composed of three parts: the left DBR, the NPC and the right DBR. The DBR mirrors possess the same unit-cell structure as the central NPC, except for different thickness of air layers. When the DBR mirrors are designed to have high reflectivity to the FW and high transparency to the SHW, a cavity will form and offer effective confinement to the FW field. To understand and optimize this nonlinear optical structure, in particular, to see the role played by the DBR mirrors, we have utilized a nonlinear transfer matrix method (TMM) [21

–24

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003). [CrossRef] [PubMed]

] that can handle any nonlinear multilayer structures to calculate the SHW generated from the new structure with different geometric parameters. The aim of these extensive numerical studies is to find appropriate structural parameters so that giant enhancement of SHG can take place.

The geometrical and physical parameters of the NPC are as follows. The thicknesses of the layer A1, A2 and B are denoted as

d_{A 1}

d_{A 2}

, and

d_{B}

. The refractive index of LiNbO₃ layer A1 and A2 at room temperature can be expressed as follows [25

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). [CrossRef]

n_{e}^{2} (λ, T) = a_{1} + b_{1} f + \frac{a_{2} + b_{2} f}{λ^{2} - {(a_{3} + b_{3} f)}^{2}} - a_{4} λ^{2},

(1)

where

a_{1} = 4.5820

a_{2} = 0.09921

a_{3} = 0.21090

a_{4} = 0.021940

b_{1} = 2.2971 \times 10^{- 7}

b_{2} = 5.2716 \times 10^{- 8}

b_{3} = - 4.9143 \times 10^{- 8}

f = (T - T_{0}) (T + T_{0} + 546)

, and

T_{0} = {24.5}^{\circ} C

. λ is the vacuum wavelength in unit of micrometer.

The incident FW is designated at 1.064 μm, and the corresponding SHW has a wavelength of 0.532 μm. The second-order nonlinear coefficient is 27.2 pm/V [26

V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals (Springer, Berlin, 1997), Vol. 64, p. 125.

], the refractive index for FW and SHW is 2.156 and 2.2342, respectively, from which the coherence length can be calculated to be 3.4016 μm. In calculation, we assume a weak light with amplitude of

E_{0} = 950

V/m is incident upon the structure. The total period number of the NPC structure is set to be

N = 29

. Now the key point is to find appropriate structural parameters that allow for realization of QPM at PBGEs. To achieve this, we first solve the photonic band structures of 1D photonic crystal and pursue a series of

d_{A 1}

(with

d_{A 2} = d_{A 2}

) and

d_{B}

values, under which the corresponding 1D NPC has two PBGEs matched with 1.064 μm and 0.532 μm. Then we calculate the SHG conversion for each group of parameters by means of the nonlinear TMM. Around each group of parameters, fine tuning to

d_{A 1}

and

d_{B}

is made and the corresponding SHG conversion efficiency is calculated. From all these data we can determine the maximum SHG conversion efficiency. The corresponding value of (

d_{A 1}

d_{B}

) should precisely correspond to the desirable NPC structure that realizes QPM at PBGEs. By following this procedure we have found that in the NPC with parameters of

d_{A 1} = d_{A 2} = 2.7919

μm, which is below the coherence length, and

d_{B} = 0.3256

μm, the FW and SHW both get close to the PBGEs, and the maximum conversion efficiency occurs at 1.06401 μm. Obviously, this NPC is the structure that we pursue. On the other hand, we find that if

d_{A 1} = d_{A 2} = 2.7919

μm and

d_{B} = 0.75

μm, the structure is highly reflective to the FW but transparent to the SHW, leading to much lower conversion efficiency. For this reason, we choose the unit cell with

d_{B} = 0.3256

μm for the central NPC and

d_{B} = 0.75

μm for the DBR mirrors.

As is depicted in Fig. 1(a), the whole structure is comprised of three parts: the left DBR consisting of

D_{1}

unit cells, the central NPC consisting of N unit cells, and the right DBR consisting of

D_{2}

unit cells. The whole structure is fully characterized by the parameters of N,

D_{1}

and

D_{2}

. Four cases are considered to investigate the benefit of DBR mirrors: (1)

D_{1} \times D_{2} = 0 \times 0

(without DBR); (2)

D_{1} \times D_{2} = 0 \times 8

(with the right DBR mirror); (3)

D_{1} \times D_{2} = 3 \times 3

(with two DBR mirrors); (4)

D_{1} \times D_{2} = 3 \times 5

(with two DBR mirrors). The reflectivity of three-, five- and eight-period DBR is found to be 94.65%, 99.68% and 99.99%, respectively for the FW. Obviously by introducing the DBR mirrors, a resonant cavity is formed for the FW, but not for the SHW. Notice that SHG primarily takes place in the central NPC, while the DBR mirrors basically play the role of selective feedback of the FW signal.

We have calculated the external conversion efficiency of the forward-propagating SHW versus the scanning wavelength for the four cases and the results are summarized in Fig. 2 . In addition, we have looked into the details of the intensity distribution of the FW in the whole structure in order to have clarified physical pictures of different mechanisms for enhancing SHG. The results are displayed in Fig. 3 for the four cases. Suppose that

η_{i}

and

I_{i}

(i = 1, 2, 3, 4) respectively represent the conversion efficiency of the forward-propagating SHW and the average intensity of FW fields within the whole structure. From Figs. 2 and 3, we can see that

η_{1} < η_{2} < η_{3} < η_{4}

and

I_{1} < I_{2} < I_{3} < I_{4}

. Because of the DBR mirrors, the conversion efficiency can be further enhanced by two orders of magnitude compared with the NPC that realizes QPM at PBGEs.

Fig. 2 Calculated conversion efficiency of the forward SHG from different NPC structures as a function of the wavelength of the incident FW. Panels (a)-(d) correspond to the case (1)-(4) structure, respectively. All the curves exhibit a very sharp peak with a width of about 0.02 nm.

Fig. 3 Calculated local field intensity distribution of FW within different NPC structures. Panels (a)-(d) correspond to the case (1) - (4) structure, respectively.

In order to have a more direct concept on to what extent SHG is enhanced in the NPC with DBR mirrors, we further consider SHG from two more samples. In Sample 1 only the pure QPM exists, while in Sample 2 only the PBGE effect exists. In comparison, in the case (1) structure shown in Fig. 2(a), both the QPM and PBGE effect coexist. Sample 1 is just a poled LiNbO₃ multilayer structure characterized by

d_{A 1} = d_{A 2}

and

d_{B} = 0

in Fig. 1(b). Obviously, it has only periodic modulation on

χ^{(2)}

. Calculations show that the parameters of unit cell for optimum SHG conversion efficiency in Sample 1 are

d_{A 1} = d_{A 2} = 3.3981

μm, where the unit-cell size is close to the coherence length. The period number of Sample 1 is

N = 24

. The structure of Sample 2 is the same as the case (1) structure except that all the ferroelectric domains are polarized along the same direction. In this structure, only the PBGE effect contributes to the SHG. In this regard, Sample 2 is a conventional photonic crystal. Note that to promise a basically equal total length of nonlinear material for the sake of fair comparison, we have assumed

N = 24

for Sample 1 and

N = 29

for Sample 2 and the case (1) structure. All samples are placed in the air background.

The calculated results of external conversion efficiency of the forward SHG under the same FW pump as in Fig. 2 are displayed in Figs. 4(a) and 4(b) for Sample 1 and Sample 2, respectively. We see that the conversion efficiency in Sample 2 is several times larger than in Sample 1, indicating that the pure PBGE effect contributes more than the pure QPM. The fluctuation in Fig. 4(a) is ascribed to multi-reflection at the boundary between the LiNbO₃ wafer and the air background. On the scale of 1.5 to 1.8 μm, Fig. 4(a) seems to exhibit a sawtooth wave package with broad width. Similar to Fig. 2, the spectral width of the peak can also be resolved in Fig. 4(b) by rescaling the wavelength axis. Comparing Fig. 4 with Fig. 2(a), it is obvious that the associated action of the QPM and PBGE effect can enhance the SHG by three to four orders of magnitudes compared with the situations when these two mechanisms act individually. Further introduction of DBR mirrors into the NPC can achieve an extremely huge (up to seven orders of magnitude) enhancement of SHG upon the conventional NPC working under the QPM scheme.

Fig. 4 Calculated conversion efficiency of forward SHG as a function of the incident FW wavelength for (a) a NPC where only the pure QPM acts, (b) a NPC where only PBGE effect plays a role.

The enhancement of SHG by introduction of DBR mirrors is closely related to the local field enhancement. The reason is that the nonlinear polarization, which is the radiation source of SHW, is determined by the local field intensity of FW, following a simple relation of

P_{i}^{(2)} (z) = - χ_{i}^{(2)} {[E_{i}^{} (z)]}^{2}

, where

E_{i} (z)

is the local field for FW in the ith layer of the NPC structure. A larger local field means a stronger radiation of SHW. If all these SHW radiations can become coherent and interfere constructively, they will create a very strong SHG out of the NPC sample. In the case (1), QPM can be realized at the PBGE. Compared with Fig. 3(a), Fig. 3(b) [for case (2) structure] reveals that the local intensity for the FW field in the multilayer structure is enhanced by several times. This obviously benefits from the right DBR mirror that is highly reflective to the FW and transparent to SHW. Consequently, the nonlinear optical frequency conversion efficiency grows almost ten times, as can be found in the Fig. 2(a) and 2(b). In the case (3) and (4), two DBR mirrors are both designated at the two sides of the structure, so that a simple cavity can be formed to confine more energy than the previous cases. This point has also been clearly reflected in the field patterns of Figs. 3(c) and 3(d). Because a thicker DBR mirror presents higher reflectivity to the FW, the cavity in the case (4) works better than the one in the case (3). This can well explain the phenomenon in Figs. 3(c) and 3(d), and also the result in Figs. 2(c) and 2(d). From this comparative analysis we can summarize a law to find an optimized structure for enhanced SHG: Increase the thickness of the right DBR mirror of a cavity to enlarge the conversion efficiency of SHG. It should be noted that the local filed enhancement in the cavity formed by the DBR mirrors is closely related to the quality factor. In general, thicker DBR mirrors result in larger values of quality factor, and then lead to higher field enhancement. In practice, it is hard to achieve infinitely large quality factor because of fabrication accuracy limitations. On the other hand, thicker mirrors will reduce coupling efficiency of FW into the cavity region. In the cases of (3) and (4), we intentionally do not increase the unit-cell number of the left DBR. The reason is to promise a relatively high coupling of the FW light into the central NPC while at the same time to maintain a high enough quality factor of cavity. As a result, the left and right DBR mirrors are asymmetric in thickness. We believe that there also exists an optimal left DBR structure, which allows the incident light to efficiently pass through and at the same time forms a good cavity together with the right DBR mirror to enhance the FW local field intensity to the maximum extent, where the impedance matching condition is satisfied [14

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Second-harmonic generation of a continuous wave diode-pumped Nd:YAG laser using an externally resonant cavity,” Opt. Lett. 12(12), 1014–1016 (1987). [CrossRef] [PubMed]

From what have been discussed above, we can see that our designed structures have realized slow light effect in cavity, and this remarkably enhances optical nonlinearity. From Fig. 3 (b)-(d), we also find, the intensity of FW in the DBR mirrors is much smaller than in the central NPC, which means that SHW generated from the DBR mirrors can be negligible. To show this, we have intentionally only considered the situation where the nonlinear effect only exists in the central NPC by setting the nonlinear coefficient of the DBR mirrors as zero. The consequent simulation results for the four cases mentioned above are almost the same as what have been reflected in Fig. 2. This confirms that SHW most generates from the central NPC and SHW generated from DBR mirrors is quite small and inessential. Therefore, despite different sample length between the bare NPC and the NPC with DBR mirrors, the comparison between them is relevant as all the structures have the same length of effective nonlinear interaction. Simply speaking, SHW generated from DBR mirrors contribute very little to enhancement of SHG in Case (2)-(4) [Fig. 2 (b)-(d)].

Now we are at a position to present a simple physical picture for the above observations. Basically, the SHG from a NPC is determined by two factors. The first factor is the SHW radiation source intensity proportional to the amplitude of the nonlinear polarization,

P_{i}^{(2)} (z) = - χ_{i}^{(2)} {[E_{i}^{} (z)]}^{2}

. Generally speaking, a stronger source leads to stronger radiation. To achieve this, a flexible way is to enlarge the local field intensity of FW for a fixed nonlinear optical material. The second factor is the interference of all these radiations out of the NPC sample, which is basically related to the propagation nature of optical waves. The best way for enhancement of SHG is to make all the radiations of SHW interfere constructively. The key for this is to adjust the phases of SHW from all parts of the nonlinear optical structure so that they travel always in-phase with the FW. The traditional phase matching in nonlinear optical crystals and QPM in usual NPCs are adopted essentially for this purpose. Introduction of photonic band gaps into the NPC structure and placing FW at the PBGEs will significantly slow down the transport group velocity of the pump light and lead to great accumulation of light energy in the NPC structure. As a result, the local field intensity of FW is enlarged greatly and makes much stronger radiation of SHW. That is why Sample 2 (with only PBGE effect) has several times stronger SHG than Sample 1 (with only QPM). Introduction of cavity effect into the NPC by means of DBR mirrors can further increase the accumulation effect of light energy and results in an even larger local field intensity of FW within the NPC structure.

The above analyses clearly indicate that three major mechanisms, the QPM, PBGE effect, and cavity effect can work and contribute to enhance SHG in nonlinear optical structures. It is then natural to expect that bringing these three mechanisms together in a single nonlinear optical structure should be able to enhance SHG to the maximum extent. The essential point is to make all SHWs radiated from every nonlinear polarization source that has been greatly enlarged by local field enhancement interfere constructively. On the contrary, a destructive interference will not generate large SHG no matter how strong the radiation sources are.

Nonetheless, it is never obvious and easy on how to bring these three factors together. In our above scheme, we set the configuration of the NPC such that both the FW and SHW are located at the edge of two different band gaps of the 1D NPC. With this, the FW and SHW will transport along the nonlinear structure in approximately the same group velocity, which is greatly reduced because of the slow light effect. As a result, the FW and SHW are basically in phase with each other. Introduction of inverted ferroelectric domains with suitable geometric parameters, such as their width, can bring a further fine tuning so that the FW and SHW now propagates exactly in phase and under the same group velocity. This is the basic point of the NPC that realizes QPM at PBGEs, which can increase the conversion efficiency of SHG by four orders of magnitude compared with the NPC where only QPM exists. Further introduction of two DBR mirrors at the two ends of this NPC simply brings in the cavity effect without any other side effect that can poison the QPM and PBGE effects. By this simple configuration, the three mechanisms have been brought together to contribute collectively and constructively to SHG. As a result, up to seven orders of magnitude enhancement of the conversion efficiency of SHG is found in the NPC with DBR mirrors compared with the NPC where only QPM exists. In comparison, the usual way to bring the cavity inside the NPC under the scheme of defect modes will make things very complicated and is difficult to bring all the three mechanisms together.

In order to realize slow light effect in a cavity to facilitate giant enhancement of nonlinear interaction, it is important to find a way to realize the proposed NPC structures in experiment. Generally, the electric poling technology can be utilized to fabricate the stripe-poling LiNbO₃, and the nanotechnology, such as focused ion beam lithography, can be employed to etch the poling LiNbO₃ for air groove with sufficient depth. The central NPC and DBR can be distinguished by the thickness of air groove. After these operations, the proposed NPC structures can be obtained. The unavoidable fabrication uncertainty will induce nonuniformity, imperfection, and even defects in the NPC structure and these factors will degrade the efficiency of enhanced nonlinear interaction because they will affect both the slow light effect and cavity resonance effect. On the other hand, optical losses can also be present due to light scattering from interface roughness. However, we still believe that a sufficiently large enhancement (although not so large as for the ideal situations) of SHG under the combination of QPM, PBGE and cavity effect can still be observed in these designed structures.

In summary, we have theoretically investigated the optical problems of SHG from different 1D NPC structures. We have found that introduction of appropriate DBR mirrors at the two ends of a NPC structure that realizes QPM at PBGE can lead to a giant enhancement to the conversion efficiency of SHG by seven orders of magnitude compared with the traditional NPC structure by using QPM. The reason is that the NPC structure can bring together the three mechanisms of QPM effect, PBGE effect and cavity effect within a single structure and make them contribute collectively and constructively to enhance SHG from the structure. Such a structure will make it possible to realize high efficiency nonlinear optical frequency conversion in a size scale that is several orders of magnitude smaller than the traditional nonlinear crystals and QPM structures. This will greatly facilitate success of all-optical integration by using photonic crystals.

Acknowledgments

This work was supported by the National Natural Science Foundation (Nos. 10634080 and 10525419) and the State Key Development Program for Basic Research of China (No. 2007CB613205).

References and links

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13.	S. Nakagawa, N. Yamada, N. Mikoshiba, and D. E. Mars, “Second-harmonic generation from GaAs/AlAs vertical cavity,” Appl. Phys. Lett. 66(17), 2159–2161 (1995). [CrossRef]
14.	W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Second-harmonic generation of a continuous wave diode-pumped Nd:YAG laser using an externally resonant cavity,” Opt. Lett. 12(12), 1014–1016 (1987). [CrossRef] [PubMed]
15.	V. Pellegrini, R. Colombelli, I. Carusotto, F. Beltram, S. Rubini, R. Lantier, A. Franciosi, C. Vinegoni, and L. Pavesi, “Resonant second harmonic generation in ZnSe bulk microcavity,” Appl. Phys. Lett. 74(14), 1945–1947 (1999). [CrossRef]
16.	H. Cao, D. B. Hall, J. M. Torkelson, and C. Q. Cao, “Large enhancement of second harmonic generation in polymer films by microcavities,” Appl. Phys. Lett. 76(5), 538 (2000). [CrossRef]
17.	B. Shi, Z. M. Jiang, and X. Wang, “Defective photonic crystals with greatly enhanced second-harmonic generation,” Opt. Lett. 26(15), 1194–1196 (2001). [CrossRef]
18.	F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70(24), 245109 (2004). [CrossRef]
19.	Y. Zeng, X. S. Chen, and W. Lu, “Optical limiting in defective quadratic nonlinear photonic crystals,” J. Appl. Phys. 99(12), 123107 (2006). [CrossRef]
20.	L. M. Zhao and B. Y. Gu, “Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material,” Appl. Phys. Lett. 88(12), 122904 (2006). [CrossRef]
21.	J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007). [CrossRef] [PubMed]
22.	J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007). [CrossRef]
23.	J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008). [CrossRef]
24.	Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003). [CrossRef] [PubMed]
25.	G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). [CrossRef]
26.	V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals (Springer, Berlin, 1997), Vol. 64, p. 125.

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 3, 2009
Revised Manuscript: July 16, 2009
Manuscript Accepted: July 16, 2009
Published: August 3, 2009

Citation
Ming-Liang Ren and Zhi-Yuan Li, "Giant enhancement of second harmonic generation in nonlinear photonic crystals with distributed Bragg reflector mirrors," Opt. Express 17, 14502-14510 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14502

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References

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]
V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998). [CrossRef]
P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003). [CrossRef]
P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004). [CrossRef] [PubMed]
A. Piskarskas, V. Smilgevi Ius, A. Stabinis, V. Jarutis, V. Pasiskevi Ius, S. Wang, J. Tellefsen, and F. Laurell, “Noncollinear second-harmonic generation in periodically poled KTiOPO4 excited by the Bessel beam,” Opt. Lett. 24(15), 1053–1055 (1999). [CrossRef]
S. Kawai, T. Ogawa, H. S. Lee Robert, C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73(6), 768–770 (1998). [CrossRef]
A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Masselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24(12), 793–795 (1999). [CrossRef]
G. Vecchi, J. Terres, D. Coquillat, M. L. V. d’Yerville, and A. M Malvezzi “Enhancement of visible second harmonic generation in epitaxial GaN-based two-dimensional photonic crystal structures,” Appl. Phys. Lett. 84(8), 1245–1247 (2004). [CrossRef]
Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78(20), 3021–3023 (2001). [CrossRef]
G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ 2 interactions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(1 ), 016609 (2001). [CrossRef] [PubMed]
Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, C. Me’riadec, and A. Levenson, “χ(2) semiconductor photonic crystals,” J. Opt. Soc. Am. B 19(9), 2094–2101 (2002). [CrossRef]
T. M. Liu, Ch. T. Yu, and Ch. K. Sun, “2 GHz repetition-rate femtosecond blue sources by second harmonic generation in a resonantly enhanced cavity,” Appl. Phys. Lett. 86(6), 061112 (2005). [CrossRef]
S. Nakagawa, N. Yamada, N. Mikoshiba, and D. E. Mars, “Second-harmonic generation from GaAs/AlAs vertical cavity,” Appl. Phys. Lett. 66(17), 2159–2161 (1995). [CrossRef]
W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Second-harmonic generation of a continuous wave diode-pumped Nd:YAG laser using an externally resonant cavity,” Opt. Lett. 12(12), 1014–1016 (1987). [CrossRef] [PubMed]
V. Pellegrini, R. Colombelli, I. Carusotto, F. Beltram, S. Rubini, R. Lantier, A. Franciosi, C. Vinegoni, and L. Pavesi, “Resonant second harmonic generation in ZnSe bulk microcavity,” Appl. Phys. Lett. 74(14), 1945–1947 (1999). [CrossRef]
H. Cao, D. B. Hall, J. M. Torkelson, and C. Q. Cao, “Large enhancement of second harmonic generation in polymer films by microcavities,” Appl. Phys. Lett. 76(5), 538 (2000). [CrossRef]
B. Shi, Z. M. Jiang, and X. Wang, “Defective photonic crystals with greatly enhanced second-harmonic generation,” Opt. Lett. 26(15), 1194–1196 (2001). [CrossRef]
F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70(24), 245109 (2004). [CrossRef]
Y. Zeng, X. S. Chen, and W. Lu, “Optical limiting in defective quadratic nonlinear photonic crystals,” J. Appl. Phys. 99(12), 123107 (2006). [CrossRef]
L. M. Zhao and B. Y. Gu, “Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material,” Appl. Phys. Lett. 88(12), 122904 (2006). [CrossRef]
J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007). [CrossRef] [PubMed]
J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007). [CrossRef]
J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008). [CrossRef]
Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003). [CrossRef] [PubMed]
G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). [CrossRef]
V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals (Springer, Berlin, 1997), Vol. 64, p. 125.

Appl. Phys. Lett.

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003). [CrossRef]
S. Kawai, T. Ogawa, H. S. Lee Robert, C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73(6), 768–770 (1998). [CrossRef]
G. Vecchi, J. Terres, D. Coquillat, M. L. V. d’Yerville, and A. M Malvezzi “Enhancement of visible second harmonic generation in epitaxial GaN-based two-dimensional photonic crystal structures,” Appl. Phys. Lett. 84(8), 1245–1247 (2004). [CrossRef]
Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78(20), 3021–3023 (2001). [CrossRef]
T. M. Liu, Ch. T. Yu, and Ch. K. Sun, “2 GHz repetition-rate femtosecond blue sources by second harmonic generation in a resonantly enhanced cavity,” Appl. Phys. Lett. 86(6), 061112 (2005). [CrossRef]
S. Nakagawa, N. Yamada, N. Mikoshiba, and D. E. Mars, “Second-harmonic generation from GaAs/AlAs vertical cavity,” Appl. Phys. Lett. 66(17), 2159–2161 (1995). [CrossRef]
V. Pellegrini, R. Colombelli, I. Carusotto, F. Beltram, S. Rubini, R. Lantier, A. Franciosi, C. Vinegoni, and L. Pavesi, “Resonant second harmonic generation in ZnSe bulk microcavity,” Appl. Phys. Lett. 74(14), 1945–1947 (1999). [CrossRef]
H. Cao, D. B. Hall, J. M. Torkelson, and C. Q. Cao, “Large enhancement of second harmonic generation in polymer films by microcavities,” Appl. Phys. Lett. 76(5), 538 (2000). [CrossRef]
L. M. Zhao and B. Y. Gu, “Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material,” Appl. Phys. Lett. 88(12), 122904 (2006). [CrossRef]
J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007). [CrossRef]

IEEE J. Quantum Electron.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]

J. Appl. Phys.

Y. Zeng, X. S. Chen, and W. Lu, “Optical limiting in defective quadratic nonlinear photonic crystals,” J. Appl. Phys. 99(12), 123107 (2006). [CrossRef]

J. Opt. Soc. Am. B

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, C. Me’riadec, and A. Levenson, “χ(2) semiconductor photonic crystals,” J. Opt. Soc. Am. B 19(9), 2094–2101 (2002). [CrossRef]

Opt. Lett.

Opt. Quantum Electron.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). [CrossRef]

Phys. Rev. B

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008). [CrossRef]
F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70(24), 245109 (2004). [CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ 2 interactions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(1 ), 016609 (2001). [CrossRef] [PubMed]
Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003). [CrossRef] [PubMed]
J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007). [CrossRef] [PubMed]

Phys. Rev. Lett.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998). [CrossRef]
P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004). [CrossRef] [PubMed]

Other

V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals (Springer, Berlin, 1997), Vol. 64, p. 125.

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