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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 5 — Mar. 8, 2004
  • pp: 841–846
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Aperiodic 1-dimensional structures for quasi-phase matching

Andrew H. Norton and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 12, Issue 5, pp. 841-846 (2004)
http://dx.doi.org/10.1364/OPEX.12.000841


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Abstract

We describe a method for designing 1-dimensional aperiodic poled grating structures of finite length that quasi-phase match multiple χ(2) processes. The poling functions for such gratings are best aligned, in terms of the dot product in Fourier space, with a design target. No restrictions are placed on the quasi-phase matching wave numbers. A grating designed for third harmonic generation is simulated.

© 2004 Optical Society of America

1. Introduction

One of the challenges in the field of wavelength conversion and harmonic generation is to design 1-dimensional poled grating structures that can simultaneously quasi-phase match several different χ (2) processes [1

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

7

7. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” http://arxiv.org/abs/nlin.PS/0311013 (2003).

]. Designing a 1-dimensional poled χ (2) grating structure entails the specification of a poling function, p:RR, of the general form

p(x)={±1forx[L2,L2],0otherwise,
(1)

where L is the grating length. The poling function specifies the sign of the χ (2) nonlinearity. For ferro-electrics such as LiNbO3, the sign of χ (2), as a function of position, can be engineered using the fabrication technique of electric field poling.

A χ (2) process involves the nonlinear mixing of two waves to produce a third wave at the sum or difference frequency. The process proceeds efficiently if the three wave vectors are quasi-phase matched by a reciprocal lattice vector associated with a strong Fourier coefficient of the grating. For example, the frequency sum process ωA +ωB =ωC in general requires quasi-phase matching (QPM) by a reciprocal lattice vector G=k C-k A-k B. The Fourier transform of the poling function is therefore required to have a strong component (G).

If only a single χ (2) process is to be quasi-phase matched then the efficiency of the process is highest for the poling function that has the largest QPM Fourier coefficient. This is well known to be a square wave with period 2π/|G|. If, however, QPM is required for several different χ (2) processes then it is not obvious how best to choose the poling function. At least four different grating design approaches can be found in the literature. These include approaches based on Fibonacci (or quasi-crystal) structures [1

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

]; modulation of the grating period [2

2. O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999). [CrossRef]

]; numerically optimized phase modulation [3

3. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO3 wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. 28, 558–560 (2003). [CrossRef] [PubMed]

]; and phase reversal approaches [4

4. M. L. Bortz, “Quasi-Phasematched Optical Frequency Conversion in Lithium NiobateWaveguides,” Ph.D. thesis, Stanford University (1994).

, 5

5. M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999). [CrossRef]

, 6

6. Z.-W. Liu, Y. Du, J. Liao, S.-N. Zhu, Y.-Y. Zhu, Y.-Q. Qin, H.-T. Wang, J.-L. He, C. Zhang, and N.-B. Ming, “Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,” J. Opt. Soc. Am. B 19, 1676–1684 (2002). [CrossRef]

], in which the poling function is taken to be a product of the form p(x)=p 1(x)…pn (x), where each pj (x) is a square-wave function.

The idea common to all these grating design approaches is that of introducing several parameters into the Fourier transform of the poling function, and then adjusting these parameters so as to obtain strong Fourier coefficients at the required phase matching wave vectors. It is clear that in such approaches the Fourier transform of the poling function may include many peaks that are irrelevant for QPM. This is potentially wasteful because by Parseval’s theorem ∫|(k)|2 dk=2πp(x)2 dx=2πL. Thus, unnecessary peaks in (k) at other than the required QPM wave vectors can only reduce the efficiency of the grating.

In Section 3 we illustrate this design approach by finding a 1-dimensional χ (2) poling function for third harmonic generation (THG) based on the two nonlinear processes [7

7. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” http://arxiv.org/abs/nlin.PS/0311013 (2003).

],

k1+k1+G1=k2,k1+k2+G2=k3.
(2)

Here k 1, k 2, k 3 are wave vectors for the fundamental, second harmonic and third harmonic respectively, and G 1, G 2 are quasi-phase matching wave vectors. In this example the χ (2) grating structure is required to have a Fourier transform with strong G 1, G 2 components. So the target transform (k) was chosen to be the piecewise constant function shown in Fig. 1, having two square peaks centered at G 1, G 2. Also shown in Fig. 1 is the Fourier transform (k) for the aperiodic grating of length L=1cm that is best aligned with the prescribed target. The discrepancy between (k) and (k) is of course due to the constraint that the poling function be of the form (1).

2. Notation

In this section we give our notation for Fourier transforms and for the real dot product that we need later. Let V(x) be the vector space with complex coefficients and basis vectors {uk (x)},

uk=12πeikx.
(3)
Fig. 1. (a) The target Fourier transform (k) (red) and the Fourier transform of the best aligned poled grating (k) (blue) of length L=1cm. The two QPM peaks were chosen for THG (see Section 5). (b) Close-up of the QPM peak at k=G 1. (c) The peak at k=G 2. The target peaks have half-widths ΔG=2π/L.
Fig. 2. Part of a THG poled grating defined by p(x)=sign(n(x)) where n(x)=w 1 cos(G 1 x)+w 2 cos(G 2 x). Values for wj and Gj are given in the text (Section 5). The grating length is L=1cm, of which 0.5mm is shown. Domains below fabrication resolution (≈1 µm for LiNbO3) can be deleted by inversion before the grating is simulated.

Vectors in V(x) are complex valued functions of the real variable x. Let fV(x) have components (k) with respect to the basis (3). Then

f(x)=f̂(k)ukdk=12πf̂(k)eikxdk,
(4)

and the components (k) define the Fourier transform of f (x). The complex valued inner product 〈,〉 is defined for f 1, f 2V(x) by

f1,f2=2πf1(x)f2(x)*dx.
(5)

The basis {uk (x)} is orthonormal with respect to Eq. (5), 〈uk, uk ′〉=δ(k-k′). The Fourier transform of f is then

f̂(k)=f,uk=f(x)eikxdx.
(6)

The space V(x) may also be considered as a vector space V′(x) with real coefficients and basis vectors {uk, iuk }, since for any fV(x) one may also write fV′(x) as

f=Re{f̂(k)}ukdk+Im{f̂(k)}iukdk.
(7)

A real valued dot product in V′(x) can be defined using the complex valued inner product in V(x),

f1·f2=12(f1,f2+f2,f1)
=2π(Re{f1(x)}Re{f2(x)}+Im{f1(x)}Im{f2(x)})dx.
(8)

The basis {uk, iuk } of V′(x) is orthonormal with respect to the dot product (8). Thus, in terms of the Fourier coefficients of f 1 and f 2, one has

f1·f2=(Re{f̂1(k)}Re{f̂2(k)}+Im{f̂1(k)}Im{f̂2(k)})dk.
(9)

3. The best aligned poling function for a given target

Let a target Fourier transform (k) be given. We say the poling function p(x) is best aligned with (k) if the dot product p·n is maximal amongst all possible poling functions of the form Eq. (1). From expression (9) for the dot product, it is clear that the property that p be best aligned with n is a natural way to specify that the peaks in (k) appear at the values of k required for QPM—one need only specify a suitable target. Moreover, Eqs. (1) and (8) lead to an explicit expression for the best aligned poling function. Since,

p·n=2πL2L2p(x)Re{n(x)}dx,
(10)

and p(x)=±1 for x∈[-L/2, L/2], the maximum value of Eq, (10) is achieved when

p(x)={sign(Re{n(x)})forx[L2,L2],0otherwise.
(11)

If (k) is chosen so that its inverse Fourier transform n(x) can be easily calculated, then Eq. (11) provides a simple expression for the poling function best aligned with n. In the following section we evaluate Eq. (11) in the case that the target (k) is a piecewise constant function with an arbitrary number of QPM peaks.

The Fourier transform of p(x) generally has no simple expression. However, (k) is easily calculated numerically by solving for the positions of the sign changes in Eq. (11). Let xj, j=2,…, r-1 be those roots of Re{n(x)} that lie in the interval [-L/2,L/2], and let x 1=-L/2, xr=L/2. Then

p̂(k)=j=1r1pjĥj(k),
(12)

where pj=p((xj +x j+1)/2) is the sign of the jth poled region and ĥ j(k) is the Fourier transform of the unit rectangular function with support on the jth poled region,

hj(x)={1forx(xj,xj+1),0otherwise,
(13)
ĥj(k)=ik(exp(ikxj+1)exp(ikxj)).
(14)

The curve for (k) shown in Fig. 1 was calculated using Eqs. (12) and (14).

4. Piecewise constant target

Here we evaluate expression (11) for the best aligned poling function in the case that the target transform (k) is a piecewise constant function with N different QPM peaks. We suppose QPM peaks are required at wave numbers Gj, j=1,…, N, with corresponding bandwidths 2ΔGj, j=1,…, N. A target Fourier transform (k) that encodes these design requirements in a simple manner is given by a weighted sum of rectangular functions Hj (k) centered at k=Gj ,

Hj(k)={1forGjΔGj<k<Gj+ΔGj,0otherwise,
(15)

From Eq, (11) it is clear that we may take n(x) to be real, in which case Re{(k)} is an even function of k and Im{(k)} is an odd function of k. Thus, we take

n̂(k)=j=1N(aj(Hj(k)+Hj(k))+ibj(Hj(k)Hj(k))),
(16)

where aj, bjR are weights that should be chosen optimally for the particular QPM problem (see Section 5 for an example). Applying the inverse transform (4) to (16) gives

n(x)=12πj=1N(aj(Hj(k)+Hj(k))+ibj(Hj(k)Hj(k)))eikxdk
=1πj=1N0(ajcos(kx)bjsin(kx))Hj(k)dk
=j=1N2sin(ΔGjx)πx(ajcos(Gjx)bjsin(Gjx))
=j=1N2sin(ΔGjx)πxwjcos(Gjx+ϕj)
(17)

where wj =(aj2+bj2)1/2 and cosϕj=aj/wj , sinϕ j=bj/wj . The poling function that is best aligned with the target transform (16) is therefore

p(x)={sign(Σj=1Nsin(ΔGjx)xwjcos(Gjx+ϕj))forx[L2,L2],0otherwise.
(18)

In practice, we find that if ΔGj is too large then the QPM peaks in (k) simply split rather than become wider. The most satisfactory results seem to be obtained for ΔGj =2π/L, in which case the first zeros of the sinc factor in (18) are at xL/2. Expression (18) then simplifies to

p(x)={sign(Σj=1Nwjcos(Gjx+ϕj))forx[L2,L2],0otherwise.
(19)

This is the form of the poling function used in the THG example of the following section.

5. An aperiodic poled grating for THG

Our poled grating design method is illustrated here for THG based on the two χ (2) processes (2). The grating is designed to operate at 140ΔC with a fundamental wavelength of λ=1550nm. We suppose the grating material is LiNbO3 with quadratic susceptibility χ (2)(x)=χp(x), where χ=41×10-12 m/V. The refractive indices at the first, second and third harmonics are respectively n 1=2.1430, n 2=2.1848 and n 3=2.2487 [8

8. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22, 1553–1555 (1997). [CrossRef]

]. The wave numbers are then kq=qωnq/c, for q=1,2, 3. Using (2) one finds that G 1=0.33865×106m-1 and G 2=0.94639×106m-1. Taking ϕj =0 in (19) we set p(x)=sign(w 1 cos(G 1 x)+w 2 cos(G 2 x)) for x∈(-L/2,L/2) and 0 otherwise, for grating length L=1cm. The weights w 1=5.2874×10-3 and w 2=4.6951×10-3 are normalized so w12+w22=L 2/2. The ratio w 1/w 2 was adjusted so as to maximize THG efficiency (for THG there exists a critical ratio of QPM Fourier coefficients [9

9. C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000). [CrossRef]

, 10

10. A. H. Norton and C. M. de Sterke, “Two-dimensional poling patterns for 3rd and 4th harmonic generation,” Opt. Express 11, 1008–1014 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008. [CrossRef] [PubMed]

]). Figure 1 shows the QPM peaks in (k) and (k) and Fig. 2 shows n(x) and p(x).

Operation of the THG grating was simulated by numerical integration of the slowly varying amplitude THG equations. For an arbitrary poling function p(x), one has

da1dx=iωn1cχp(x)(a2a1*eiG1x+a3a2*eiG2x),
(20)
da2dx=iωn2cχp(x)(2a3a1*eiG2x+a12eiG1x),
(21)
da3dx=3iωn3cχp(x)a1a2eiG2x,
(22)

where aq (x) is the complex amplitude of the q-harmonic, q=1, 2, 3. Figure 3 shows the relative energy fluxes for the three waves. The boundary conditions were a 1(0)=1×107V/m, and a 2(0)=a 3(0)=0. Equations (20)(22) were integrated using a 4th order Runge-Kutta (RK) method with the step size dynamically adjusted so that the boundaries of the poled domains (discontinuities in p(x)) coincided with an RK step boundary. The THG curves obtained for grating phases ϕj ≠0, were visually indistinguishable from those of Fig. 3 for ϕj =0. The removal of small grating features (e.g., domains of width <1 µm) had a small effect on the THG curves, that could be compensated for by small adjustments to the ratio w 1/w 2.

Fig. 3. THG simulation. Relative energy of the fundamental, 2nd and 3rd harmonic waves (red, green and blue curves respectively) as a function of distance through the poled grating.

6. Conclusion

We have described a new method for designing 1-dimensional aperiodic poled grating structures that support multiple QPM processes. The method has been illustrated with the design of a 1cm long THG grating. This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program. CUDOS (the Centre for Ultrahigh bandwidth Devices for Optical Systems) is an ARC Centre of Excellence.

References and links

1.

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

2.

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999). [CrossRef]

3.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO3 wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. 28, 558–560 (2003). [CrossRef] [PubMed]

4.

M. L. Bortz, “Quasi-Phasematched Optical Frequency Conversion in Lithium NiobateWaveguides,” Ph.D. thesis, Stanford University (1994).

5.

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999). [CrossRef]

6.

Z.-W. Liu, Y. Du, J. Liao, S.-N. Zhu, Y.-Y. Zhu, Y.-Q. Qin, H.-T. Wang, J.-L. He, C. Zhang, and N.-B. Ming, “Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,” J. Opt. Soc. Am. B 19, 1676–1684 (2002). [CrossRef]

7.

S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” http://arxiv.org/abs/nlin.PS/0311013 (2003).

8.

D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22, 1553–1555 (1997). [CrossRef]

9.

C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000). [CrossRef]

10.

A. H. Norton and C. M. de Sterke, “Two-dimensional poling patterns for 3rd and 4th harmonic generation,” Opt. Express 11, 1008–1014 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008. [CrossRef] [PubMed]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4400) Nonlinear optics : Nonlinear optics, materials

ToC Category:
Research Papers

History
Original Manuscript: January 21, 2004
Revised Manuscript: February 19, 2004
Published: March 8, 2004

Citation
Andrew Norton and C. de Sterke, "Aperiodic 1-dimensional structures for quasi-phase matching," Opt. Express 12, 841-846 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-841


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References

  1. K. Fradkin-Kashi and A. Arie, �??Multiple-wavelength quasi-phase-matched nonlinear interactions,�?? IEEE J. Quantum Electron. 35, 1649�??1656 (1999). [CrossRef]
  2. O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, �??Engineering competing nonlinearities,�?? Opt. Lett. 24, 1413�??1415 (1999). [CrossRef]
  3. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, �??Multiple quasi-phase-matched LiNbO3 wave length converter with a continuously phase-modulated domain structure,�?? Opt. Lett. 28, 558�??560 (2003). [CrossRef] [PubMed]
  4. M. L. Bortz, �??Quasi-Phasematched Optical Frequency Conversion in Lithium Niobate Waveguides,�?? Ph.D. thesis, Stanford University (1994).
  5. M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, �??Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,�?? Opt. Lett. 24, 1157�??1159 (1999). [CrossRef]
  6. Z.-W. Liu, Y. Du, J. Liao, S.-N. Zhu, Y.-Y. Zhu, Y.-Q. Qin, H.-T. Wang, J.-L. He, C. Zhang, and N.-B. Ming, �??Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,�?? J. Opt. Soc. Am. B 19, 1676�??1684 (2002). [CrossRef]
  7. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, �??Multistep parametric processes in nonlinear optics,�?? <a href="http://arxiv.org/abs/nlin.PS/0311013"> (2003).http://arxiv.org/abs/nlin.PS/0311013</a>
  8. D. H. Jundt, �??Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,�?? Opt. Lett. 22, 1553�??1555 (1997). [CrossRef]
  9. C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, �??Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,�?? Opt. Lett. 25, 436�??438 (2000). [CrossRef]
  10. A. H. Norton and C. M. de Sterke, �??Two-dimensional poling patterns for 3rd and 4th harmonic generation,�?? Opt. Express 11, 1008�??1014 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008</a>. [CrossRef] [PubMed]

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