Abstract
We propose a new type of semiconductor lasers by implementing the concept of parity–time symmetry in a twodimensional circular Bragg grating structure, where both the real and imaginary parts of the refractive index are modulated along the radial direction. The laser modal properties are analyzed with a transfermatrix method and are verified with numerical simulation of a practical design. Compared with conventional distributedfeedback lasers with modulation of only the real part of refractive index, the parity–timesymmetric circular Bragg lasers feature reduced threshold and enhanced modal discrimination, which in combination with the intrinsic circularly symmetric, large emission aperture are clear advantages in applications that require modehopfree, highpower, singlemode laser operation.
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Introduction
Semiconductor lasers are an important building block in fiberoptic communication, where lasers of pure output spectrum, compact size, high reliability, and low cost are usually desired^{1}. Many efforts have been made to obtain highperformance lasers. For example, distributedfeedback and distributedBraggreflector structures have been developed for creating large discrimination in threshold gain among different oscillation modes of a laser cavity, which facilitates the realization of singlemode lasers^{2,3}. In the meantime, quantum well and quantum dot structures have been employed for improving power efficiency and thermal stability^{4}. Circular Bragg lasers constructed by cylindrical distributed Bragg reflectors were studied decades ago, where highQfactor, largearea, singlemode laser emission can be obtained in a broad operation range^{5,6,7,8}. With the intrinsic circular aperture and lowdivergence emission angle, such lasers have advantages in coupling their emitted light directly into an optical fiber or to onchip photonic components, thus lending themselves to a wide range of applications in integrated photonics, optoelectronics, and fiberoptic communication.
The concept of parity–time (PT) symmetry was first developed by Bender et al. in quantum mechanics^{9}. A Hamiltonian is called PT symmetric if it commutes with the PT operator, which requires that the real (imaginary) part of the complex potential be an even (odd) function of the coordinates. It was found later that this concept also applies to optical systems due to the resemblance between the Schrödinger equation and the wave equations^{10}. PT symmetry in optics can be realized similarly by introducing modulation to both the real and imaginary parts of the refractive index, where the modulation pattern follows an even and odd function respectively^{11}. This trick has been implemented in several photonic structures to achieve otherwise unattainable functionalities, such as lasers and laser amplifiers^{12,13,14,15,16,17,18}, coupled nanobeam cavities^{19}, unidirectional reflectionless^{20} and nonreciprocal transmission^{21} optical components.
In this paper, we for the first time introduce the PT symmetry into the design of circular Bragg lasers. By using a transfermatrix method, we first analyze the reflection and transmission properties of the PTsymmetric circular Bragg reflectors (CBRs), from which the PTsymmetric circular Bragg lasers are constructed. A comparison between the modal properties of the PTsymmetric circular Bragg lasers and their conventional counterparts concludes that the former possess a significantly lower threshold and larger modal discrimination for the targeted lasing mode, both of which contribute crucially to the development of modehopfree, singlemode lasers for highpower applications. Numerical results from finitedifference timedomain simulation of a practical design show good agreement with those from the transfermatrix method.
Results
Structural description of the proposed PTsymmetric circular Bragg lasers
Figure 1(a) illustrates the twodimensional PTsymmetric circular Bragg laser on a chip. Such lasers can be fabricated from a III–V epiwafer which is able to provide optical gain under optical or electrical pumping. The PT symmetry is obtained in the CBR by introducing modulation to both the real and imaginary parts of the refractive index along the radial direction (r) as shown in Fig. 1(b), which can be realized respectively by selective etching and metal deposition on the III–V epiwafer^{12}. More specifically, the complex refractive index of the CBR is expressed by
where r_{0} is the starting radius of the CBR and n_{0} is the average effective refractive index. ∆n_{r} and ∆n_{i} are the modulation depths of the real and imaginary part of the refractive index, respectively. l is an integer starting from 0. Δr is the thickness of each modulated layer. The modulation period of the refractive index is Λ = 4·Δr. With N periods of modulation, the CBR spans a length of N·Λ in the radial direction. It should be noted that the layer thickness Δr generally should not be set as a constant for two reasons: First, the phase of the eigenmodes of traveling waves in the cylindrical coordinates, i.e., the Hankel functions, does not follow a linear dependence with r. Therefore, Δr should follow the local period of the Hankel functions for obtaining perfect phase matching, thus rendering chirped modulation along the radial direction^{5,8}. Second, the wavelength in each modulated layer is different and thus Δr should be inversely proportional to the real part of the refractive index of the respective layer. In this study, the above two effects are negligible for large radius under the weakmodulation condition (∆n_{r} ≪ n_{0}). The PT symmetry is satisfied under the condition that the real and imaginary parts of the refractive index are respectively an even and odd function of position along the radial direction (r).
Reflection and transmission properties of the PTsymmetric circular Bragg reflectors
We develop a transfermatrix method similar to that in ref. 22 for analyzing the PTsymmetric CBR. It is convenient to consider the optical field components satisfying the Helmholtz equation in cylindrical coordinates, which can be expressed by the z component of the electric and magnetic fields
where r, φ, and z are the radial, azimuthal, and axial coordinates respectively, and k_{0} is the wavevector in vacuum. By using the effective medium approach, we can simplify the problem from three dimensional into two dimensional so that the refractive index n(r, z) reduces to n(r) as defined in Eq. (1) and ∂^{2}/∂z^{2} can be dropped from Eq. (2). Assuming that the r and φ dependence of the field can be separated, we obtain E_{z}(r, φ) = E_{z}(r) exp(jmφ), where m is an integer representing the azimuthal modal number. Introducing E_{z}(r, φ) into Eq. (2) leads to
where k(r) = k_{0} n(r) is constant within each modulated layer. The solution of Eq. (3) can be expressed as a linear combination of the Hankel functions of the first and second kinds^{23}
where k_{q} and r_{q} denote respectively the wavevector and radius of the qth layer. and represent the outward and inwardgoing cylindrical modes, with their amplitudes denoted by and respectively. We can rewrite Eq. (4) and its derivative in a matrix form
where is defined as the coefficient matrix. Based on the continuity conditions of the electric and magnetic fields at each interface, the relation between Layer q and q + 1 can be expressed as
where is the transfer matrix from Layer q to q + 1. As a result, by multiplying the transfer matrices of each layer we establish a relation between the amplitudes in Layer 1 and N + 1:
Let us consider an outwardgoing cylindrical wave impinging on the 1st layer of the CBR with an amplitude . By setting (r_{0} + L) = 0 in Eq. (7), we obtain the reflection coefficient R = /^{2} = −U_{21}/U_{22}^{2} and the transmission coefficient T = (r_{0} + L)/(r_{0})^{2} = U_{11} − U_{12} × U_{21}/U_{22}^{2}.
We aim at designing a circular Bragg laser which emits circularly symmetric beam (m = 0) at a vacuum wavelength λ_{0} of 1550 nm. Without loss of generality, we may assume the average effective refractive index n_{0} to be 1.55 and the resulting effective wavelength inside the CBR λ_{eff} = λ_{0}/n_{0} = 1.00 μm. The thickness of each modulated layer Δr is set to be 125 nm, and thus the modulation period Λ is 4Δr = 500 nm. We choose the number of the modulation periods N to be 500 and the corresponding radial length L to be N·Λ = 250 μm. We investigate the reflection (R) and transmission (T) response of the PTsymmetric CBR to an outwardgoing wave impinging on the innermost layer. Figure 2(a) shows the calculation results when ∆n_{r} = ∆n_{i} = 1.0 × 10^{−3} or 1.5 × 10^{−3}. It is clear that R can be larger than 1 at the designed wavelength and stronger modulation of the refractive index leads to enhanced R. These behaviors do not contradict with the conservation of energy because the PTsymmetric CBR structure forces more electric field to be distributed in the gain regions. Therefore, we can control the reflection strength of the CBR by designing an appropriate modulation depth. Meanwhile, the transmission T remains wavelength independent and always equal to 1. As a comparison, Fig. 2(b) plots the results for a traditional CBR with ∆n_{i} = 0 and ∆n_{r} = 1.0 × 10^{−3} or 1.5 × 10^{−3}. Under the same modulation depth ∆n_{r}, the reflection at the targeted wavelength is much weaker than that in Fig. 2(a) and is always smaller than 1. Moreover, R and T add up to 1 in accordance with the conservation of energy in the traditional sense.
It is interesting to study the behavior of R and T when ∆n_{r} and ∆n_{i} are unequal. Figure 2(c) shows the results for ∆n_{r} = 1.0 × 10^{−2} and ∆n_{i} = 1.0 × 10^{−3}, where the modulation to the real part of the refractive index dominates. The reflection and transmission spectra are similar to those of the traditional counterpart (∆n_{i} = 0) as shown in Fig. 2(d), where the reflection for the side modes is enhanced due to the strong ∆n_{r}. The only difference is that the PTsymmetric CBR provides overall stronger reflection, which can exceed 1 at the peak, than the traditional CBR owing to the additional modulation ∆n_{i}. Figure 2(e) shows the results for ∆n_{r} = 1.0 × 10^{−3} and ∆n_{i} = 1.0 × 10^{−2}, where the modulation to the imaginary part of the refractive index dominates. In this case, the reflection and transmission spectra take similar patterns where R is greatly suppressed at the targeted wavelength and enhanced for the side modes. These results also resemble those of a structure with pure gain modulation (∆n_{r} = 0) as shown in Fig. 2(f), although the PTsymmetric CBR provides overall stronger reflection owing to the additional modulation ∆n_{r}. The results in Fig. 2(c–f) have clearly shown that when ∆n_{r} and ∆n_{i} are unequal, the larger of the two determines the reflection and transmission characteristics. The imbalance between ∆n_{r} and ∆n_{i} results in reflection reduction at the targeted wavelength and enhancement for the side modes, leading to worse discrimination between the designed and unwanted modes. Therefore, it is crucial to balance the ∆n_{r} and ∆n_{i} in a PTsymmetric CBR for designing robust singlemode lasers.
It is important to note that, under different modulation schemes in Fig. 2, the devices operate in different phases (PTsymmetric phase or brokenPTsymmetric phase). In a recent work^{24}, Ge et al. proposed a generalized conservation relation between the transmittance and reflectance T − 1 = (R_{L}·R_{R})^{1/2}, which can be adopted to determine the presence of PT symmetry and PTsymmetric breaking transitions: the system is in the PTsymmetric phase when T < 1, in the brokenPTsymmetric phase when T > 1, and at the spontaneous PTsymmetric breaking point (i.e., the exceptional point) when T = 1. Therefore, in Fig. 2(a) the CBRs operate at the spontaneous PTsymmetric breaking point with T = 1, because the modulation ∆n_{i} is balanced with ∆n_{r}. In Fig. 2(b,c,d) the CBRs operate in the PTsymmetric phase with T < 1, because the modulation ∆n_{i} is trivial compared with ∆n_{r}. In Fig. 2(e,f) the CBRs operate in the brokenPTsymmetric phase with T > 1, because the modulation ∆n_{i} is larger than ∆n_{r}.
Modal analysis of the PTsymmetric circular Bragg lasers
Now we analyze a laser structure constructed from the PTsymmetric CBRs. The laser structure consists of a central diskshaped gain or loss region surrounded by a PTsymmetric CBR. It should be noted that the proposal of PTsymmetric laser structures does not have limitation on the average effective refractive index n_{0}. One can always design the structural parameters (e.g., the CBR’s starting radius r_{0} or the thickness of each modulated layer Δr) based on a specific material system to satisfy the laser oscillation condition and obtain perfect phase matching at the targeted wavelength. The laser oscillation condition is r_{CBR} · r_{ctr} · δ_{disk} = 1, where r_{CBR} = (r_{0})/(r_{0}) is the complex reflection coefficient of the CBR which depends on the modulation depths of the real and imaginary parts of the refractive index. r_{ctr} is the reflection coefficient at the center of the disk which must be exactly 1 in order to keep the finiteness of the total field. δ_{disk} is a complex propagation factor expressed as exp[2 g(λ)·r_{0} + 2j·ϕ(λ)], which contains the amplitude and phase information of light propagating radially in the central disk. g(λ) and ϕ(λ) represent the wavelengthdependent gain/loss coefficient and the phase change, respectively. To satisfy the laser oscillation condition, the radius of the central disk region must be chosen such that light at the targeted wavelength λ_{0} experiences a phase change ϕ of multiple integers of 2π. Therefore, we choose r_{0} to be 380 nm which corresponds to the first zero of the Bessel function of the first kind^{25}. The light propagating in the central disk region experiences either a gain or a loss depending on the sign of g(λ) in δ_{disk}. From the laser oscillation condition we can obtain g(λ) for each mode, which is the threshold gain required for lasing. The threshold gain of the first five lasing modes under different refractive index modulation is plotted in Fig. 3(a), while their one and twodimensional modal field distributions are presented in Fig. 3(b–f). For lasers constructed from the PTsymmetric CBRs as shown in ① and ② where ∆n_{r} = ∆n_{i} = 1.0 × 10^{−3} or 1.5 × 10^{−3}, we find a negative threshold gain for the targeted wavelength, indicating that no additional gain is necessary for the targeted mode to lase and the lasing can occur even when the central disk region is lossy. The difference between the threshold gain of the targeted mode and its adjacent modes is as high as 3.99 × 10^{4} cm^{−1}, yielding excellent modal discrimination for singlemode laser operation. Moreover, increase in the modulation depths leads to a uniform reduction of threshold gain for all the modes, and thus the large modal discrimination is maintained. In contrast, lasers constructed from conventional CBRs with ∆n_{i} = 0 as shown in ③ and ④ always require a positive threshold gain at the targeted wavelength, no matter how strong the modulation depth ∆n_{r} is. For ∆n_{r} = 1.0 × 10^{−3} as shown in ③, the threshold gain is 1.47 × 10^{4} cm^{−1} and the modal discrimination is 3.68 × 10^{4} cm^{−1}. Although the threshold gain of the targeted mode can be reduced by increasing ∆n_{r}, e.g., from 1.0 × 10^{−3} to 4.0 × 10^{−3}, this results in greater reduction of threshold gain of the unwanted modes, causing worse modal discrimination (e.g., 1.81 × 10^{4} cm^{−1} in ④) and thus less robust singlemode laser operation. Therefore, we conclude that PTsymmetric circular Bragg lasers have clear advantages over their conventional counterparts because the former possess much lower threshold gain and larger modal discrimination, both of which facilitate the realization of singlemode lasers.
In order to verify the modal analysis from the transfermatrix method, we simulated a practical design of PTsymmetric circular Bragg laser based on the parameters of a quantum well wafer used previously^{26}. We set the refractive index n_{0} and the modulation depths (∆n_{r}, ∆n_{i}) to be 3.40 and 0.006 respectively to satisfy the requirement of PT symmetry. The CBR’s starting radius r_{0} is 175 nm and the thickness of each modulated layer Δr is 57 nm. The number of the modulation periods N is set to be 100, and thus the corresponding radial length L is 22.8 μm. It should be noted that the choice of the number of the modulation periods is related to the preset modulation depths. Smaller modulation depths can also be adopted at the expense of increased number of the modulation periods with correspondingly longer radial length^{20}. Figure 4(a) shows the simulated reflection spectrum of the PTsymmetric CBR by using the finitedifference timedomain (FDTD) method in Lumerical Solutions^{27}, which is in good agreement with that calculated from the transfermatrix method (TMM) in Fig. 4(b). This indicates that a practical PTsymmetric CBR structure can be engineered for realizing singlemode lasers. We also obtained the optical field distribution from the FDTD simulation and the TMM as shown in Fig. 4(c) and (d) respectively. It is clear that light of the targeted wavelength (λ_{0} = 1550 nm) is confined to the central disk region thus facilitating lowthreshold lasing.
Conclusion
In conclusion, we have proposed twodimensional parity–timesymmetric circular Bragg lasers and analyzed their modal properties including threshold gain and field distribution. Such lasers are constructed from a type of circular Bragg reflectors whose refractive index is modulated in both the real and imaginary parts along the radial direction. By setting balanced modulation depth to the real and imaginary parts we can obtain significantly reduced threshold gain with large modal discrimination for the targeted mode, facilitating robust singlemode laser operation. To demonstrate the feasibility for real applications, we also performed finitedifference timedomain simulation of a laser structure with practical design parameters, and obtained the results in good agreement with those from the transfermatrix method. Featuring low threshold and robust singlemode operation in addition to the intrinsic circular aperture and lowdivergence emission angle, such parity–timesymmetric circular Bragg lasers will find wide applications in integrated photonics, optoelectronics, and fiberoptic communication.
Methods
The proposed PTsymmetric circular Bragg lasers can be fabricated from a III–V epiwafer. The modulation of both the real (∆n_{r}) and imaginary (∆n_{i}) parts of the refractive index along the radial direction can be realized respectively by selective etching and metal deposition on the III–V epiwafer. The performance of the PTsymmetric CBRs with various modulation depths (∆n_{r}, ∆n_{i}) is investigated with a transfermatrix method derived in Eqs (1)–(7). To realize a practical PTsymmetric circular Bragg laser, the refractive index n_{0} and the modulation depths are set to be 3.40 and 0.006, respectively. The CBR’s starting radius is 175 nm and the thickness of each modulated layer Δr is 57 nm. The number of the modulation periods N is set to be 100, and the corresponding radial length L is 22.8 μm. We employ the FDTD method in Lumerical Solutions and the transfermatrix method to obtain the reflection spectrum of an outwardgoing cylindrical wave impinging onto a PTsymmetric CBR as well as the E^{2} field distribution of the lasing mode (λ_{0} = 1550 nm). In the FDTD simulation, perfectly matched layers are set as the boundary condition for the computation in Lumerical Solutions.
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How to cite this article: Gu, J. et al. Parity–timesymmetric circular Bragg lasers: a proposal and analysis. Sci. Rep. 6, 37688; doi: 10.1038/srep37688 (2016).
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Acknowledgements
This work was partially supported by a grant from the NSFC/RGC Joint Research Scheme sponsored by the Research Grants Council of the Hong Kong Special Administrative Region, China, and the National Natural Science Foundation of China (Project No. N_CUHK415/15), and partially supported by Project No. BMEp515 of the Shun Hing Institute of Advanced Engineering and Direct Grant for Research of the Faculty of Engineering, The Chinese University of Hong Kong.
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X.S. conceived the project; J.G. developed the transfermatrix method, performed FDTD numerical simulation, and analyzed the data under the supervision of X.S.; X.X., J.M., and Z.Y. contributed to numerical simulation and figure generation; J.G. and X.S. wrote the manuscript, which was reviewed and commented by all the authors.
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Gu, J., Xi, X., Ma, J. et al. Parity–timesymmetric circular Bragg lasers: a proposal and analysis. Sci Rep 6, 37688 (2016). https://doi.org/10.1038/srep37688
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