Parity time-symmetric vertical cavities: intrinsically single-mode regime in longitudinal direction

Hugh F. Jones; Mykola Kulishov; Bernard Kress

doi:10.1364/OE.24.017125

1. Introduction

Vertical cavities that confine light to small volumes with the longitudinal direction perpendicular to the substrate are very important structures for many applications, either for fundamental science, or for applied studies in numerous technological fields. They are already indispensable for research in light-matter interaction in quantum electrodynamics [1], in various applications including single-photon light sources [2], in vertical-cavity surface-emitting lasers (VCSEL) [3], and Si-integrated on-chip lasers [4]. Applications of these remarkable devices are as diverse as their geometrical and resonant properties.

The cavity mode that evolves into the first lasing mode is basically defined by the geometry and dielectric distribution of the cavity in the absence of gain. The specific spatial and spectral overlap of the cavity modes with the gain distribution normally determines which mode evolves into the first (fundamental) lasing mode. In order to improve mode selectivity, cavities with a frequency-dependant feedback mechanism are used, such as Distributed Feedback (DFB) structures, where the loss is different for different longitudinal modes. As Kogelnik and Shank [5] demonstrated, the so-called index-coupled DFB lasers intrinsically have two degenerate longitudinal modes, symmetrically placed about the Bragg wavelength. A periodic modulation in the gain of the active medium in DFB dramatically changes the lasing pattern. The necessary feedback provided by a gain/loss modulation in the pure gain-coupled DFB devices results in single-mode operation, with the lowest threshold mode arising at the Bragg frequency [5]. It has been shown theoretically [6–8] and confirmed experimentally that even a small amount of gain-coupling added to a mainly index-coupled DFB laser improves the single-mode operation considerably. Advances in fabrication technology in the 1990s produced pure-gain-coupled DFB lasers, as well as mixed (index and gain)-coupled DFB lasers from GaAlAs [9], InGaAsp/InP [10] etc. In the pure gain-coupled DFB structure, its corrugated active layer produces a periodic perturbation in the modal gain coefficient, and the index modulation generated at the same time is canceled out by the underlying grating with the opposite spatial phase [9].

The first studies (in the 1990s) of combined index and gain/loss coupled structures [6, 11] with an arbitrary phase shift between the index-of-refraction grating and the gain/loss grating of DFB structures were mostly focused on reducing the threshold current, achieving higher single-mode suppression ratios, i.e. improving the laser operating characteristics. Surprisingly enough, there have not been many studies until recently on the grating combination with a quarter-period shift between index and gain/loss modulations, which can be represented by Δncos(Kz) for the index and jΔσ sin(Kz) for gain/loss along the longitudinal coordinate z, where K is the grating wave number, j = (−1)^1/2, Δn and Δσ are amplitudes of the index and gain/loss modulation, respectively, i.e. the real part of the index has an even spatial distribution, while the imaginary part is odd, i.e. n(z) = n*(-z). Most likely the lack of interest at that time can be attributed to the fact that such a grating is not reflective from one of its sides. Therefore it does not form a resonator by itself, and cannot function as a DFB cavity without an additional reflector. Later such systems were called Parity-Time (PT) symmetric, the terminology coming to optics from quantum mechanics after the theoretical findings of C. Bender et al. [12] published in 1998.

Their theoretical investigations indicated that even non-Hermitian Hamiltonians can exhibit entirely real spectra as long as they satisfy the condition of PT-symmetry. Due to the equivalence between the Schrödinger equation in the temporal domain and the paraxial approximation to the Helmholtz equation in the spatial domain, the concept of PT-symmetry was brought into optics by Ruschhaupt et al. [13] in 2005, although such structures were studied in optics long before [14, 15].

It turned out that judicious engineering of the gain/loss distribution along with the index profile in accordance with PT-symmetry produced a series of intriguing optical phenomena, such as unidirectional invisibility [15], negative refraction and planar focusing [16], trapping light in a ring resonator [17]. Applying PT-symmetry to laser cavities produced many unusual lasing effects. For example, a cavity divided between gain and loss such that PT-symmetry is preserved [18, 19] produces very unusual lasing modes. Such cavities, with zero net single-pass gain or loss at threshold can simultaneously function as laser amplifiers and as coherent perfect absorbers (CPA); they are called CPA-lasers.

Laser cavities typically support a large number of closely-spaced modes, which results in mode competition, random fluctuations, bad monochromaticity and laser beam quality. Single-mode operation is critical for many laser applications. Traditional DFB lasers (index coupled or gain-coupled) can support single longitudinal mode operation only within a limited range of pumping conditions. However, other longitudinal modes start to lase as the pump level is increased. On the other hand, distributed Bragg reflector (DBR) lasers support single longitudinal mode operation only for very short cavity length, as in the case of VCSELs. To achieve stable single-mode preparation with the injected current even higher above the threshold it is critical to improve the modal discrimination against other competing modes as much as possible.

Recently it has been demonstrated theoretically and experimentally [20–25] that by exploiting the concept of PT-symmetry, new laser devices with intrinsic single-mode operations regardless of the gain spectral bandwidth can be realized. Originally this was predicted theoretically in a cavity formed by two PT-symmetric gratings [20, 21], and then in the DBR structure with an embedded PT-symmetric grating inside [22], where intrinsically single-mode lasing can be supported if a uniform gain area between the Bragg reflectors is replaced by a balanced PT-symmetrical grating without any restrictions on the cavity length. An intrinsically single-mode laser was also explained theoretically and built experimentally [23–25] in a ring resonator with a built-in PT-symmetric grating. Our objective is to design a laser based on a pure reflective volume PT-symmetric grating that radiates perpendicular or at an angle to the grating slab, generating one and only one fundamental mode. This non-Hermitian laser is not capable of supporting any higher-order longitudinal mode. It is important to emphasize that VCSELs also operate in single-mode regime due to a restriction on their cavity length. They are very short, with a limited number of amplifying quantum wells (3-5 for electrical pumping). The proposed structure does not require such restrictions; we will show how it can operate in intrinsically single mode longitudinal regime without any restriction on the cavity length.

2. Statement of the problem

In this paper we extend our recent studies [26, 27] of a slab with a pure reflective PT-symmetric grating to configurations that support lasing. The proposed configuration is shown in Fig. 1(a) in a schematic form. It consists of alternating layers that form a PT-symmetric grating of spatial period Λ on a substrate with a perfectly-reflecting mirror between the grating and substrate. The grating is composed of harmonic modulation of the relative dielectric permittivity and gain/loss modulation with the same spatial frequency but shifted by a quarter period Λ/4 with respect to each other. It is important to note that any one-dimensional periodic refractive index can be approximated by its first-order Fourier representation [21].

Fig. 1 Schematic view of the PT-symmetric vertical emitting laser based on a planar purely reflective volume grating with index (black color fringes) and gain/loss (red color fringes) modulations in the form decribed by Eq. (1).

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The following complex distribution of the dielectric constant in the region from z = -L/2 to z = + L/2 can be presented in the following form:

ε (x, z) = ε_{2} + Δ ε_{Re} \cos (2 β z + φ) + j Δ ε_{Im} \sin (2 β z + φ)

where ε₂ is the average relative permittivity of the grating, Δε_Re is the amplitude of the real part of relative permittivity modulation, Δε_Im is the amplitude of the gain/loss periodic distribution, and β = π/Λ. In Eq. (1) we have introduced a phase ϕ, which corresponds to a possible shift in z. This particular combination [15] of index and gain/loss modulation has been the subject of intensive theoretical [28] and experimental [29, 30] study in recent years. A passive PT-symmetric volume grating (with index and loss modulation) was manufactured recently [30] by alternating thin film depositions of amorphous Si and silica layers on a cleaned glass wafer using plasma-enhanced chemical vapor deposition (PECVD). By implementing active layers with optical pumping such an approach can be used to build PT-symmetric gratings with index and gain/loss modulation.

3. Pure reflective PT-symmetric grating in amplifying and lasing modes

In our recent publications [26, 27] we developed techniques to accurately analyze the diffraction of H-mode polarized light (E_x = E_z = 0, E_y ≠0) on such PT-symmetric gratings. The pure reflective PT-symmetric grating is very asymmetric in its diffraction; it has a diffractive side where incident light experiences strongly amplified back diffraction, whereas light incident on opposite (non-diffractive) side of the grating passes through the grating without any diffraction. For achieving lasing the non-diffractive side should be the side of incidence, with strong reflection from the opposite (diffractive) side provided, for example, by placing a highly-reflective metal layer there, as shown in Fig. 1(a). Only in this configuration is a positive feedback provided that might lead to lasing when a threshold is reached. Indeed, incident light under any incident angle [only normal incidence is shown in Fig. 1(a)] reaches the mirror without any significant reflection. After being reflected from the mirror the light is reflected back again toward the mirror, experiencing strong back-diffraction with amplification. This process is repeated again and again, generating more and more light back into free space, as shown schematically in Fig. 1(a) for normal incidence.

A recently obtained closed-form solution [27] for pure reflective volume PT-symmetric gratings with balanced gain/loss and index modulation (Δε_Re = Δε_Im) embedded on a substrate, with different refractive indices on either side, is perfectly suited to the analysis of the situation when such a grating diffracts with strong amplification below and above the lasing threshold.

Generally the DFB structure under consideration diffracts light with strong amplification into two directions of the internal angle of refraction, θ = ± θ_B, where θ_B is the Bragg angle: θ_B = arcos(λ/(2Λ(ε₂)^1/2) where lasing can occur at the threshold condition. Here λ is the free-space wavelength. In Fig. 2 we show |R_L|² as a function of θ [Fig. 2(b) and 2(d)] and the wavelength [Fig. 2(a) and 2(c)] for values of the balanced index and loss/gain modulation ξ = 2Δε/ε₂ below threshold and at threshold. As we can see, this exhibits extremely sharp peaks at ~ ± 13.72° for λ ≈632.072 nm.

Fig. 2 Reflective wavelength (a, c) and angular (b, d) spectra for ε₁ = 1; ε₂ = 2.4; ε₃ = ∞; L = 72Λ; Λ = 0.21 µm below the threshold (ξ = 0.0078) (a, b) and near the threshold condition: ξ = ξ_th = 0.0104 (c, d).

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Our objective is to design the PT-symmetric grating with the bottom reflector in such a way that this structure is able to lase perpendicular to the substrate at one and only one longitudinal fundamental mode and is unable to lase at any higher-order mode. This can be achieved by carefully engineering gain/loss and index modulations along with their phases.

4. The threshold condition using the modified Kogelnik approach

The electric field outside the slab and within the slab can be represented as a sum of forward (f) and backward (b) propagating waves (Fig. 1(b)):

\begin{array}{l} E_{1} = \exp (j k_{1} z) + R_{L} \exp (- j k_{1} z) \\ E_{2} = E_{f} (z) \exp (j k_{2} z) + E_{b} (z) \exp (- j k_{2} z) \end{array}

where k₁ = k(ε₁)^1/2, k = ω/c is the wave-vector in free space, related to the free-space wavelength λ by k = 2π/ λ, c is the velocity of light in vacuo, ω is the angular frequency of the wave, and we have normalized the amplitude of the forward-propagating wave to 1.The boundary conditions for the electric field at the slab boundary (z = -L/2), namely continuity of E and E’, lead to the following transfer matrix for the boundary:

M^{(1, 2)} = \frac{1}{2} (\begin{matrix} (1 + γ) \exp (j \frac{k_{2} - k_{1}}{2} L) & (1 - γ) \exp (j \frac{k_{1} + k_{2}}{2} L) \\ (1 - γ) \exp (- j \frac{k_{2} + k_{1}}{2} L) & (1 + γ) \exp (j \frac{k_{1} - k_{2}}{2} L) \end{matrix})

where γ = k₁/k₂ = (ε₁)^1/2/(ε₂)^1/2.

Within the well-known Kogelnik coupled-wave approximation the transfer matrix for the grating itself has the following form [15]:

M^{(G R)} = (\begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix})

where

m_{11} = (\cos h (ρ L) + j \frac{δ}{ρ} \sin h (ρ L)) \exp (- j δ L)

m_{12} = j \frac{κ_{12}}{ρ} \sin h (ρ L)

m_{21} = - j \frac{κ_{21}}{ρ} \sin h (ρ L)

m_{22} = (\cos h (ρ L) - j \frac{δ}{ρ} \sin h (ρ L)) \exp (j δ L)

where

ρ = {(κ_{12} κ_{21} - δ^{2})}^{1 / 2}

;

κ_{12} = κ_{n} + κ_{σ}

,

κ_{21} = κ_{n} - κ_{σ}

, where

κ_{n}

and

κ_{σ}

are the coupling coefficients related to the index and gain/loss modulation, respectively, and δ = k₂-β is the detuning from the Bragg condition.

The electric field amplitudes at the mirror (z = + L/2) can then be found from the following matrix equation:

(\begin{matrix} E_{f} (L / 2) \\ E_{b} (L / 2) \end{matrix}) = M^{(G R)} M^{(1, 2)} (\begin{matrix} 1 \\ R_{L} \end{matrix})

Using the reflection condition: E_f( + L/2)exp(jk₂L/2) + E_b( + L/2)exp(-jk₂L/2) = 0 at a perfect mirror, we can obtain the final expression for the reflection coefficient:

R_{L} = - \frac{(M_{11} \exp (j k_{2} L) + M_{21}) (1 + γ) + (M_{22} \exp (- j k_{2} L) + M_{12}) (1 - γ)}{(1 - γ) (M_{21} + M {}_{11}e x p (j k_{2} L)) + (M_{22} \exp (- j k_{2} L) + M_{12}) (1 + γ)} e^{j k_{1} L}

The lasing threshold condition occurs when the denominator of Eq. (13) is zero.

4.1 Perfectly balanced PT-symmetric grating

The PT-symmetric grating is balanced when it provides equal contribution from the index and gain/loss modulations (Δε_Re = Δε_Im), and Eq. (1) takes the following form:

ε (x, z) = ε_{2} (1 + \frac{ξ}{2} \exp (j (2 β z + ϕ)))

For this particular case the transfer matrix M^(GR) (4) reduces to the following form:

M^{(G R)} = (\begin{matrix} 1 & m_{12}^{(P T)} \\ 0 & 1 \end{matrix})

where

m_{12}^{(P T)} = j k_{2} L \frac{\sin (δ L)}{2 δ L} ξ \exp (j φ)

In general the phase ϕ in the expression for m₁₂^(PT) could be arbitrary, but we consider here only two cases: (i) ϕ = 0, when m₁₂^(PT) is purely imaginary, and (ii) ϕ = π/2, when m₁₂^(PT) is real. In the following discussion we will take the grating length L to be an integral number of grating periods, i.e. L = mΛ.

The very special characteristics of the balanced grating, namely complete transparency for light incident from one side and enhanced reflection for light incident from the other side, are fundamentally due to the fact that when Δε_Re = Δε_Im the system is precisely at the threshold (exceptional point) for PT-symmetry breaking [28]. For Δε_Re < Δε_Im propagation within the grating is sinusoidal, while for Δε_Re > Δε_Im it involves exponential growth and decay.

4.1.1 ϕ = 0, m₁₂^(PT) pure imaginary

The lasing condition is obtained by equating the real and imaginary parts of the denominator to zero.

\cos (k_{2} L) = 0

\frac{2 γ}{1 + γ} \sin (k_{2} L) = k_{2} L \frac{ξ}{2} \frac{\sin (δ L)}{δ L}

The real part of the denominator, Eq. (14a), gives us the frequency lasing condition, while the imaginary part, Eq. (14b), provides us with the threshold value. The real part, Eq. (14a), is equal to zero only when k₂L = (i + 1/2)π, which, on writing i = m + r, is equivalent to k₂ = π/Λ + (r + 1/2)π/L. Then the detuning is δ = k₂-β = (r + 1/2)π/L, where r = 0, ± 1, ± 2, … . In other words, the non-Hermitian vertical-emitting cavity with a PT-symmetric grating of an integral number of periods can lase at discrete wave-numbers away from the Bragg condition (δ = 0), i.e. δ = ± π/2L; ± 3π/2L; ± 5π/2L, ….with the fundamental lasing mode occurring at δ = ± π/2L.

The threshold value for the PT-symmetric grating strength can be found from Eq. (14b) under the condition that sin(k₂L) = (−1)^{m + r}:

ξ_{t h}^{(r)} = {(- 1)}^{m} \frac{2 γ}{1 + γ} \frac{(2 r + 1)}{(m + r + 1 / 2)}

The threshold should have a positive value, so if m is odd, r must be negative, leading to a “blue” shift of the lasing wavelength, δ = -π/2L; −3π/2L; −5π/2L, …, [Fig. 3(c) and Fig. 3(d)] while if m is even we get a “red” shift, δ = π/2L; 3π/2L; 5π/2L, …. [Fig. 3(a) and Fig. 3(b)]. As a result, for the even-period gratings the lowest threshold lasing starts when r = 0 with the shift δ = π/2L or Δν = c/4n₂L from the Bragg frequency, as shown in Fig. 3(a) and Fig. 3(c). The thresholds have slightly different expressions for gratings with even and odd numbers of periods, but in fact the PT-symmetric gratings with m = 72 periods (even) and with m = 73 periods (odd) have the same threshold 2γ/(72.5(1 + γ)), but with opposite shifts with respect to the Bragg wavelength (λ_B = 2Λ/(ε₂)^1/2 = 632.071 nm). The frequency spacing to the next lasing mode is c/(2(ε₂)^1/2L). This frequency spacing is equal to the longitudinal mode spacing of a traditional two-mirror cavity of the same length L as the PT-symmetric grating. Figures 3(b) and 3(d) demonstrate that when the pumping level is higher than the threshold for the first mode (r = 1), the lasing hops to the next lasing mode (r = 2) and so on. Clearly, such a structure does not produce the desirable single-mode operation.

Fig. 3 Reflection spectra for the PT-symmetric grating with ϕ = 0 for even number of periods L = 72Λ (a) and (b) and for odd number of periods L = 73Λ (c) and (d) below the threshold for the first longitudinal mode: ξ< ξ _th⁽¹⁾ for (a) and (c) and ξ _th⁽¹⁾ <ξ< ξ _th⁽²⁾ for (b) and (d). The blue (dashed) vertical lines show the Bragg wavelength λ_B = 2Λ/(ε₂)^1/2 = 632.071 nm when δ = 0. The other parameters: ε₁ = 1; ε₂ = 2.4; Λ = 204 nm.

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4.1.2 ϕ = π/2, m₁₂^(PT) real

It is easy to notice that the mode spacing in k-space Δk = δ = π/2L corresponds exactly to the spacing between the zeros of the sinc-function that is a specific feature of this PT-symmetric grating. If the lasing modes can be shifted so that the fundamental mode occurs at δ = 0, the structure will support only a single lasing mode coinciding with the main lobe of the sinc-function, as illustrated in Fig. 4(b).

Fig. 4 Possible lasing modes for (a) ϕ = 0, (b) ϕ = π/2. The red bars indicate the possible values of k₂L, while the blue curve is the sinc function of Eq. (16).

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This can be achieved by taking ϕ = π/2, i.e. phase-shifting the PT-symmetric grating with respect to the mirror at the bottom and the ε₁/ε₂ interface by a quarter period, Λ/4. This results in m₁₂^(PT) becoming real, and corresponds to the following complex dielectric constant modulation:

ε (x, z) = ε_{2} (1 + j (ξ / 2) \exp (2 j β z))

By engineering the complex modulation in this way the threshold condition takes the following form:

\sin (k_{2} L) = 0

\frac{2}{1 + γ} \cos (k_{2} L) = k_{2} L \frac{ξ}{2} \frac{\sin (δ L)}{δ L}

As a result of the phase change, it is now the imaginary part, Eq. (17a), that determines the admissible values of k₂, namely k₂L = sπ, which, upon writing s = m + r, is equivalent to δ = k₂-β = rπ/L, where r = 0, ± 1, ± 2, ….

The phase condition is satisfied for all these r values. However, Eq. (17b) has a solution only for the fundamental mode, r = 0, because sinc(δL) is zero for all higher order modes. Thus the PT-symmetric cavity in this configuration supports only the fundamental mode. The cavity becomes completely transparent at the wave-numbers for any higher order modes, and such modes simply cannot be amplified. The threshold condition is

ξ_{t h}^{(0, m Λ)} = {(- 1)}^{m} \frac{4}{π m (1 + γ)}

with m even, as seen from Fig. 5(a). At the same time, for m odd, the interaction becomes completely destructive, with light suppression at the Bragg wavelength, as shown in Fig. 5 (b). If instead we took ϕ = -π/2, lasing would occur for m odd [Fig. 5(d)].

Fig. 5 Reflection spectra for the PT-symmetric grating with ϕ = π/2 (a) and (b) and ϕ = -π/2 (c) and (d) for even number of periods L = 72Λ (a) and (c) and for odd number of periods L = 73Λ (b) and (d) below the threshold for the fundamental longitudinal mode. The blue (dashed) vertical lines show the Bragg wavelength λ_B = 2Λ/(ε₂)^1/2 = 632.071 nm when δ = 0. The other parameters: ξ = 0.008; ε₁ = 1; ε₂ = 2.4; Λ = 204 nm.

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Bearing in mind that the coupling coefficient of the PT-symmetric grating is κ = π(2Δε)/ ((ε)^1/2λ), the threshold condition can be presented in the more traditional form: κL = 2/(γ + 1), or in the case of zero Fresnel reflection κL = 1, which exactly coincides with the threshold value in the case of two concatenated PT-symmetric gratings with their non-diffractive sides oriented outwards from the cavity [20]. This threshold value, κL = 1, is 2/π times lower than the threshold value for a traditional DFB laser according to Kogelnik and Shank [5]. The constructive contribution of the Fresnel reflection can be used to reduce the threshold even lower based on the factor 2/(γ + 1) when ε₁ > ε₂.

4.1.3 Dielectric contrast and the threshold

It is important to point out that in the configuration described there is no need for a top reflector (mirror). The Fresnel reflection from the ε₁/ε₂ interface affects the threshold, but it is not critical in achieving lasing. The external dielectric constant, ε₁, strongly influences the threshold condition through Fresnel reflection. Although there is no need for an external reflector because the PT-symmetric grating provides this reflectivity and the reflective feedback, the more light is reflected back into the cavity by the interface in addition to the light reflected by the grating, the more power is involved in the positive feedback and the lower the threshold should be if these two reflected light waves interfere constructively. This is exactly what occurs in the case ε₂ > ε₁. The higher the ε₂ /ε₁ ratio, the lower the threshold compared with the case of zero Fresnel reflection, when ε₂ = ε₁, and $ξ_{t h}^{(0)} = 2 / (π m)$ . On the other hand, when ε₁ > ε₂, the Fresnel reflected light and the grating reflected light interfere destructively, so the threshold increases. In Fig. 6 the red curve shows the threshold dependence on the external dielectric constant for ε₂ = 2.4 and m = 72 based on Eq. (17).

Fig. 6 The threshold value as a function of the external dielectric constant; the red and blue curves are based on the approximate Kogelnik method [Eqs. (18) and (19) respectively] and the magenta diamonds and black circles are numerically calculated values based on the exact Bessel function method [27]. The vertical dashed line shows the threshold for zero Fresnel reflection, when ε₁ = ε₂.

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It is possible to adjust the phase of the PT-symmetric grating within our vertical cavity laser in such a way that its interaction with the $ε_{1} / ε_{2}$ interface will be opposite to the situation shown in Fig. 6 by the red curve. This follows from Eq. (17) when, instead of consisting of a whole number of grating periods, the PT-symmetric grating has a half-integral number of periods, i.e. L = (m + 1/2)Λ. In that case it is easy to show that such a configuration also provides intrinsically single-mode lasing at the Bragg wavelength with the threshold condition:

ξ_{t h}^{(0, (m + 1 / 2) Λ)} = \frac{2 γ}{π (1 + γ)} \frac{2}{(m + 1 / 2)}

The As a result of the change in phase, the Fresnel contribution now interferes constructively with the PT-symmetric grating reflection for $ε_{1} < ε_{2}$ and destructively for $ε_{1} > ε_{2}$ , as shown by the blue curve in Fig. 6, with a few calculated values from the exact method. For zero Fresnel reflection ( $ε_{1} = ε_{2}$ ) the threshold is clearly the same for the both configurations. The blue curve in Fig. 3 shows the threshold dependence on the external dielectric constant for $ε_{2}$ = 2.4 and m = 72 based on Eq. (19).

4.2 Structure robustness of the intrinsically single-mode operation

The intrinsically single longitudinal mode of operation is clearly demonstrated for a perfectly balanced PT-symmetric grating, i.e. when the contributions from the index modulation and from the gain/loss modulation are equal, $κ_{n} = κ_{σ}$ , and a quarter period shift between these two modulations results in the vanishing of one of the off-diagonal element, m₂₁, in Eq. (12). However, such a perfect balance could be difficult to achieve in practice. We can expect that the real PT-symmetric grating could be slightly unbalanced, i.e. $κ_{n} \neq κ_{σ}$ ; $κ_{21} = κ_{n} - κ_{σ} \neq 0,$ , $κ_{21} < < κ_{12}$ . So, the question is how this unbalanced PT-symmetric grating will support the intrinsically single mode behaviour. We compare the reflection spectra of the balanced, $κ_{n} = κ_{σ} = κ_{0}$ , and misbalanced PT-symmetric gratings with 10% misbalance towards the gain/loss modulation, i.e. $κ_{σ} = 1.1 κ_{0}$ , $κ_{N} = 0.9 κ_{0}$ , which we will call gain/loss-dominated, and 10% misbalance towards the index modulation $κ_{N} = 1.1 κ_{0}$ , $κ_{σ} = 0.9 κ_{0}$ , which we will call index-dominated. Our simulation shows that in both cases the misbalanced gratings preserve the main resonance peak with amplification at the resonance. The amplification is higher than the balanced case for the gain/loss-dominated grating ( $κ_{n} = 0.9 κ_{0}$ , $κ_{σ} = 1.1 κ_{0}$ ), and lower for the index-dominated grating ( $κ_{n} = 1.1 κ_{0}$ , $κ_{σ} = 0.9 κ_{0}$ ). Taking as an example the PT-symmetric grating with L = 72Λ, Λ = 204 nm, ε₁ = 1; ε₂ = 2.4, we can investigate the threshold conditions for the balanced, gain/loss-dominated and index-dominated configurations. Obviously the threshold is achieved first for the gain/loss-dominated grating at ξ_th^(G/L-0) = 0.01042. The balanced grating can be brought to lasing at ξ_th^(B) = 0.01075, as also follows from Eq. (19), and finally the threshold for the index-dominated is achieved at ξ_th^(IN-0) = 0.01105.

Of course, it is more intriguing to find out what will happen if we continue to increase ξ. At ξ values close to 0.03 the central peak at the Bragg wavelength recedes, and two amplifying peaks emerge at approximately 619 nm and 644 nm, as shown in Fig. 7. However, the peaks growing only for the gain/loss-dominated grating (magenta, dashed curves) simultaneously moving towards one another in the process of growing [Fig. 7(b) and 7(c)]. Finally the peaks merge, and lasing at the second longitudinal mode occurs at the Bragg frequency at ξ_th^(G/L-1) = 0.0983 [Fig. 7(d)]. Then the next (third) longitudinal mode takes place at ξ_th^(G/L-2) = 0.186. It is very important to emphasize that not only the balanced, but also the index-dominated grating does not lase at any higher longitudinal modes.

Fig. 7 Reflection spectra for the PT-symmetric balanced grating (solid, red curves), gain/loss dominated grating (magenta, dashed curves) and index-dominated grating (blue, dotted curves) for different value of the coupling coefficient ξ with ϕ = π/2, L = 72Λ, ε₁ = 1; ε₂ = 2.4; Λ = 204 nm.

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Another point to emphasize is that the lasing process of the gain/loss-dominated grating is very different from those of traditional gain-coupled DFBs described by Kogelnik and Shank [5]. They demonstrated that after the first lasing condition occurs at the Bragg resonance (δ = 0) at the threshold $κ_{σ} L = π / 2$ , the next order lasing takes place at two frequencies symmetrical about the Bragg frequency (δ≠0) at $κ_{σ} L = 3 π / 2$ . In contrast to this lasing pattern, the proposed structure even in its gain/loss-dominated form always lases at the Bragg wavelength, as shown in Fig. 7, with significant mode discrimination ξ_th^(G/L-1)/ξ_th^(G/L-0) = 0.0983/0.01042 = 9.4. The mode discrimination for the traditional gain coupled DFB structure is only 1.5π/0.5π = 3, i.e. three times lower.

Finally we reproduce the reflective characteristics in a vertical direction for the above-discussed parameters using Bessel method [27] that covers not only the wavelength spectrum, but also the angular one. For the PT-symmetric grating with period Λ = 0.204 µm and λ = 632.07 nm, the diffraction occurs at the Bragg angle θ_B = 0.85°. The spectral characteristics of such a structure are shown in Figs. 8(a) and 8(c) at normal incidence, θ = 0, for the coupling strength below the threshold, ξ = 0.005, and practically at the threshold ξ = ξ_th = 0.00685, respectively. The dashed blue vertical line shows the lasing wavelength, which coincides with the Bragg wavelength in Fig. 8(c).

Fig. 8 Reflective wavelength (a, c) and angular (b, d) spectra for ε₁ = 6; ε₂ = 2.4; ε₃ = ∞; L = 72Λ; Λ = 0.204 µm at ξ = 0.005 (a, b) and at the threshold condition: ξ = ξ_th = 0.00685 (c, d).

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The proposed PT-symmetric surface-emitting laser also exhibits a very nice angular spectrum, as shown in Fig. 8(b) below the threshold and in Fig. 8(d) at the threshold for the lasing wavelength ( $λ_{t h} = 632.07$ nm) with a divergence of less than 0.3° in the internal refraction angle, θ.

5. Conclusion

In conclusion, we have proposed and simulated a surface-emitting cavity based on a PT-symmetric DFB structure. Our analysis shows that the proposed structure exhibits unusual lasing characteristics. The structure can be configured in such a way that it becomes intrinsically single mode in its balanced configuration regardless of the gain spectral bandwidth and the cavity length. Compared with vertical cavity surface emitting lasers (VCSELs) that also operate in single-mode regime due to a very short cavity with a limited number of amplifying quantum wells (3-5 for electrical pumping, the proposed design does not impose any restriction on the length of the cavity (grating) to preserve its single-mode operation. We also showed that a deviation of the PT-symmetric grating from its perfectly balanced form will preserve its intrinsically single-mode operation when the coupling contribution from the gain/loss modulation is not equal to the coupling contribution from the index modulation.

We have considered a perfect mirror for simplicity, but it is easy to modify our formulae to accommodate a reflectivity less than one. In that case the lasing conditions remain unaltered, but the expressions for the lasing threshold change slightly.

The influence of the Fresnel reflection from the boundary between the grating slab and the superstrate was analyzed, and we demonstrated that this reflection can be used to reduce the threshold by more than 30%. Unlike the earlier proposed PT-symmetric ring laser [28], our design provides unidirectional lasing.

In this paper we have not addressed the issue of transverse modes. The cavities can be made as an array of small pillars each of which supports only one transverse mode. There are also many methods that have been developed for VCSELs to suppress higher-order transverse modes, based on their different radial profiles [32]. However, we believe that the most efficient approach, which we are planning to explore in our next paper, is to use PT-symmetry in the transverse direction [33].

The proposed structure paves a route for designing a special class of multifunctional optical active devices with completely nonsymmetrical optical responses and new mechanisms for light manipulation.

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Parity time-symmetric vertical cavities: intrinsically single-mode regime in longitudinal direction

Abstract

1. Introduction

2. Statement of the problem

3. Pure reflective PT-symmetric grating in amplifying and lasing modes

4. The threshold condition using the modified Kogelnik approach

4.1 Perfectly balanced PT-symmetric grating

4.1.1 ϕ = 0, m₁₂^(PT) pure imaginary

4.1.2 ϕ = π/2, m₁₂^(PT) real

4.1.3 Dielectric contrast and the threshold

4.2 Structure robustness of the intrinsically single-mode operation

5. Conclusion

References and Links

Cited By

Figures (8)

Equations (21)

Optics Express

Abstract

1. Introduction

2. Statement of the problem

3. Pure reflective PT-symmetric grating in amplifying and lasing modes

4. The threshold condition using the modified Kogelnik approach

4.1 Perfectly balanced PT-symmetric grating

4.1.1 ϕ = 0, m12(PT) pure imaginary

4.1.2 ϕ = π/2, m12(PT) real

4.1.3 Dielectric contrast and the threshold

4.2 Structure robustness of the intrinsically single-mode operation

5. Conclusion

References and Links

Cited By

Figures (8)

Equations (21)

Optics Express

4.1.1 ϕ = 0, m₁₂^(PT) pure imaginary

4.1.2 ϕ = π/2, m₁₂^(PT) real