## Abstract

Novel engineered step-chirped gratings (SCG) for broadband frequency converters based on quasi-phase matched second harmonic generation in MgO-doped lithium niobate waveguides have been theoretically modeled and simulated. It is shown mathematically that engineered apodized gratings can flatten the efficiency response. Also, it is verified that the bandwidth and flatness of an apodized SCG can be improved extensively with decreasing the number of segments and increasing the apodization ratio, respectively. Further, we show enhancing the minimum width of the line of the gratings to 1 micron for easing fabrication, almost all the beneficial effect on the efficiency response of an apodized SCG are maintained. The possibility of increasing the width of the poled lines and the increase in the chirp step due to use of the SCG structure, may provide more convenient route for fabrication and poling.

©2008 Optical Society of America

## 1. Introduction

Frequency converters using nonlinear optical waveguides in lithium niobate have been extensively studied over the past years. The most widely used method is quasi phase matching (QPM) in poled grating structures in waveguide frequency doublers in order to enhance the nonlinear effect [1]. There has been increasing interest in broadening the bandwidth of such waveguide-based second harmonic generation (SHG) devices [2] because there are many applications in pulse compression and ultrafast optical signal processing [3]. Some work has been done on waveguide structures to increase the bandwidth and modified QPM grating structures have been proposed to broaden the phase matching bandwidth [4,5], whilst the bandwidths are still not adequate and waveguide devices have not been demonstrated. The use of a chirped grating structure for SHG instead of one with a uniform period offers the benefit of a larger bandwidth. Although the performance of the uniform QPM devices depends strongly on the operational temperature, a chirped one can easily overcome this problem. Furthermore, periodically poled MgO-doped LiNbO_{3} waveguides have the advantage of higher optical damage threshold, and can also operate at room temperature [1]. However, there are some problems using chirped gratings. One problem is the smallness of change in the linear-chirped grating (LCG) period [6] which is typically around a few hundred picometers for a converter with a large flat bandwidth of several nanometers. We already proposed that the step-chirped gratings (SCG) can solve this problem [7]. Using the SCG enables us to increase the period change, increasing convenience for fabrication, whilst the bandwidth and efficiency remain almost the same in comparison to the LCG. Another problem is the noticeable ripples on the conversion efficiency curves for SHG. Our approach to solve this problem is to impose some apodization [6,8]. This can be done by changing the duty ratio of the poled regions which helps to remove the ripples and achieves a nearly flat response. However, the problem still exists that there is a need for higher conversion bandwidth, e.g., to generate a broadband source having desired profile with more than 30 nm near short wavelength of 775 nm for optical coherence tomography (OCT) [9]. Here, this problem can be solved with the special design of few-segment engineered SCGs in such a way to provide us with a controllable flat broadband response and to facilitate the poling and fabrication processes.

In this paper, we show mathematically that flattening of the phase matching response of engineered gratings is possible and propose an effective numerical approach to apodize and flatten the second harmonic conversion efficiency bandwidth. By design of inverted domains in the form of few-segment step-chirped gratings in poled MgO-doped lithium niobate waveguides, the bandwidth may be broadened with a smaller number of segments and larger step chirp. Further, we show that by increasing the starting width of the line of the poled regions in the apodized SCG, it is still possible to attain almost the same efficiency response with a small price of a slight increase in the non-uniformity of conversion efficiency over the bandwidth.

## 2. Theory of apodized chirped gratings for flattening of efficiency response

In frequency doublers based on QPM-SHG any phase mismatch between the laser wavelength and gratings should be resolved. However, uncertainty in the propagation constant, error in the fabrication process and fluctuation of the laser wavelength and temperature variations may increase the mismatch. A chirped grating enables us to solve these problems by broadening the phase matching bandwidth but it leads to ripples in the broadband conversion efficiency curve. The ripple feature is created by the interference among the constituent phase matching spectra originating from different positions in the gratings. In the following, we consider mathematically how to remove these noticeable ripples. We use the wave equation by taking the Fourier transform and invoking the slowly varying envelope approximation as [10]

The electric field *E*̃(*z*,*ω*) propagates in the presence of nonlinear polarization *̃ _{NL}*(

*z*,

*ω*) in the nonlinear medium where the polarization is only a perturbation to the system, and both the electric field and nonlinear polarization are scalar plane waves. Also,

*ε*is the linear permittivity and

*µ*

_{0}is the permeability. Describing the fields in complex notation as $\tilde{E}(z,\omega )=\frac{1}{2}\{\epsilon (z,\omega ){e}^{-\mathrm{jk}\left(\omega \right)z}+{\epsilon}^{*}(z,-\omega ){e}^{jk\left(-\omega \right)z}\},$, and assuming the slowly varying envelope approximation, Eq. (1) can be simplified approximately as

where *n*(*ω*) is the refractive index and *c* is the light speed. It is possible to define the spectrum of nonlinear polarization leading to second harmonic field with the relation

where ⊗ stands for convolution and _{eff}*d* is the effective nonlinear coefficient determined by the field polarizations and the second-order tensor of the SHG medium. Thus, Eq. (2) gives the equation governing the second harmonic (SH) field as

where
$\kappa =\frac{{\omega}_{0}}{{n}_{2\omega}c}$ and *Δk*(Ω,*ω*′)=*k*(Ω)-*k*(*ω*′)-*k*(Ω-*ω*′) which is a function of the frequencies at the second harmonic (Ω) and fundamental harmonic (*ω*′). By integrating Eq. (4) over the total length, the SH spectrum at the output of the converter of length *L _{t}* is

In Eq. (5), the phase matching spectrum can be engineered by controlling effective nonlinear coefficient. Here, the SH power spectrum is of great importance. In other words, the effort should be concentrated to generate a broad power spectrum controlling *d _{eff}*(

*z*) while we alter the QPM grating period, such that every local period Λ(

*z*) contributes to a constituent phase matching spectrum. Nevertheless, the ripple structure in the efficiency curve versus wavelength for chirped gratings with a constant effective SHG coefficient is associated with sharp edges in the effective SHG coefficient. This problem can be solved by introducing some form of apodization on the effective SHG coefficient. Apodization, i.e. changing the nonlinearity slowly such that the propagating light does not notice the change impacts conversion efficiency. Therefore, proposing the apodized chirped gratings, we can engineer the effective nonlinear coefficient to flatten the efficiency curve of broadband SHG-based frequency doublers. Further reduction of fluctuation in the efficiency response is also anticipated by modifying the duty ratios of inverted domains in apodized chirped gratings.

## 3. Engineering of apodized step-chirped gratings

A step-chirped grating is a structure which enables one to obtain a wide phase matching bandwidth with a reasonable chirp step. It is formed by dividing the length of the gratings into several uniform grating sections of constant segments (periods) with a small period increment between successive sections called a chirp step. To achieve a flattop wide-bandwidth response, an apodized step-chirped grating (ASCG) structure with few segments in MgO-doped lithium niobate waveguides is proposed as shown in Fig. 1(a). In this structure, the total grating length *L _{t}* has been divided into several sections of constant period but slightly chirped, each with a chirp step ΔΛ between the adjacent sections. The central region with the length

*L*is composed of

*p*sections with equal duty-cycle ratios. Each of the two adjacent sides have apodized regions with the lengths

*L*′ and

*L*″ consisting of

*p*′ sections. At the input, a constant duty-cycle ratio for each section is

*a*=

_{s}*s*/2

*p*′, (

*s*=1, 2,…,

*p*′) and changes symmetrically towards the exit. Each section,

*m*consists of

*n*segments, so

*L*=

_{m}*n*Λ

*where Λ*

_{m}*=Λ*

_{m}_{1}+ΔΛ(

*m*-1), (

*m*=1,2,…,

*p*+2

*p*′) with Λ

_{1}as the period of the first section. To obtain the total second harmonic (SH) wave and efficiency, we can consider

*p*+2

*p*′ uniform gratings. The condition for QPM in the first section is

*β*

_{2ω}-2

*β*=2

_{ω}*π*/Λ

_{1}where

*β*and

_{ω}*β*

_{2ω}are the mode propagation constants of the fundamental and second harmonic waves, respectively. The QPM period of the grating is calculated by finding the effective indexes of waveguides and with the help of the Sellmeier expression for the MgO-doped LiNbO

_{3}waveguide [11]. As the waveguide dispersion is only important in waveguides with very small effective mode areas, it is neglected here. Assuming the grating has a binary modulation, the effective SHG coefficient of the region within a grating period in each section can be written as

*d*=(2

_{m}*d*

_{33}/

*π*)sin(

*π*

*a*) where

_{m}*d*

_{33}is the SHG coefficient of MgO-doped LiNbO

_{3}. Thus, in each section we have a uniform grating but with slightly different

_{m}*d*from its adjacent section by changing the duty-cycle ratio

_{m}*a*which denotes the ratio of one region to the period. Consequently, this model of the typical domain-inverted QPM gratings has increasing and decreasing duty-cycle ratios and normalized effective SHG coefficients, at the beginning and at the end of the structure.

According to Fig. 1(a), *r*=(*L*′+*L*″)/*L _{t}* is the apodization ratio which is the ratio of the total length with varying effective SHG coefficients to the total length of the structure. Moreover,

*r*=0 reveals an unapodized device while for 0<

*r*<1 one can find partially apodized devices and

*r*=1 manifests a fully apodized device. Here, the governing coupled mode equations are solved numerically with full pump depletion and the SHG conversion efficiency is defined as the power ratio of output to input,

*η*=

*P*/

_{out}*P*. The effective cross section, total interaction length, period of the first section and refractive index difference in the MgO-doped lithium niobate waveguide are assumed to be 20

_{in}*µ*m

^{2}, 50mm, 18.4

*µ*m and 0.01, respectively. Also, we consider the starting width of the line of the poled domain of

*a*

_{1}Λ

_{1}=100 nm, a chirp step of ΔΛ=2nm and an input power of 50mW in

*P*=50mW. Figure 1(b) depicts the efficiency of a six-segment SCG (

_{in}*r*=0) and ASCGs versus the fundamental harmonic wavelength. Based on the apodization approach, the flattening of conversion efficiency and reduction of ripples in the efficiency response of SHG-based frequency doubler by the change in the duty-cycle ratio of inverted domains at the beginning and end parts of the few-segment ASCGs are demonstrated. It shows how the ripples and spectral ears of the ASCG may be suppressed with increasing

*r*. For a small apodization ratio,

*r*=0.25, the efficiency is nearly ripple-free with the bandwidth of about 55 nm except for the two residual “ears” at the edges. The maximum flat efficiency of about -21.6 dB can be achieved. Further suppression of these spectral ears can be achieved by introducing longer apodization regions.

The maximum flat efficiency of about -21.7 dB for *r*=0.5 and 0.75 with the nearly flat bandwidth of about 45 nm and 35 nm can be achieved, respectively whilst there is only a small fluctuation of ±0.05 dB in the response. It is seen that with increasing *r*, it is possible to obtain improved forms of efficiency curves which are appropriate for some applications, e. g., the shape of the spectrum of the light source is an important issue in OCT because it affects the dynamic range of the scanner [9]. For *r*=1 a fully apodized structure leads to a Gaussianlike efficiency curve. However, the reason for changing the form of the efficiency response is that the efficiency is proportional to the effective SHG coefficient and correspondingly the duty-cycle ratio. Further, for the same waveguide length in MgO-doped lithium niobate, the ASCG increases the step chirp in comparison to that of the ALCG. To achieve similar results, the ASCG (6-segment) structure needs a larger chirp step of around 2 nm in comparison with that of the ALCG (1-segment) structure of about 330 pm which may prove to be more convenient for the poling process [7]. The small values of domain size may be difficult to fabricate, as kindly indicated by the Reviewer. However, we note that the poled region less than 1 micron [12] and a few nanometers of chirp between the poling periods of adjacent domain gratings [6,13] have been demonstrated in QPM devices. Furthermore, chirp below the accepted resolution of the e-beam writing machine has also been previously demonstrated in linearly chirped fiber Bragg gratings [14].

Figure 2(a) shows the efficiency of ASCG structure for *r*=0.40 to obtain the maximum flat bandwidth, assuming different large chirp steps, waveguide lengths and input power parameters. Fig. 2(a) demonstrates that keeping the chirp step around 2 nm leads to a near flat response and the enhancement of the chirp step leads to a slanted response with increasing bandwidth, and a decrease in efficiency. Nevertheless, the slight slope seen in the response for ΔΛ=2.5nm and 3 nm is basically a result of nonlinear wave-number variation 2*π*/Λ(*z*). Moreover, the noteworthy feature of the ASCG structure is that when the input power is halved (25 mW), the efficiency is also halved and the bandwidth is nearly unchanged. On the other hand, decreasing the waveguide length to one half (25 mm), decreases the bandwidth to almost one half while the efficiency remains the same. The reasons for this are based on the structure of the ASCG which consists of several unchirped sections with nearly equal lengths (due to chirped structure the periods change between sections and therefore they are different in length, e.g., about 12 nm for 6-segment ASCG), each of which produces almost the same efficiency for the phase-matched wavelength of that section (and this is proportional to the squared length of section) and the similar bandwidth with a small shifted central wavelength. Thus, as we increase the sections with a small step chirp between them, we consequently increase the length and the equivalent bandwidth of the whole step-chirped structure while the mean efficiency remains almost unchanged. At the same time as the length of each section is very small, its efficiency changes linearly with power, as in unchirped phasematching. These results are very useful for the design of the ASCG as they show that the bandwidth and efficiency are almost linearly proportional to the length and input power, respectively.

One challenge in the previous design is the smallness of the initial width of the line of the poled regions in the apodized parts which is assumed 100 nm. In order to examine the effect of enhancing the width of the lines, we consider a 1000-nm starting line width (*a*
_{1}Λ_{1}). Figure 2(b) illustrates the efficiency curves versus input wavelength for the 5-, 6- and 7-segment ASCG structures for *r*=0.40 and *L _{t}* ≈ 50mm. It shows that using

*a*

_{1}Λ

_{1}=1

*µ*m introduces a small fluctuation in the response of ±0.15 dB. Nonetheless, increasing the minimum line width may also make the fabrication of the few-segment apodized step-chirped gratings with the large chirp steps more convenient.

Further, there are other notable points in Figs. 2 which are useful for controlling the flat-top bandwidth of the phase matching response of the ASCG structure. Fig. 2(a) shows for a constant length, raising the chirp step, increases the bandwidth (and the slant in the response) and decreases the efficiency. On the other hand, Fig. 2(b) demonstrates that decreasing the number of segments for the same length, increases the bandwidth and decreases the efficiency. Thus, it is possible to achieve higher bandwidth when there are fewer segments in more sections in the same length and it is a good technique to design highly-broadband frequency converters. Consequently, there is a trade-off among the acceptable slant, bandwidth and efficiency for choosing the parameters and a compromise must be done for a design. However, the efficiency can still be improved with some techniques, e.g., using the waveguides with smaller effective cross sections [7], which are under further investigation using the aforesaid substrate.

## 4. Conclusion

We have shown that using engineered inverted domains in the form of step-chirped gratings in MgO-doped poled lithium niobate waveguides, the efficiency curves of frequency doublers have been broadened and flattened significantly. The efficiency curves with the bandwidth as large as 65 nm (which can be increased) can be smoothed dramatically and flattened with the ripples being reduced to less than ±0.15 dB, even with 1000-nm starting line width of poled region, and the spectral ears can be significantly suppressed at the cost of longer apodization lengths. We believe that the use of the engineered ASCG in this substrate with the modified larger starting line width and step chirp is a highly flexible technique for design, and for easing the fabrication requirements of highly-broadband frequency converters.

## Acknowledgments

This research is supported by a National Science and Engineering Research Council of Canada Strategic Grant, a Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT) Team grant and the Canada Research Chairs Programs. The authors would like to thank Prof. C.-Q. Xu for helpful advice.

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