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Abstract
In the second-harmonic generation processes involving Laguerre-Gaussian (LG) beams, the generated second-harmonic wave is generally composed of multiple modes with different radial quantum numbers. To generate single-mode second-harmonic LG beams, a type of improved quasi-phase-matching method is proposed. The Gouy phase shift has been considered in the optical superlattice designing and an adjustment phase item is introduced. By changing the structure parameters, each target mode can be phase-matched selectively, whose purity can reach up to 95%. The single LG mode generated from the optical superlattice can be modulated separately and used as the input signals in the mode division multiplexing system.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The optical superlattice (OSL) has been widely used in nonlinear optics fields to change the frequency of the laser beam. By modulating the structure of OSL, many nonlinear optical processes can be realized, such as the second-harmonic generation (SHG) process [1,2], nonlinear optical imaging [3], and nonlinear beam shaping [4]. Some particular beams, such as the Airy beam, can also be generated from OSLs in nonlinear optical processes [4–6]. In 1992, the orbital angular momentum (OAM) of light was proposed by Allen and his co-workers [7], which brought the Laguerre-Gaussian (LG) beam back into focus. As a typical OAM beam, the LG beam carries the azimuthal quantum number l and can be applied in many fields [8–11]. Furthermore, the LG beam features another quantum number called the radial quantum number p [12–19]. This quantum number has potential applications in the mode division multiplexing system to expand the communication capacity and attracts increasing attentions of researchers. The OSL can be used to generate the LG beam with a specific wavelength [20,21]. The conversion of the quantum numbers of LG beam in nonlinear processes has been researched widely [22–26]. It is demonstrated that both the value of l and p may change in the nonlinear process and the OSL can affect the quantum numbers of the output beam. For example, the azimuthal quantum number of the crystal can be brought into the output beam [20]. Thus different OSLs can be utilized to manipulate the generated nonlinear LG beam. In our previous research [27], a periodic OSL is used and the second-harmonic (SH) LG wave is demonstrated to be multi-mode with different radial quantum numbers p when the value of p of the fundamental wave (FW) is non-zero. However, in the mode division multiplexing system, the single-mode signal beam is necessary. The periodic structure is appropriate to generate a SH wave with single azimuthal quantum number l [24], but how to design OSLs to generate a SH wave with a single radial quantum number p is still unsolved. The traditional quasi-phase-matching (QPM) method with a periodic structure [24,27] is demonstrated to be unable to generate the single-mode SH wave efficiently.
2. Method
In this paper, a type of improved QPM method is proposed to generate single-mode second-harmonic LG beams. The influence of Gouy phase shift is fully considered in our method and an adjustment phase item is introduced. The adjustment coefficient and length of OSL can be used to purify the SH mode and improve the nonlinear conversion efficiency simultaneously.
We take the FW with p = 1 mode as an example to generate a single-mode SH wave with the target radial quantum number. When the FW is $LG_1^1$ mode propagating along the z-axis, the electric fields for the SH wave can be written as:
Under the condition of small signal approximation, the propagation characteristics of each mode can be described as:
Here, Cj (j = 0, 1, 2) is a constant coefficient, $f(z )$ is the structure function of the 1D OSL, B1 is the amplitude of the FW, zR is the Rayleigh distance, $\Delta k = {k_2} - 2{k_1}$ denotes the wave-vector mismatch, and k1, k2 are the wave vectors for the FW and SH wave respectively. The Eq. (2) is deduced from the nonlinear paraxial equations for the SHG process in the condition of small signal approximation [27], and it can be used describe the propagation characteristics of each LG component mode in SH wave individually. Under the guidance of this equation, we can modulate each component mode wave separately, instead of treating the SH wave as a whole.
In practical application, modulating each single-mode signal separately and then multiplexing them is a common practice [28]. Thus it is necessary to develop a method to generate a single-mode SH wave. However, in the SHG process the SH wave is usually multi-mode as shown in Eq. (1) and the traditional QPM structure is unable to generate the single-mode SH wave efficiently.
In the traditional QPM method, a periodic structure is used whose structure function can be written as $f(z )= {\mathop{\rm sgn}} \{{\textrm{Re} [{\textrm{exp} ({ - i\Delta kz} )} ]} \}$, and $\Delta k$ is the wave-vector mismatch. Numerical calculations have been performed to simulate the propagation of the SH wave and each component mode. In our simulation, the FW wave is a single $LG_1^1$ mode at the wavelength of 1064nm, and has a beam waist of 10µm. The lithium tantalate crystal is used to induce the SHG effect. The coherence length is 3.9µm at 25°C, and the Rayleigh distance is 631.9µm. As shown in Fig. 1(a), the range of the proportion of each mode is fixed, which is 1.9% to 43.2% for p = 0 mode and 56.3% to 97.8% for p = 2 mode. In other words, the purity of the output single-mode SH wave is limited in the range mentioned above. Especially in the case of generating the single p = 0 SH mode, the target mode is even not the major component mode, whose purity is below 50%. (For convenience, we call the desired single LG mode in SH wave as the target mode, and call the other mode as the disturbance mode.) Meanwhile, to generate the high purity single-mode SH wave, the length of the crystal is also limited. Thus, the traditional QPM structure is inappropriate to produce a pure single-mode SH wave.
Fig. 1. (a) Simulation of the proportion of each mode in traditional QPM structure. The proportion of the p = 0 mode is ranged from 1.9% to 43.2%, and the proportion of the p = 2 mode is ranged from 56.3% to 97.8%. (b) The variation of the Gouy phase shift of the p = 0 and p = 2 mode. The variation of the total additional phase while the adjustment coefficient is (c) 0.0033 µm−1 and (d) −0.0033 µm−1. The position of Rayleigh distance is marked. As shown in Fig. 1(b), the Gouy phase shift in p = 0 mode reaches π when the SH wave propagates to the Rayleigh distance in the traditional QPM structure.
From Eq. (2), we find that three modes share the same coherence length and have the same factor: $\frac{\textrm{1}}{{1 - i\frac{z}{{{z_R}}}}}$. This factor also exists in the Gaussian beam and should be compensated properly by the OSL, otherwise the nonlinear conversion efficiency will not reach the expected level [29–33]. A properly designed chirped structure can be utilized to compensate for this factor in the nonlinear process of Gaussian beams. Here, we can use a similar method to compensate for it in the nonlinear process of LG beams, because this factor has the same form in every component mode and will not affect the separation task. Different additional phases as shown in the last factor of each equation are the only difference in the propagation of component modes. (This factor causes by the Gouy phase shift of the high-order LG mode, so we still call this factor as Gouy phase shift for convenience in this paper).
In the SHG processes, the Gouy phase shift in p = 0 mode reaches π when the SH wave propagates to the Rayleigh distance. The additional π phase shift can be equivalent to a domain inversion, which will break the QPM condition, thus the proportion of p = 0 mode decreases. This property is utilized to generate the single-mode SH wave in our improved QPM method. We introduce an adjustment phase as a new controllable parameter, which can be written as $\textrm{exp} ({i\alpha z} )$, and α is the adjustment coefficient. Thus the total additional phase is the sum of the Gouy phase shift and the adjustment phase, and can be written as $4\arctan \frac{z}{{{z_R}}} + \alpha z$ for p = 0 mode but only $\alpha z$ for p = 2 mode. With the help of the adjustment phase, the difference between the propagation of two modes is controllable.
The appropriate value of the adjustment coefficient can reduce the proportion of disturbance modes. Here, we take α = 0.0033 µm−1 and α = −0.0033 µm−1 as examples to present the separation ability of the adjustment phase. The total additional phases of two modes are simulated as shown in Figs. 1(c)–1(d). These phases will directly decide the propagation of two modes. We suppose that both the factor $\frac{\textrm{1}}{{1 - i\frac{z}{{{z_R}}}}}$ and the phase mismatch, which are the same in the each component mode’s wave equation as shown in Eq. (2), can be completely offset by the OSL structure. Thus the total additional phase is the phase difference between two wavelets generated at different distances. If the phase difference between two wavelets is between π and 2π, two wavelets will interfere destructively. In other words, if the total phase shift changes more than π, the mode will reach an inflection point.
As shown in Fig. 1(d), the appropriate negative adjustment coefficient can neutralize the influence of the Gouy phase to some extent. The total phase shift of the p = 0 mode changes less than π through the whole propagation, while p = 2 mode oscillates periodically. Thus we can design the OSL with a particular length, where the total additional phase of p = 2 mode is 2π, to eliminate the p = 2 mode and ensure a high intensity of p = 0 mode at the same time. That is, with the help of the adjustment coefficient, we can generate a SH wave with single p = 0 mode. By similar methods, we can also generate the SH wave with another single mode. The Gouy phase shift of the disturbance mode should be compensated to ensure that the trough value of the disturbance mode is 0.
It is necessary to state that we ignore the influence of p = 1 mode in the discussion in previous paragraph, because the corresponding coefficient C1 is almost zero [27]. As a consequence, the intensity of p = 1 mode in SH wave is far less than other modes and the p = 0 and p = 2 mode are two main modes. But in the simulation in following paragraphs, we take the influence of the p = 1 mode into consideration for accurate results.
After the above discussion, considering the binarization of structure function, we can get the corresponding structure functions for different target modes:
Here, n represents the radial quantum number of the target mode. This structure function results in a slightly chirped structure which is different from the traditional periodic QPM structure. By changing the adjustment coefficient α and the length of the crystal, we can find the optimal design schemes to generate the target mode.
3. Results
3.1 Generation of the single p = 0 mode
On the one hand, we discuss the case that the target mode is p = 0 mode. To generate a single p = 0 mode SH wave, the structure function should be written as:
We simulate the dependence of the intensity, purity and efficiency ratio of the p = 0 mode and SH wave on the adjustment coefficient α and length of OSL d as shown in Figs. 2(a)–2(c). Here, the purity, which can be written as $\frac{{{{|{B_{\textrm{20}}^{im}} |}^2}}}{{{{|{B_{\textrm{20}}^{im}} |}^2} + {{|{B_{\textrm{22}}^{im}} |}^2}}}$, is represented by the ratio of the intensity of p = 0 mode to the intensity of the whole SH wave. The efficiency ratio of p = 0 mode, which is $\frac{{{{|{B_{\textrm{20}}^{im}} |}^2}}}{{{{|{B_{\textrm{20}}^{tr}} |}^2}}}$, is represented by the ratio of the intensity of component p = 0 mode in the designed structure to that in the traditional QPM structure and this parameter can be used to describe the intensity increase of p = 0 mode. The efficiency ratio of SH wave expressed as $\frac{{{{|{B_{\textrm{20}}^{im}} |}^2} + {{|{B_{\textrm{22}}^{im}} |}^2}}}{{{{|{B_{\textrm{20}}^{tr}} |}^2} + {{|{B_{\textrm{22}}^{tr}} |}^2}}}$ is the ratio of the intensity of multi-mode SH wave, which is the sum of the intensity of every component mode, in the designed structure to that in the traditional QPM structure and this parameter can be used to describe the loss of the conversion efficiency. The superscript “im” or “tr” means the corresponding value in the improved QPM structure or traditional QPM structure respectively.
Fig. 2. Simulation of the dependence of (a) the purity and the efficiency ratio of (b) p = 0 mode and (c) SH wave on the adjustment coefficient α and the length of the structure d.
From Figs. 2(a)–2(c), we find some appropriate optimal design schemes, for example, where α = −0.0045 µm−1 and d = 1398.5 µm. With these parameters, the purity of the p = 0 mode can reach 95.3% and the nonlinear conversion efficiency can still maintain 64.5% of that in the QPM structure. It is worth to mention that the intensity of p = 0 mode in improved QPM structure is 32.2 times the intensity in the QPM structure. Thus both the proportion and intensity of the p = 0 mode are improved greatly.
Figures 3(a) and 3(b) show the simulations of the dependence of the intensity and the proportion of p = 0 mode on the propagation distance in the improved QPM structure with α = −0.0045 µm−1 and in the traditional QPM structure. In these figures, we can find the difference between the propagation characteristics of p = 0 mode in two structures. When d = 1398.5 µm, both the proportion and intensity of p = 0 mode in improved QPM structure are near the crest, while the proportion of the mode in the traditional QPM structure is only less than 5%. It is worth mentioning that the purity can reach up to 90% or 95% when d is range from 1135.9 µm to 1654.2 µm or 1320.0 µm to 1446.2 µm respectively. It means the purity is relatively insensitive to crystal length, and a relatively big length error can be tolerated in experiment.
Fig. 3. The simulation of the dependence of (a) the intensity and (b) the proportion of p = 0 mode on the propagation distance. The amplitude profile simulated by the finite difference method in (c) QPM structure and (d) improved QPM structure at d = 1398.5 µm, and the phase profile in (e) QPM structure and (f) improved QPM structure at d = 1398.5 µm. (g) The simulated distributions at center line of the SH wave in two structures.
Figures 3(c)–3(f) show the amplitude and phase profiles and the distributions at the center line of the SH wave simulated by the finite difference method [34] in two structures at d = 1398.5 µm. The influences of the p = 1 mode, binary structure and the backflow between the FW and SH wave are all taken into consideration. By contrast, the purifying effect of the improved QPM structure is obvious. In the amplitude profiles of the SH wave, the number of the dark rings between two bright rings is related to the radial quantum number. There is no obvious dark ring in the amplitude profile of the SH wave in designed structure, because the p = 0 mode is the absolutely main mode in the SH wave. On the contrary, more than two bright rings can be found in the amplitude profile of the SH wave in the QPM structure, which indicates the existence of p = 2 mode. By analyzing the amplitude and phase distribution of the SH wave, the proportion of the p = 0 mode is 94.2% by calculation, which is perfectly compatible with our calculation before.
3.2 Generation of the single p = 2 mode
On the other hand, we discuss the case that the target mode is p = 2 mode. To generate a single p = 2 mode SH wave, the structure function should be written as:
We simulate the dependence of the purity, efficiency ratio of the p = 2 mode and the SH wave on the adjustment coefficient α and the length of OSL d as shown in Figs. 4(a)–4c. To reflect the promoting effect on the purity, only the fields where the purity is more than 97.8%, which is the maximum of purity in the QPM structure, are shown in the Fig. 4(a). By analyzing these figures, we can easily find many design schemes to increase the proportion of the target mode. The purity reaches a high value especially in the line where αd = 2π. The purity and the efficiency ratio of the SH wave in this line with d ranged from 200 µm to 1000µm are shown in Fig. 3(c) and Fig. 3(d). The purity maintains up to 98% and the loss of the nonlinear conversion efficiency decreases with the increase of d. Thus we can generate a single p = 2 mode with high purity and intensity through the improved QPM structure with these parameters in this line. It is worth mentioning that the introduction of the adjustment phase can extremely unfreeze the degree of freedom of the length of OSL. In the traditional QPM structure, the proportion of the p = 2 mode reaches up to 90% only when the length of OSL is more than 760 µm. The limitation of the length of OSL may limit the practical application of the QPM structure in some cases. But in the improved QPM method, we can design OSL structures with different lengths to purify the SH mode, which may be useful under some extreme conditions in experiments.
Fig. 4. Simulation of the dependence of (a) the purity and the efficiency ratio of (b) p = 2 mode and (c) SH wave on the adjustment coefficient α and the length of OSL d. To reflect the promoting effect on the purity, only the fields where the purity is more than 97.8%, which is the maximum of purity in the QPM structure, are shown in the figure. (d) The purity and (e) the efficiency ratio in the line, where αd = 2π, with d ranged from 200 µm to 1000 µm.
4. Summary
To conclude, we design a type of improved QPM structure to generate single-mode LG beams in the SHG processes. By tuning the structure parameters, both single p = 0 mode and p = 2 mode with high purity and intensity can be generated. Especially in the case of generating single p = 0 mode, the purity can reach up to 95%, which is only less than 50% in traditional QPM structure, and the intensity of p = 0 mode in the improved QPM structure is far stronger than that in traditional QPM structure. In the case of generating single p = 2 mode, with the help of adjustment phase, the length of OSL is no longer limited and the purity can reach up to 98% with the length ranged from 200 µm to 1000 µm. Numerical simulations based on the finite difference method have been performed to verify the theory. Our work provides an efficient way to manipulate the component modes of LG beams in nonlinear processes.
Funding
Natural Science Foundation of Jiangsu Province (BK20200701); Priority Academic Program Development of Jiangsu Higher Education Institutions; National Natural Science Foundation of China (11774165, 11874214, 12004177, 91950103); National Key Research and Development Program of China (2017YFA0303700).
Disclosures
The authors declare no conflicts of interest.
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