Quasi-phase matching using frozen waves without periodic poling

Fahad S. Alghannam

doi:10.1364/OE.502902

1. Introduction

Quasi-phase matching is a widely utilized technique in nonlinear optics. Typically, quasi-phase matching requires periodic poling, where one of the crystalline axes of a nonlinear material is inverted periodically. The most common way to achieve periodic poling is ferroelectric domain engineering, where a strong electric field is applied to the nonlinear material to permanently invert the crystalline axis at the required domains [1]. Other methods for periodic poling include poling by scanning probe microscopy [2,3], poling by electron beam [4], fabrication of oriented-patterned semiconductors [5–8], all-optical techniques [9,10], and 2D and 3D nonlinear photonic crystals [11–14]. Other techniques for phase matching include modal phase matching [15,16] and quantum well intermixing [17].

In perfect phase-matching conditions, the generated field of a nonlinear optical process adds constructively in the forward direction. Therefore, the generated field amplitude grows with the propagation distance in the crystal. However, in the presence of wavevector mismatch Δk, the wave generated from different regions of the crystal will add with a phase that increases with the propagation distance. The wave generated from two domains separated by the coherent buildup length L_coh will have a phase difference of π. Therefore, the generated field amplitude will oscillate with the propagation distance having a period of 2L_coh. As an example, for the second harmonic generation (SHG) process, L_coh =π/Δk.

In quasi-phase matching, the effect of the wavevector mismatch is countered by reversing the sign of the second-order susceptibility χ⁽²⁾ periodically with a period of 2L_coh to ensure the continuous buildup of the generated field amplitude with the propagation distance. In practice, this is achieved using periodic poling. In this article, we propose that quasi-phase matching can also be achieved by manipulating the pump fields without periodic poling. We will only consider the interaction of frozen periodic patterns with planewaves in nonlinear materials. Nonlinear interactions of non-frozen periodic patterns and more complex manipulations of the pump beams will have interesting effects but will not be considered in this study [18–20].

The concept of frozen waves was introduced in 2004 [21], where it was shown that one can generate stationary localized wave fields with a longitudinal intensity pattern that can assume any desired shape. Subsequently, this concept was demonstrated experimentally and implemented for many applications [22–26]. Moreover, a more complex 3D control of the optical fields was demonstrated [27,28]. The interference pattern of two coherent beams can generate a simple sinusoidal periodic pattern that can achieve the required quasi-phase matching in a way similar to that of periodic poling. Instead of reversing the sign of χ⁽²⁾, the sign flip is introduced by the field itself. In this article, we will only analyze the case where one of the interacting beams has a periodic pattern and the others are assumed to be planewaves. Note that with the assumption of planewaves, analytical solutions are attainable where the desired resemblance to quasi-phase matching is apparent. However, for practical implementations, focused beams might be necessary. In such a case, it might be better to utilize numerical simulations to study the effects of different pumping patterns on the efficiency of nonlinear processes.

2. Periodically patterned pump field for sum-frequency generation (SFG)

Let us first consider the process of SFG, where two pump fields E₁ and E₂, with frequencies ω₁ and ω₂, respectively, interact in a nonlinear material to generate field E₃ with a frequency ${\omega _3} = \; {\omega _1} + \; {\omega _2}$. Now, let field E₁ have the form:

(1)$${E_1} = {A_1}\cos ({{k_m}z} ){e^{i({{k_1}z - \; {\omega_1}t} )}} + c.c.\; $$

and let fields E₂ and E₃ be planewaves. The generated field amplitude A₃ is given by [29]:

(2)$${A_3} = \; {\kappa _{3}}\; \mathop \int \nolimits_0^L cos({{k_m}z} ){e^{i\Delta kz}}\; dz$$

where L is the propagation distance in the crystal, and the phase mismatch

$\Delta k = {k_3} - {k_1} - {k_2}$ and ${\kappa _{3}}$ is a constant given by:

{\kappa _{3}} = \; \frac{{8\pi i{\omega _3}^2{d_{eff}}}}{{{k_3}\; {c^2}}}\; {A_1}{A_2}

where A₁ and A₂ are constants (i.e., the undepleted pump approximation) and d_eff is a constant related to the second-order susceptibility for the nonlinear process in a given geometry, as defined in [29]. The integral (Eq. (2)) can be solved analytically to give:

(3)$${A_3} = \; \frac{{{\kappa _3}}}{2}\left[ {\; \frac{{{e^{i({\Delta k + {k_m}} )L}} - 1}}{{i({\Delta k + {k_m}} )}} + \; \frac{{{e^{i({\Delta k - {k_m}} )L}} - 1}}{{i({\Delta k - {k_m}} )}}} \right].\; $$

We calculate the intensity of the generated field ${I_3} = \; {|{{A_3}} |^2}$, which gives:

{I_3} = \; \frac{{{{|{{\kappa_3}} |}^2}}}{4}\{ {L^2}sin{c^2}\left( {\frac{{({\Delta k + {k_m}} )L}}{2}} \right) + \; {L^2}sin{c^2}\left( {\frac{{({\Delta k - {k_m}} )L}}{2}} \right) + \; \frac{2}{{\Delta {k^2} - {k_m}^2}}

(4)$$[{\cos ({2{k_m}L} )- \cos ({\; ({\Delta k + {k_m}} )L} )- \cos ({({\Delta k - {k_m}} )L} )+ 1} ]\}$$

As expected for SFG, quasi-phase matching occurs near the points ${k_m} ={\pm} \Delta k$. Figure 1 illustrates the shape of ${I_3}$ as a function of ${k_m}$ for different values of Δk. Next, we need to compare the efficiency of this technique with that of conventional periodic poling. Here, we must consider that the average intensity of the patterned field is half that of a planewave; therefore, the pump field in Eq. (1) must be multiplied by a factor of $\sqrt 2 $. This factor also appears naturally in the experimental design discussion in Section 4 (Eq. (14)). Figure 2 shows a comparison of the generated field from both techniques as a function of propagation distance. Here, we see that our technique is even more efficient than a typical periodic poling. Difference frequency generation (DFG) will have very similar results to SFG.

Fig. 1. The intensity of the generated field for the SFG process for two different values of phase mismatch. Quasi-phase matching is achieved for ${k_m} = \; \pm \Delta k$.

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Fig. 2. The intensity of the generated field for the SFG process with quasi-phase matching achieved through periodic poling and through manipulation of the pump field.

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3. Periodically patterned pump field for second harmonic generation

At this point, it is worth repeating the above calculations for SHG. In SHG, a strong pump field with frequency ${\omega _p}$ interacts with a nonlinear crystal to generate the second harmonic beam with frequency $\omega = \; 2{\omega _p}$. Again, we assume the pump field to have the form:

(5)$${E_p} = {A_p}\cos ({{k_m}z} ){e^{i({{k_p}z - \; {\omega_p}t} )}} + c.c.\; $$

and the generated field to be a planewave:

(6)$${E_{sh}} = {A_{sh}}\; {e^{i({kz - \; \omega t} )}} + c.c$$

However, since “both input fields” have sinusoidal patterners, we obtain

(7)$${A_{sh}} = \; {\kappa _{sh}}\; \mathop \int \nolimits_0^L co{s^2}({{k_m}z} ){e^{i\Delta kz}}\; dz$$

where

{\kappa _{sh}} = \; \frac{{8\pi i{\omega ^2}{d_{eff}}}}{{k\; {c^2}}}\; {A_p}^2

and

\Delta k = \; k - 2{k_p}.

Now, we calculate the intensity of the generated field ${I_{sh}} = \; {|{{A_{sh}}} |^2}$, which gives:

{I_{sh}} = \; \frac{{{{|{{\kappa_{sh}}} |}^2}}}{{16}}\; \{ \; {L^2}sin{c^2}\left( {\frac{{({\Delta k + 2{k_m}} )L}}{2}} \right) + \; {L^2}sin{c^2}\left( {\frac{{({\Delta k - 2{k_m}} )L}}{2}} \right) + \; {L^2}sin{c^2}\left( {\frac{{\Delta kL}}{2}} \right) + \;

\frac{2}{{\Delta {k^2} - 4{k_m}^2}}\; [{\cos ({4{k_m}L} )- \cos ({\; ({\Delta k + 2{k_m}} )L} )- \cos ({({\Delta k - 2{k_m}} )L} )+ 1} ]+ \;

\frac{4}{{\Delta k({\Delta k + 2{k_m}} )}}\; [{\cos ({2{k_m}L} )- \cos ({\; ({\Delta k + 2{k_m}} )L} )- \cos ({\Delta kL} )+ 1} ]+

(8)$$\frac{4}{{\Delta k({\Delta k - 2{k_m}} )}}\; [{\cos ({2{k_m}L} )- \cos ({\; ({\Delta k - 2{k_m}} )L} )- \cos ({\Delta kL} )+ 1} ]\} $$

It turns out that quasi-phase matching is also possible in this case, and, as expected for SHG, it occurs near the points ${k_m} ={\pm} \Delta k/2$. Figure 3 illustrates the shape of ${I_{sh}}$ as a function of ${k_m}$ for different values of Δk. However, unlike SFG, SHG with the sinusoidal pump pattern is less efficient than the conventional periodic poling method. Figure 4 shows a comparison of the generated field from both techniques as a function of propagation distance.

Fig. 3. The intensity of the generated second harmonic field for two different values of phase mismatch. Quasi-phase matching is achieved for ${k_m} = \; \pm \Delta k/2$.

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Fig. 4. The intensity of the generated field for the SHG process with quasi-phase matching achieved through periodic poling and through manipulation of the pump field.

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To increase the efficiency of the frozen wave technique, we will have to use a more complicated periodic pattern. For example, consider the following pump field:

(9)$${E_p} = {A_p}\left[ {\cos ({{k_m}z} )+ \frac{1}{3}\cos ({3{k_m}z} )} \right]{e^{i({{k_p}z - \; {\omega_p}t} )}} + c.c.\; $$

First, we need to correct for the average intensity by multiplying the field by 3/√5. This factor is obtained by comparing the average intensity of the field in Eq. (9) with that of a planewave. If we compare this with the conventional method, we find that our method is again more efficient. Figure 5 shows a comparison of the intensities of the fields generated with the conventional method and that generated by the pump pattern in Eq. (9). Adding the term $\frac{1}{3}\textrm{cos}({3{k_m}z} )$ was motivated by the idea that we need to “quench” the pump field in the out-of-phase regions, which is only partially achieved with the pattern in Eq. (1). Adding more terms might further improve the performance, but this will come at the cost of more challenging experimental setups.

Fig. 5. The intensity of the generated field for the SHG process with quasi-phase matching achieved through periodic poling and through a pump field with the pattern in Eq. (9).

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4. Experimental generation of a sinusoidal periodic pattern

Although the complete experimental analysis of this technique is beyond the purpose of this article, it can be illustrative to show an experimental design for achieving the simple sinusoidal pattern described in Eq. (1). Figure 6 shows the suggested experimental diagram. Let us start with a planewave polarized perpendicular to the page of the form

(10)$$E = \; {A_0}\; {e^{i({kx - \; {\omega_1}t} )}} + c.c.\; $$

A planewave is an ideal state that cannot be generated in the lab but can be approximated by a greatly expanded Gaussian beam. Then, this beam is split with a beam splitter and recombined to generate the required pattern in Eq. (1). After the beam splitter and before the two beams cross each other at the “striped region” in the figure, the two beams have the form

(11)$${E_{a,b}} = \; \frac{{{A_0}}}{{\sqrt 2 }}\; {e^{i({k{x_{a,b}} - \; {\omega_1}t} )}} + c.c.\; $$

where the directions ${x_a}$ and ${x_b}$ are shown in Fig. 6. At any point within the “striped region,” the field is given by

(12)$$E = \; {E_a} + {E_b} = \; \frac{{{A_0}}}{{\sqrt 2 }}\; {e^{ - i\; {\omega _1}t}}({{e^{ik{x_a}}} + \; {e^{ik{x_b}}}} )+ c.c.\; $$

If we insert a nonlinear crystal in the striped region along direction z, which forms an angle ϑ with ${x_a}$, as illustrated in Fig. 6(b), then the field in the crystal will have the following form:

(13)$$E = \; \; \frac{{{A_0}}}{{\sqrt 2 }}\; {e^{ - i\; {\omega _1}t}}({{e^{ikzcos\vartheta }} + \; {e^{ikzsin({\vartheta + \varphi } )}}} )+ c.c.\; $$

where angle φ is defined in Fig. 6. Then, Eq. (13) can be rearranged to

(14)$$E = \; \sqrt 2 \; {A_0}{e^{ - i\; {\omega _1}t}}\; {e^{ikz\left( {\frac{{\sin ({\vartheta + \varphi } )\; + \; cos\vartheta }}{2}} \right)}}\cos \left( {kz\frac{{\sin ({\vartheta + \varphi } )\; - \; cos\vartheta }}{2}} \right) + c.c.\; $$

We can simply set

{k_1} = \; k\frac{{\sin ({\vartheta + \varphi } )+ \; cos\vartheta }}{2}

{k_m} = k\; \frac{{\sin ({\vartheta + \varphi } )\; - \; cos\vartheta }}{2},

and Eq. (1) will be recovered. Note that ${k_m}$ can be chosen by simply tuning the angles ϑ and φ.

Fig. 6. Suggested setup for generating the sinusoidal periodic pattern in Eq. (1).

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5. Practical considerations

Throughout this article, we compared the performance of our technique with that of conventional periodic poling to illustrate the potential of the former. However, it should be noted that such ideal comparisons are hardly sufficient to determine the actual performance of the frozen wave technique in practice. Here, we will point out some practical limitations.

First, as mentioned in the introduction, the calculations made in this study are only valid for planewaves. In this case, analytical solutions are attainable, and a simple proof-of-principle experiment can be designed, as we did in the previous section. However, for many applications, the pump beams need to be focused. The periodic-patterned beam will be a superposition of Bessel beams in the manner explained in [21]. Furthermore, since fields ${E_2}$ and ${E_3}$ are not generally planewaves in this case, Eq. (2) will have to be revised. An analytical solution might not be attainable, and numerical simulations are needed.

We also point out some experimental limitations associated with the frozen wave technique and their counterparts in the periodic poling technique. First, the frozen wave structure can only be maintained for a limited distance. This distance would correspond to the length of the periodically poled structure in the conventional method. Second, the pattern shape will depend on some parameters of the light source. For example, in the proof-of-principle design in the previous section, the visibility and phase stability of the interference pattern depend on the spatial and temporal coherence of the light source. On the other hand, an “equivalent” limitation also exists in periodic poling, where physical parameters deviate from the ideal design or where some regions at the periodically poled crystal are not reversed at all [30].

One clear advantage of pump manipulation techniques, such as our technique, over conventional quasi-phase matching is that they can be applied to any nonlinear material. Different periodic poling methods, like the ones cited in the introduction, only work for specific types of crystals. Furthermore, once an experimental setup, like the one presented in Fig. 6, is built, quasi-phase matching in different types of nonlinear crystals can be studied easily compared with the experimentally challenging techniques for periodic poling.

6. Conclusion

On top of the efficiency improvement, our technique does not require a permanent modification of the nonlinear crystal. In conventional periodic poling, the period needs to be chosen at the manufacturing step. Each periodically poled crystal will be manufactured for a specific nonlinear process and wavelength. In a research environment, it is quite useful to be able to repurpose the same crystal for different applications. With our method, this is possible since we only need to modify the pump field, and no permanent alteration of the crystal is required. Finally, it should be noted that the purpose of this article is to introduce the concept of quasi-phase matching with frozen waves. For some applications, the practical implementation of this method requires the pump beams to be focused rather than planewaves.

In summary, we have discussed the idea of quasi-phase matching without periodic poling using frozen waves. We have analyzed the cases of SFG and SHG, where one of the pump beams has a simple periodic pattern. In the case of SFG, the quasi-phase matching achieved with a sinusoidal pattern is more efficient than the conventional method with periodic poling. In SHG, quasi-phase matching with a simple sinusoidal pattern is possible. However, to enhance efficiency, one needs to use more complicated periodic patterns like the one described in Eq. (9). We have also suggested a design for a simple proof-of-principle experiment for the proposed technique and finally discussed some experimental limitations of our method and related them to limitations in the conventional method.

Disclosures

The author declares no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Quasi-phase matching using frozen waves without periodic poling

Abstract

1. Introduction

2. Periodically patterned pump field for sum-frequency generation (SFG)

3. Periodically patterned pump field for second harmonic generation

4. Experimental generation of a sinusoidal periodic pattern

5. Practical considerations

6. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (23)

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