arXiv:1701.03649v1 [physics.optics] 13 Jan 2017

Analytical coupled-wave model for photonic crystal quantum cascade lasers Zhixin Wang1, Yong Liang2,∗, Xuefan Yin1 , Chao Peng1,+ , Weiwei Hu1, and J´erˆome Faist2 1:State Key Laboratory of Advanced Optical Communication Systems and Networks, Peking University, Beijing, 100871, China 2:ETH Zurich, Institute of Quantum Electronics, Auguste-Piccard-Hof 1, Zurich 8093, Switzerland E-mail: *[email protected];+:[email protected]

A coupled-wave model is developed for photonic-crystal quantum cascade lasers. The analytical model provides an efficient analysis of full three-dimensional large-area device structure, and the validity is confirmed via simulations and previous experimental results.

1 Introduction Quantum cascades lasers (QCLs) [1] are unique semiconductor emitters that cover the entire mid-infrared (MIR) region of the electromagnetic spectrum. During the last two decades, significant advances have been made in the many aspects of the device, such as layer processing and laser mounting [2, 3], thermal and optical loss management [2, 3], as well as the active region design and optimization [3, 4]. Nowadays, QCLs are becoming the most promising light sources for many laser-based applications, e.g., trace gas spectroscopy [5], process control [6] and biological sensing[7]. However, a number of important MIR applications prefer singlemode operation, high output power, and good beam quality, including infrared countermeasures 1

[8], remote sensing, and photo-acoustic spectroscopy [7]. A promising approach to keep a single-mode lasing even over a very large area is to use a two-dimensional (2D) periodic structure, i.e., photonic crystal (PhC) [9]. This is because PhC enables flexible control of the cavity loss of the laser mode candidates and therefore allows finer control of the mode selection compared to the conventional DFB lasers. Combining advantages of both PhC and intersubband transitions, PhC-QCL has attracted much attention in the recent years [10, 11, 12]. Despite the experimental advances, systematic theoretical investigation has yet been lacking. This is because it is very challenging to use the commonly-used brute-force simulation tools such as finite-difference time-domain (FDTD) or finite-element methods to simulate this type of large-area lasers. In recent years, our group has developed an analytical Coupled Wave Theory (CWT) framework to investigate the physics of 3D periodic photonic structures, including PhC [13, 14, 15, 16, 17] and gratings [18]. This framework provides comprehensive and reliable modeling of photonic crystal surface-emitting lasers (PC-SELs) [9] and bound states in continuum (BIC) interpretation [19, 20]. The finite-size effects have been studied on TE polarization [21, 22, 23]. For TM mode, K. Sakai’s has worked on 2D model [24]. Our previous CWT work provides preliminary insights in physics [25, 19], however it cannot model the finite-size structure, and the accuracy is not good enough, as we will show later. In this work, we present an analytical model, the improved 3D-CWT to model surfaceemitting PhC-QCLs. This model provides various laser properties, including band structure, mode pattern, intensity profile, mode frequency, cavity loss, and far-field pattern (FFP), etc. The presented 3D-CWT works effectively for large-area PhC-QCL with ordinary computation resources. Comparison with FDTD [26] and previous experimental results demonstrates the validity of our model.

2

2 3D CWT Models a)

b)

B

E1

A

E2

c) κ 2 : Direct 2D coupling κ 3 : Direct 1D coupling

Surface Emission

Top electrode Frequency (c/a)

0.5

InP

InP

Ry

0.33 0.32

0.4

B

0.3

A

0.3

E2

0.29

0.2

0.28 -0.05

0

E1 0.05

0.1 0 -0.5

Bo!om electrode

0

X - Γ- M

κ2

Sx

0.31

Rx

κ3 Sy

0.5

Figure 1: (a) Device schematic of PhC surface-emission QCLs. The square-lattice PhC pillars (red color) are formed by dry-etching. Surrounding medium are semi-insulating InP (dark gray). (b) Typical band structure of the PhC for TM mode. The right panel shows the in-plane mode pattern of the PhC layer depicted in (a). (c) Two major coupling mechanism inside PhC slab, κ2 : direct 2D coupling and κ3 : direct 1D coupling. A schematic of a surface emitting PhC-QCL is displayed in Fig. 1 (a). The square lattice PhC is constructed by circular shaped active material, surrounded by InP with low refractive index. Since intersubband transition does not provide gain parallel to growth direction, the PhC-QCL works only with TM polarization. A typical band structure of TM-polarized PhC is shown in Fig. 1 (b). At the 2nd-order Γ-point, there exist four band edge modes, A, B, E1 and E2 [indicated in the inset of Fig. 1 (b)]. The magnetic field inside the PhC follows Maxwell’s equation: ∇×[

1 ∇ × H(r)] = k 2 H(r), ε(r)

(1)

where ε(r) is a periodic function, representing the permittivity distribution of the structure, and k is the wavenumber. For TM mode, H(r) = (Hx (r), Hy (r), 0), can be expanded folP lowing Bloch’s theorem: Hi (z) = Hi,mn (z)e−imx β0 x−iny β0 y , i = x, y, where β0 = 2π/a (a is the lattice constant), mx = m + ∆x , ny = n + ∆y and ∆x , ∆y are deviations from 3

the Γ point [14]. Similarly, the Fourier transform of 1/ε(r) can be written as: 1/ε(r) = P ′ ′ ξ00 + ξm′ n′ e−im β0 x−in β0 y , where κ00 represents the average refractive index of the material. In the PhC layer, κ00 = f f (1/εa ) + (1 − f f )(1/εs), where εa and εs are the permittivity of the active material and the surrounding InP, respectively, and f f represents the filling factor. For simplicity, vertical sidewall is assumed (solution for tilted case is discussed in Ref. [13]). In the perspective of CWT [14], the Bloch waves inside the PhC slab can be classified √ into three groups: basic waves ( m2 + n2 = 1); radiative waves (m = n = 0) and high-order √ waves( m2 + n2 > 1). At 2nd-order Γ point, the basic waves, (m, n) = {(1, 0), (−1, 0), (0.1), (0, −1)}, dominate the energy inside the PhC slab. The amplitude of the four basic waves are denoted, in order, as Rx (x, y), Sx (x, y), Ry (x, y) and Sy (x, y). With the procedure described in Appendix, the coupling equations can be obtained and simplified as :  ∂R      − ∂xx Rx Rx  ∂Sx   Sx   Sx  ∂x   + i  (δ + iα)  ∂Ry  = C   Ry  Ry  − ∂y  ∂Sy Sy Sy ∂y

(2)

where δ and α represent the frequency deviation from guided mode and the cavity loss, respectively. The 2nd term on the left-hand side depicts the varying envelope of the basic waves, induced by the finite-size effect [21]. The matrix C comprehensively describes the couplings between the basic waves. Inheriting the concept of 1D distributed feedback (DFB) structure [27], the boundary condition is adopted as: L L L L Rx (− , y) = Sx ( , y) = Ry (x, − ) = Sy (x, ) = 0 2 2 2 2

(3)

By solving the coupling equation, we are able to obtain the electromagnetic field of a mode, together with real and imaginary parts of the corresponding frequency. Further, various laser 4

properties of the PhC-QCL can be interpreted, including the mode frequency, cavity loss, threshold gain, and intensity profile, FFP, etc.

3 Results and Discussions 0.40

b) 3D−FDTD 3D−CWT

0.35

0.30

0.25 −0.2 (X)

Frequency (ωa/2πc)

Frequency (ωa/2πc)

a)

0.330 0.325 0.320 0.315 0.310 0.30

0 (Γ) 0.2 (M) k (2π/a)

3D−FDTD Mode A 3D−FDTD Mode B 3D−FDTD Mode E 3D−CWT Mode A 3D−CWT Mode B 3D−CWT Mode E

0.40 0.50 Filling factor

0.60

Figure 2: (a) Band structure of the PhC-QCL calculated by 3D-CWT and 3D-FDTD near 2nd order Γ point. Filling factor (FF) = 0.3. (b) Band edge mode frequency dependence on filling factors (FFs). First, we calculate infinitely periodic structure to validate the proposed 3D-CWT. The band structure is shown in Fig. 2 (a). The parameters of the model are: a = 2.7µm, εa = 11.169, εs = 9.386, thickness of PhC slab tg = 2.5µm, filling factor (FF) = 0.3. Top cladding and substrate material are also InP (εs ). In addition, the mode frequency dependence on filling factors (FFs) is shown in Fig. 2 (b). The comparison between 3D-CWT and 3D-FDTD indicates an excellent agreement between these two techniques. It is demonstrated on Fig. 2 (a) that the current 3D-CWT shows a significant improvement on accuracy. In previous results [19], a discrepancy of approximately 0.5% already occurs on k = 0.02. In contrast, Fig. 2 (a) 5

exhibits a much better consistence (δω/ω0 ≈ 0.05%, δω is the frequency deviation and ω0 is the center frequency) within k = 0.2 (one order of magnitude). This is because a new technique is introduced, including an iteratoin algorithm (this will be describe elsewhere). Even though the new technique is adopted, the calcuation time is negliably small (less than 1 minute).

1.5 αL

X

Γ

-10

-5

0

200 a

B

0 δL

5

1

Mode B

0

200 a

Mode E1,2

0.5 0

200 a

10

1 0.5

200 a

200 a

0.5 0

Mode A

0.5

M

A E1,2

1.0

1

Relative intensity

Relative intensity

Frequency

2.0

a

b)

2.5

Relative intensity

a)

200 a

Figure 3: (a) Mode spectrum (αL − δL) and (b) mode intensity profiles of PhC-QCL with 200 × 200 periods at FF = 0.50. Period a = 2.7µm. Inset of (a) shows the band structure near 2nd Γ point of PhC at FF = 0.50. The range of inset axises is: k : −0.1 - 0.13, frequency: 0.29 - 0.35. Next, we calculate the finite PhC-QCL. Compared with 3D-FDTD which is hardly applicable for a large device even on a super computer, the presented 3D-CWT finite-size program only costs several seconds on a personal computer. It provides solutions of various modes, of which the frequency deviation δL and cavity loss αL is displayed on a mode spectrum figure. Fig. 3 (a) is an example at FF = 0.50. The corresponding mode intensity profiles and patterns obtained with 3D-CWT are shown in Fig. 3 (b). The intensity profile, i.e. the field envelope of the mode, is determined by [21]: I(x, y) = |Rx (x, y)|2 + |Ry (x, y)|2 + |Sx (x, y)|2 + |Sy (x, y)|2. According to the intensity profile, it is quite straightforward that the cavity loss of mode B (6.48 cm−1 ) is less than mode A (21.85 cm−1 ) and mode E (11.48 cm−1 ) at FF = 0.50. Since the

6

model is based on 3D analysis, the cavity loss consists of both vertical radiation and in-plane leakage through boundaries. Therefore, the 3D-CWT offers an effective approach to calculate the threshold gain and to investigate the properties of lasing mode.

0.325 0.320 0.315

3D-CWT Mode A 3D-CWT Mode B 3D-CWT Mode E

0.310 0.305 0.300 0.10

b)

c)

0.330

Frequency (ωa/2πc)

Frequency (ωa/2πc)

a)

0.20

0.30 0.40 0.50 Filling factor

0.60

0.330

0.320 0.315 0.310 0.305 0.300 0.10

0.70

2D FDTD Mode A 2D FDTD Mode B 2D FDTD Mode E

0.325

0.20

0.30 0.40 0.50 Filling factor

0.60

0.70

0.60

0.70

d)

102

Cavity loss (cm−1)

Cavity loss (cm−1)

102

101 3D-CWT Mode A 3D-CWT Mode B 3D-CWT Mode E

100 0.10

0.20

0.30 0.40 0.50 Filling factor

0.60

0.70

101

100 0.10

2D FDTD Mode A 2D FDTD Mode B 2D FDTD Mode E

0.20

0.30 0.40 0.50 Filling factor

Figure 4: Frequency and cavity loss dependence of finite-size PhC-QCL on filling factors, obtained by: (a,b) 3D-CWT, L = 200a; (c,d) 2D-FDTD, L = 150a. The refractive index for active material and InP are 3.342 and 3.0637, respectively. Further, mode frequency and cavity loss of PhC-QCL with 200 × 200 periods is obtained with 3D-CWT, as shown in Figs. 4 (a) and (b). Since a 3D-FDTD simulation for such a large device is not applicable on our server, 2D-FDTD is adopted for comparison with 3D-CWT, as shown in Figs. 4 (c) and (d). Here a relatively smaller structure for 2D-FDTD is chosen to compensate the vertical confinement difference between 2D and 3D cases. Even though there is no vertical radiation in 2D case, our 3D-CWT and 2D-FDTD results more or less show 7

similar trend. This implies that the in-plane loss dominates the cavity loss. At FF= 0.50, for example, 59% percent of the energy is concentrated inside the PhC slab. But in 2D case, it is exactly 100%. Moreover, the percentage of vertical energy concentration inside the PhC slab varys under different FFs. This is why the mode gap between the individual modes in 3D case is always much smaller than that in 2D case. At larger FFs, the two results agrees better. In contrast, at small FFs, the two results deviate much more. This is because at small FFs the vertical confinement is too small, leading to a much weaker refractive index contrast compared to the 2D case. Specifically, the lasing mode (the lowest threshold mode) switches from mode A to E, then to B, and finally back to A, in accordance with the increasing FF. The mode switching points roughly correspond to the FFs where κ2 and κ3 equal to 0.

a) Mode A

Polarization Direction

b) Mode E

c) Mode B

1°

Figure 5: FFPs and polarization characteristics of lasing modes of PhC-QCL with 200 × 200 periods. Modes and parameters: (a): mode A at FF = 0.16, (b): mode E at FF = 0.25, (c): mode B at FF = 0.50. Far-field pattern (FFP) and polarization properties are crucial features for lasers, and the flexible control of them is one of the most remarkable advantages of PhC-QCLs [28]. The presented CWT is able to provide FFP and its polarization features of each single mode, by implementing Fourier transform of the radiation field at the surface of the device [21, 29]. For a PhC-QCL with 200 × 200 periods, Fig. 5 illustrates the calculated FFP and radiatial-polarized beam of the individual lasing modes at different FFs. The beam divergence angle is less than 2◦ , reflecting the large-area lasing of 2D PhC. Mode A and mode B exhibit doughnut-shaped

8

far-field beam pattern, whereas mode E is optimistic to be utilized for generating single-lobed beam. In particular, the polarized FFP of mode B perfectly reproduces previously reported experimental results [30]. a)

b) −1

Cavity loss (cm )

Frequency (ωa/2πc)

2

10

0.324 0.322 0.32 0.318

0

10

−2

10

−4

10

2000 4000 6000 8000 10000 12000 L (µm) Mode A Mode A (infinite)

Mode B Mode B (infinite)

2000 4000 6000 8000 10000 12000 L (µm) Mode E Mode E (infinite)

Figure 6: Frequency (a) and cavity loss (b) dependence on device length L (FF = 0.50 and lattice constant a = 2.70µm). Moreover, the frequency and cavity loss dependence is investigated with 3D-CWT, as shown in Fig. 6. With increasing device length, the mode frequency converges to the infinite case. At small lengths, the cavity loss decreases rapidly with increasing device length, indicating the domination of in-plane loss. For an infinitely periodic structure, the surface emission of mode A and mode B are canceled due to the desctructive interference of the basic waves [14]. Therefore, the cavity loss of mode A and mode B is declining towards zero, with increasing device length. In contrast, the radiation loss of mode E still exists on infinite structure, caused by the constructive interference of the basic waves. In this case, the cavity loss of mode E is asymptotic to the radiation loss of infinite case.

4 Conclusion In this work, an analytical model, the improved 3D-CWT is proposed for investigation of PhCQCLs. The model not only offers a physical picture to understand the feedback coupling mech9

anism inside the device, but also provides numerical solutions to finite-size surface emitting PhC-QCLs. The reliability of 3D-CWT is validated by comparison with FDTD simulation and previous experimental results. The 3D-CWT physically explores the electromagnetic fields inside the PhC-QCL by calculating the coupling between basic waves, radiative waves and high-order waves. The finite-size effect is investigated by considering the variation of field envelope. The 3D-CWT provides a variety of properties of surface-emission PhC-QCL, including band structure, mode pattern, intensity profile, mode frequency, cavity loss, and radiative beam patterns, etc. It enables valid 3D analysis of large-area PhC-QCL at ordinary computational resources. We believe this work will facilitate the optimization design of PhC-QCLs.

Acknowledgments This work is supported by National Natural Science Foundation of China (NSFC) (61320106001). Yong Liang is supported by the ETH Zurich Postdoctoral Fellowship Program and the Marie Curie Actions for People COFUND Program (No. FEL-27 14-2).

Appendix: Derivation of Coupling Equation By substituting the Fourier expansions of H and ξ into Eq. (1) and collecting terms multiplied by the same factor, we obtain ∂ ∂2 − ξ00 n2y β02 + δ(±d)∆ξ ]Hx,mn + ξ00 mx ny β02 Hy,mn 2 ∂z ∂z ∂ ∂ ∂ + ny )Hy,mn −ξ00 2iny β0 Hx,mn + ξ00 iβ0 (mx ∂y ∂y ∂x 2 ∂ ∂ ξm−m′ ,n−n′ {[− 2 + ny n′y β02 + δ(±d)∆ξ ]Hx,m′ n′ − m′x ny β02 Hy,m′ n′ ∂z ∂z [k 2 + ξ00

=

X

m′ ,n′ 6=m,n

+i(ny + n′y )β0

∂ ∂ ∂ Hx,m′ n′ − im′x β0 Hy,m′ n′ − iny β0 Hy,m′ n′ } ∂y ∂y ∂x 10

(4)

∂ ∂2 [k + ξ00 2 − ξ00 m2x β02 + δ(±d)∆ξ ]Hy,mn + ξ00 mx ny β02 Hx,mn ∂z ∂z ∂ ∂ ∂ + mx )Hx,mn −ξ00 2imx β0 Hy,mn + ξ00 iβ0 (ny ∂x ∂x ∂y 2 X ∂ ∂ ξm−m′ ,n−n′ {[− 2 + mx m′x β02 + δ(±d)∆ξ ]Hy,m′ n′ − mx n′y β02 Hx,m′ n′ = ∂z ∂z m′ ,n′ 6=m,n 2

+i(mx + m′x )β0

(5)

∂ ∂ ∂ Hy,m′ n′ − in′y β0 Hx,m′ n′ − imx β0 Hx,m′ n′ } ∂x ∂x ∂y

Substituting the explicit description of basic waves into Eq. (4) and (5), with (m, n) = {(1, 0), (−1, 0), (0, 1), (0, −1)}, the coupling equations can be obtained. Take Rx as an example: ∂2 ∂ ∂Rx − ξ00 β02 + δ(±d)∆ξ ]Rx Θ0 − ξ00 2iβ0 Θ0 2 ∂z ∂z ∂x ∂ ∂2 ξ1−m′ ,−n′ {[− 2 + m′ β02 + δ(±d)∆ξ ]Hy,m′ n′ − n′ β02 Hx,m′ n′ } ∂z ∂z [k 2 + ξ00

=

X

m′ ,n′ 6=1,0

(6)

Here, the basic wave Hy,1,0 is considered to propagate only in x direction, and high orders of derivative are neglected due to the slow variation of basic wave envelope. Since high-order Fourier terms ξ1,1 , ξ1,−1 , ξ−1,1 and ξ−1,−1 are negligibly small compared with ξ00 , corresponding terms are also neglected. The δ(±d) term illustrates the unique surface coupling mechanism of TM mode [19]. By substituting the guided mode equation: [k 2 + ξ00

∂2 ∂ − ξ00 β02 + δ(±d)∆ξ ]Θ0 = 0 2 ∂z ∂z

(7)

into Eq. (6) for all the four basic waves [21], the coupling equations Eq. (2) can be obtained.

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