We have analyzed optical parametric interaction in a 2D NPC. While in general the nonlinear coefficient is small compared to a 1D NPC, we show that at numerous orientations a multitude of reciprocal vectors contribute additively to enhance the gain in optical parametric amplification and oscillation in a 2D patterned crystal. In particular, we have derived the effective nonlinear coefficients for common-signal amplification and common-idler amplification for a tetragonal inverted domain pattern. We show that in the specific case of signal amplification with QPM by both G10 and G11, symmetry of the crystal results in coupled interaction with the corresponding signal amplification by G10 and G1,-1. As a consequence, this coupled utilization of all three reciprocal vectors leads to a substantial increase in parametric gain. Using PPLN we demonstrate numerically that a gain that comes close to that of a 1D QPM crystal could be realized in a 2D NPC with an inverted tetragonal domain pattern. This special mechanism produces two pairs of identical signal and idler beams propagating in mirror-imaged forward directions. In conjunction with this gain enhancement and multiple beams output we predict that there is a large pulling effect on the output wavelength due to dynamic signal build-up in the intrinsic noncollinear geometry of a 2D NPC OPO.
© 2008 Optical Society of America
In a two-dimensional χ (2) nonlinear photonic crystal (2D NPC), the sign of χ (2) is reversed periodically in two directions . This produces a nearly infinite number of reciprocal vectors in the 2D reciprocal lattice. Each reciprocal vector corresponds to one or more phase-matching solution. The 2D structure thus provides a much greater flexibility in the quasi-phase-matching (QPM) of wavelength conversion processes, permitting simultaneous efficient multi-wavelength second harmonic and sum frequency mixing to occur [2,3]. Similarly the 2D NPC makes parametric generation and amplification interesting. The multitude of off-angle reciprocal vectors makes possible either multi-wavelength parametric generation and amplification or the emission of the same generated wavelength in more than one direction. These possibilities may find potentially interesting applications in, for example, simultaneous multiple-trace-gas detection  where several wavelengths are required at the same time or in multichannel information processing where efficient cascaded conversion of several wavelengths are necessary . However, parametric generation and amplification in a 2D NPC has not been reported even though 1D NPC QPM optical parametric oscillators (OPOs) were demonstrated from the early days of periodic poling for QPM . This is perhaps because the effective χ (2) in a 2D NPC is a lot smaller than in the 1D case due to its structure factor [1,7] so that generally the parametric gain would be too small. In view of the potential application of multi-wavelength and multi-directional emissions from 2D NPC parametric generators it is our objective to explore 2D NPC for situations where the parametric gain can be substantially enhanced to turn 2D NPCs into a competitive parametric medium.
We have analyzed optical parametric generation and oscillation in a 2D NPC. In this paper we show that in several circumstances more than one reciprocal vector can act together to provide a higher QPM optical gain. Moreover, in certain geometries the gain can approach that in the 1D case without requiring merging of the structure into a 1D structure. Furthermore a pair of signal (and the idler) beams of identical wavelength can be generated simultaneously and emitted symmetrically on opposite transverse sides of the pump beam. These are unique properties of a 2D NPC parametric device that may find interesting applications in remote sensing or information processing. Here we describe both the theoretical derivation and numerical simulation of these circumstances to guide the design of such a 2D NPC parametric generator or parametric oscillator. The analysis is for a tetragonal inverted domain structure and for the reciprocal vectors Gmn, m,n=0, ±1 which results in the highest effective parametric gain. However the analysis should be readily generalized to apply to any of the five Bravais structures of the 2D NPC and their complete set of reciprocal vectors.
2. Parametric interaction in 2D NPC
A 2D NPC with a tetragonal inverted domain structure can be represented by a reciprocal lattice shown in Fig. 1. In this lattice structure, the reciprocal vectors Gmn are
where Λx and Λy are the periodicities along the x and y axes; m and n are integers. For QPM parametric amplification, momentum conservation of the pump, signal, and idler waves gives
where kp, ks, and ki are the wave vectors of the pump, signal, and idler waves, respectively. Suppose the pump beam propagates parallel to one of the unit vectors in the structure, then the angles of kp and ks relative to the unit vector are 0 and θs respectively. For a given G, Eq. (2) can be written in scalar form as
where k=kp-G, θ and θl are the angles of k and ki respectively. A wavelength vs. θs or θl curve for G can be constructed from Eq. (3) and (4). Since every reciprocal vector Gmn contributes to a tuning curve there is an infinite set of curves for a single 2D NPC parametric generator. In this paper, we focus on the vectors G10, G11, and G1,-1 that have the highest strength and therefore show the most prominent effects. The curves associated with these three vectors are shown in Fig. 2. There are three sets of curves, each set representing a G vector indicated at the degenerate point of the set. These curves are calculated with the Sellmeier equations from reference 8 to serve as an example case and uses 1064 nm as the pump beam with Λx=Λy=29.5 µm, and the crystal temperature is 157°C. The curves has a mirror symmetry at θs=0 or θi=0 due to the symmetry of the tetragonal structure. Two features unique to 2D NPC can be discerned from these curves. The first is that for every θs or θi there can be more than one wavelength that satisfies the QPM condition, meaning multiple wavelengths can have gain in every direction. The second is that there are several locations, some of which are designated by letters C through F in Fig. 2, where the signal curves or the idler curves intersect. At these locations, two reciprocal vectors contribute together to the nonlinear interaction of generating the signal or the idler wavelength. The parametric gain there can therefore be expected to be enhanced as we shall see in the next section.
3. Theoretical derivation of the parametric gain
Assume that all of the electric fields are along the z-axis, and the waves are propagating along the x-axis, and suppose the pump, signal, and idler are plane waves and that there is no walk-off problem, i.e., the beam radiuses are large compared to the length of interaction, and further assuming slowly varying amplitudes and paraxial propagation, then the parametric interaction can be described by the following coupled equations :
where Ep(x), Es(x), and Ei(x) are the electric field amplitudes in E(r)=ẑE(x) exp[i(wt-k·r)], and Δ k=kp-ks-ki is the momentum mismatch. In a 2D NPC, the nonlinear coefficient d 33(r) can be expressed in the form of a Fourier series :
where Gmn are the reciprocal vectors. The phase term exp[-i Δk·r] in Eq. (5) is cancelled by the phase term of d 33(r) if one of the reciprocal vectors Gmn is equal to Δk. In the absence of pump depletion, Ep(x) is a constant. Eq. (5) leads to
Letting Ei(0)=0 gives the solution:
This is equivalent to the standard parametric gain expression for birefringent phase-matched interaction with an effective nonlinear coefficient deff=|amn|d 33.
Common-signal amplification: In the case when two reciprocal vectors G1 and G2 with strength a 1 and a 2 couple the pump beam to the same signal field Es such as point C in Fig. 2 so that the signal wavelength is the same but the idlers E i1 and E i2 have different propagating directions, the coupled equations become
With E i1(0)=E i2(0)=0, the amplification to Es is the solution Es(x)=Es(0)cosh(gx), where
Hence effective nonlinear coefficient is which is larger than that given in Eq. (9). The result shows that the increase is due to a simple addition of contribution from two paths of interaction.
Common-idler gain: Similarly when two reciprocal vectors G1 and G2 with strength a1 and a2 coupling the pump beam to a common idler Ei but two signals E s1 and E s2, which have the same wavelength but different propagating directions (point D in Fig. 2), the coupled equations then are
The solution to Eq. (12) assuming E s1(0)=E s2(0)=0 is Ei(x)=Ei (0) cosh(gx) where g is defined in Eq. (11). For amplification of E s1 with a common-idler gain, we have E s2(0)=Ei(0)=0 and the solution
where as in Eq. (11). Comparing this to the case of a common signal, the gain coefficient is identical to the common signal case but the signal growth is reduced by a constant factor of .
In Fig. 2, point C (θs=0°, θ i1=2.8°, and θ i2=-2.8°) has a common-signal amplification using G11 and G1,-1. Point D (θi=0°, θ i1=1.4°, and θ i2=-1.4°) has a common-idler amplification. Point E presents a special situation that is particularly more interesting. At this location the vectors G10 and G11, and G1,-1 are all involved in the amplification process. The phase-matching diagram is shown in the left of Fig. 3 with the output directions of the signal and the idler beams of an OPO shown on the right side of the figure. First, there is the common-signal amplification with G10 and G11, generating k s1, k i1, and k i2. Because of mirror symmetry, k i1 and k i2 are exactly the same as k i3 and k i4 respectively. Consequently, these two idlers generate k s2 using G10 and G1,-1. The two signals and two idlers thus interact coherently with each other using QPM from the three reciprocal vectors. The coupled equations are
Eq. (14) can be written in matrix form as
For a given d 33(r), if we can choose the origin of the unit cell such that a 10, a 11, and a 1,-1 are all real and positive, and let a 11=a 1,-1, the square roots of the eigenvalues of the matrix are a 10+a 11 and |a 10-a 11|. The effective nonlinear coefficient is the square root of the larger eigenvalue of the matrix. Therefore,
Note that this is different from , the gain due to common signal contribution from G10, G11, and G1.-1 as derived in Eq. (11). This is because the mechanism is different from a straight-forward common-signal interaction.
In general 2a 10>a 11 so that the signal gain at this point is larger than purely from common signal interaction. Note that generally the parametric gain in a 2D NPC is smaller than in the 1D case [3,7]. However, because of these enhancements to the gain we show below with a specific example that in certain configurations the gain can reach that of the 1D QPM case.
4. Numerical simulation
We illustrate the effects and consequences of the overlapping gain by way of a 2D PPLN crystal that has a hexagonal inverted domain pattern with a filling ratio of 18.75%  as shown in Fig. 4. The dimensions of the unit cell are 29.5×29.5 µm2. Also shown in Fig. 4 is the pattern of the perfect case of a square inverted domain (a filling ratio of 25%). The normalized Fourier coefficients amn in Eq. (6) for these two patterns can be calculated and the values for m=1, n=0, ±1 are shown in Table 1. For reference, the coordinates (λs, θs) for the points of intersection A–F in Fig. 2 for this pattern are given in table 2. We now proceed to calculate the optical parametric gain and the corresponding output wavelength for a singly-resonant OPO that resonates at the signal wavelength.
In a 2D NPC OPO, satisfying the QPM condition means that the pump, signal and idler are non-collinear with the exception of the case of QPM with the reciprocal vector G10. A finite pump beam size will hence lead to walk-off in the spatial overlap of the pump, signal and idler beams. In view of this the analytic form shown in Eq. (9) is not adequate in describing the parametric gain. In this simulation we take into account the beam walk-off to mimic the case of a real experiment. Because the periodic domain varies in the x-y plane and propagation of the beams lie only in that plane, the change caused by the parametric interaction to the z-dependence of the field is negligible and can be ignored in the present consideration. For every x, the fields are a function of y. We fix the pump beam to propagate in the x direction, parallel to G10 of the QPM pattern. The pump beam is assumed to be Gaussian. We include a flat-flat linear optical cavity whose axis in free space is at an angle θm relative to G10. The mirror angle θm determines the direction of the signal beam that travels at an angle relative to the pump beam inside the crystal. Since the signal direction in free space is normal to the cavity mirror, for small angles this angle in the 2D NPC will be equal to θm divided by the refractive index of the material. Therefore, we define a signal angle which for infinitely narrow beams is equal to θs. With this cavity we calculate the roundtrip signal gain of each signal wavelength. This in essence will be the single-pass parametric amplification gain for a given and a finite pump beam diameter.
We use broadband white noise as the initial input signal and idler. The pump, signal and idler fields interact in the crystal as described by Eq. (5). The waves exit the crystal and the signal is reflected by the cavity mirrors and re-enters the crystal. The mirrors and the crystal are assumed broadband and lossless. The increase in the signal intensity is calculated following each pass through the crystal to obtain the roundtrip parametric gain. In the beginning the signal build-up is small as a large portion of the “noise” signal is not phase-matched and is dissipated. After several round trips, the increase in signal intensity and therefore the gain reaches a stable value. Eventually the gain decreases due to the depletion of the pump beam. The stable value before pump depletion is then the parametric gain for the selected direction in this 2D NPC. The calculation is repeated over a range of wavelengths to map out the gain as a function of signal wavelength at a given and this leads to a gain spectrum. The wavelength with the highest gain in the spectrum is the output signal wavelength for a free-running OPO at the given .
The result of the calculation is shown in Fig. 5 which displays the gain coefficient expressed as the gain-length product gL in color-coded amplitude as a function of the signal wavelength and the angle . This gain is calculated for a Gaussian pump beam waist w0 of 160 µm (Ep(y)=Ep(0)exp(-y 2/w0 2)). The crystal length is 2.69 cm. The cavity length is 5.5 cm and the peak intensity is 90 MW/cm2. These quantities are chosen to match crystals that can readily be fabricated. To help with interpreting the result, we superimpose on the figure the tuning curve (solid lines) from Fig. 2. It is immediately obvious that there are three curves for the gain with close but not exact overlap with the tuning curves corresponding to QPM by G10, G11, and G1,-1. To help with the analysis of this figure, we plot the largest gL product for each angle in Fig. 6(a) and the wavelength that corresponds to the largest gain at the corresponding angle in Fig. 6(b). Also shown is the curve for the second largest gain peak for the angle. We see now that gL for G10 is largest at . As increases, the gain falls off because of increasing walk-off between the pump and the signal. The curve with a lower gain corresponds to QPM by G11. At this gain benefits from a common-signal enhancement (point C in Fig. 2 and Eq. (11)) and results in a peak there. But this gain is less than that for G10 because the deff is smaller and the idler is off-angle and walks off to result in a smaller gain for the signal. Away from the gain due to G11 is naturally smaller than for G10 since the Fourier coefficient for higher order QPM is smaller.
The most unique and significant part of this 2D NPC gain is in the region between and 0.8° in the vicinity that corresponds to point E in the tuning curve. We can see from both Fig. 5 and Fig. 6(a) that the two gain curves merge into one and the gain shows a prominent peak at close to that for point E. As explained in the previous section this is a consequence of the simultaneous coupled interaction of G10, G11, and G1,-1 that dominates the gain in this region. The enhancement here is substantial and results in a large increase of the calculated gL in this region to a value that is higher than that of G10 at 0°. As exceeds 0.8°, the gain of G10 continues to fall off, while the gain of G11 slightly increases because of reduced walk-off between signal and idler (point B). There is a small peak close to 1.4° that is a result of common-idler enhancement (point D and Eq. (13)).
Examination of the peak gain and the output wavelength at each in the figures shows several interesting and unanticipated phenomena. Firstly, the entire gain curve exhibits a clear deviation from the tuning curve calculated from the Sellmeier equations. Analysis of the signal build-up in the cavity shows that this is a consequence of the dynamic build-up process of the signal in a beam with a finite size in the presence of beam walk-off. The transverse intensity of the signal beam is pulled toward the direction of the idler as the signal grows inside the crystal because of the walk-off. In order to make the signal beam return to the same point after a round trip, the cavity compensates this pull by choosing a direction for the signal that is slightly away from the idler direction that would have been determined by the tuninig curve. As a result of this balancing act, the signal wavelength is shifted to a value higher than that given by the tuning curve. Note that the degree of the pull increases with the beam diameter and the amount of this shift will increase with the size of the pump beam. Secondly the location of the highest gain in the region between and 0.8° is shifted away from the point of intersection of the two tuning curves at 0.714° to 0.67°. Since G10 is a major contributor to this gain, the shift of the peak location can be explained by a rapid tailing off of the gain due to G10 as increases. Finally, in that same region the signal wavelength makes a jump from a smooth progression to a value close to that associated with the highest gain, one that is provided by the triple interaction against that from G10 or G11 alone. This is a case of frequency pulling in the presence of multiple coupled interactions in an optical cavity that is commonly seen in laser cavities . Yet the magnitude of the shift of more than 10 nm as shown here is probably unprecedented.
In order to distinguish the source of the shifts in Fig. 5 described above between walk-off of the signal beam and the pump beam that is inherent in QPM of 2D NPCs from that due to a finite pump beam size, we calculated the gain and wavelengths for the case of w0=1000 µm. The results are shown in Fig. 7 and Fig. 8. At 1000 µm, the beams remain overlapped substantially over the course of the length of the crystal. Effects limited by the beam size are nearly eliminated. This results in a rather flat gL curve for each G vector. At 0°, 0.7°, and 1.4°, the phenomena of enhanced gain and shifts in signal wavelength when multiple reciprocal vectors act together are prominently displayed: The gains rise up at these angles and there is a stronger pull on the signal wavelength toward those where the gain peaks. In Table 3 we compare the normalized deff derived from the calculated gL product for various w0 at points A~E for the inverted domains pattern of Fig. 4(a). Clearly approaches the value derived with plane-wave approximation as w0 increases. Significantly deff at point E is consistently larger than that of G10. Its value is only 22% less than that of the relative 1D QPM value of 0.61 when the pump beam size is large and this is even with a realistic filling ratio of 18.75%. A perfect square domain would give a deff that nearly matches the 1D value. And the increased deviation from the calculated tuning curve confirms the physical picture of deviation given in the above paragraph.
We have analyzed optical parametric interaction in a 2D NPC. While in general the nonlinear coefficient is small compared to a 1D NPC, we showed that at numerous orientations and wavelengths a multitude of reciprocal vectors contribute additively to enhance the gain in optical parametric amplification and oscillation in a 2D patterned crystal. In particular, we have derived the effective nonlinear coefficients for common-signal amplification and common-idler amplification for a tetragonal inverted domain pattern. We showed that in the specific case of a common-signal amplification with QPM by both G10 and G11, symmetry of the crystal results in coupled interaction with the corresponding common-signal amplification by G10 and G1,-1. As a consequence, this coupled utilization of all three reciprocal vectors leads to an effective nonlinear coefficient that is larger than that of additive common-signal amplification using G10, G11, and G1,-1. We demonstrated with a numerical example using a PPLN crystal that a gain that comes close to that of a 1D QPM crystal could be realized in a 2D NPC with an inverted tetragonal domain pattern.
This special mechanism produces two pairs of identical signal and idler beams propagating in mirror-imaged forward directions. This is not possible with traditional 1D parametric devices. In conjunction with the gain enhancement and multiple beams output we also predict that there is a large pulling effect on the output wavelength and direction due to dynamic signal build-up in a noncollinear geometry that is intrinsic to the 2D NPC OPO.
We should point out that the output wavelength in this special 2D NPC is tunable just as with other NPC OPOs by changing the temperature or the grating period of the crystal. The parameters we have chosen in the simulation were based on realistic conditions so that the results of our analysis can be readily verified by a laboratory experiment.
In addition to emitting naturally separated pump, signal, and idler beams the high gain coupled with simultaneous emission of a dual pair of signal (and idler) beams of identical wavelengths in the 2D NPC OPG/OPO might find interesting applications in quantum information processing, providing a dual pair of entangled photons.
We thank Chien-Jen Lai for helpful discussions. This work was supported by the Nanoscience and Technology Program managed by the National Science Council under grant no. NSC 95-2120-M-001-006.
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