1. Introduction

Photonic-crystal surface-emitting lasers (PC-SELs) [1–13] which possess a two-dimensional (2D) periodic refractive-index distribution adjacent to the active layer as shown in Fig. 1, have attracted a great deal of attention due to their promising characteristics. Large-area single-mode oscillation which enables watt-class high-power emission with narrow beam divergence (FWHM<1°) and high beam quality (M ²<1.1) beam has been successfully demonstrated [7]. Furthermore, unique functionalities of PC-SELs such as control of polarization, beam patterning, and beam steering have been also reported [3, 8–13].

 

Fig. 1 Schematic of photonic-crystal surface-emitting lasers and a typical photonic-band structure of square-lattice photonic crystal with circular air holes calculated by plane-wave-expansion method.

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Focusing on polarization control, PC-SELs with several polarizations has been realized by tuning in-plane shape and arrangement of air holes in photonic-crystal layers [3, 8, 10, 11]. However, there has been limited investigation into properties of polarization in previous work: phases of electric fields in all directions are same in beam plane. Beams which have phase difference of electric fields in each directions or phase distribution in beam plane, such as circularly-polarized beam and Laguerre-Gaussian beam, have not been realized with PC-SELs. To realize desired beam pattern and polarization from small one-chip laser device, controlling phase properties of each electric fields is needed.

In this paper, we report the design of PC-SELs with circularly-polarized beam toward arbitrary control of phase properties of each electric field. We expect that our work will lead to chip-sized lens-free light source with desired beam pattern and polarization, and contributes to several laser applications, such as imaging [14], photochemical reactions [15], and optical excitation of spin-polarized electrons [16].

2. Proposal of device structure

Figure 2(a) shows the schematic of photonic-crystal structure we proposed. This structure has oblique-triangular-prism-shaped air holes which are arranged in a square lattice. Focusing on the lowest frequency mode at the second Γ point, indicated as “Mode A” in Fig. 1, electric fields are distributed around air holes as shown in Fig. 2(b). Electric field distribution is calculated using rigorous coupled-wave analysis (RCWA) method (see Appendix A for details). In this case, electric fields in y direction are overlapped with tops and bottoms of air holes. It is expected that electric fields in y direction are mainly diffracted to vertical direction because electric fields in y direction should be scattered by air holes due to their overlapping with air holes. On the other hand, at the middles of air holes, electric field is distributed all direction. In this case, polarization of diffracted light is determined by in-plane air-hole shape as shown in our previous work [8]. Electric fields in x direction should be mainly diffracted at the middles of air holes by using equilateral triangle shape as shown in Fig. 2(b). In this instance, electric fields in x and y direction have phase difference due to the difference of the height of diffraction point; difference of the height of diffraction point produces the optical path difference between E_x and E_y in z direction (light output direction). The phase difference can be tuned by the height of air holes. It is expected that circularly-polarized component is emitted when phase difference becomes π/2.

 

Fig. 2 (a) Schematic of photonic-crystal structure for generation of circularly-polarized beam. (b) Electric field distribution inside a unit cell at the top (upper panel), middle (middle panel), and bottom (lower panel) of air hole.

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3. Results and Discussion

We investigated appropriate height of PC layer and tilt angle of air holes using three-dimensional coupled-wave theory (3D-CWT) [17–19]. Figure 3 shows the ellipticity of polarization vs tilt angle of air holes θ with different thickness of PC layer. Lattice constant of photonic crystal a was set as 295 nm, cavity length L was 200 μm, filling factor of air holes was 16%, and air holes were tilted in Γ-X direction. Ellipticity of polarization χ is defined as $\chi =\sqrt{I_{min}/I_{max}}$, where I_max and I_min are maximum and minimum intensity of polarization characteristics calculated by 3D-CWT (see Appendix B for details). We then calculate intensity and phase difference of electric fields Ex and Ey in the case of h = 120 nm which shows the highest ellipticity of polarization in Fig. 3 (light-blue line). Figure 4(a) and 4(b) show amplitude and phase difference of Ex and Ey. We consider PC area to be infinite. It is confirmed that amplitude of Ex and Ey crosses around θ = 30 degrees and phase difference between Ex and Ey is around 1/2π regardless of θ when h = 120 nm. Figure 4(c) shows the influence of tilt angle θ on Stokes parameters S₁, S₂, and S₃. As a result, S₃ which indicates circularly polarized component becomes dominant as θ increases. When θ = 31.6 °, S₃ reaches to 0.998. In this case, S₁ and S₂ are suppressed: S₁ = −0.035 and S₂ = −0.045, respectively.

 

Fig. 3 Ellipticity of polarization χ vs. tilt angle θ with different height of photonic-crystal layer.

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Fig. 4 (a) Amplitude of Ex and Ey vs. tilt angle θ, (b) Phase difference δ between Ex and Ey vs. tilt angle θ, (c) Stokes parameters S₁, S₂, and S₃, vs. tilt angle θ.

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We also investigate the influence of cavity length L. We calculate degree of polarization (DOP) with different cavity length L for h = 120 nm and θ = 31.6°. DOP is improved as L increases as shown in Fig. 5. DOP is over 0.9 when L is above 200 μm. Beam divergence is reduced as lasing area becomes large. Figure 6 shows results of threshold gain in the cases of L = 50 μm and L = 200 μm. These results shows that DOP increases as L increases. However, tolerance of single-mode oscillation might be deteriorated due to reduced threshold gain margin between mode A and B as L increases. However, we expected that tolerance of single-mode oscillation can be improved by further optimization of not only air-hole shape but also whole device structure such as the distance between photonic crystal layer and active layer, refractive-index contrast inside photonic crystal layer, and shape of electrode.

 

Fig. 5 DOP vs. cavity length L calculated by 3D-CWT. Two panels in the graph show far-field profiles with L = 50 and 200 μm.

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Fig. 6 Threshold gain vs. tilt angle θ with (a) L = 50 μm, and (b) L = 200 μm.

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Finally, we show the relationship between rotation directions of circular polarization and tilt direction of air holes. It is revealed that right-handed or left-handed circular polarization can be chosen by changing tilt direction of air holes as shown in Fig. 7.

 

Fig. 7 Polarization characteristics with different tilt direction.

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4. Summary

We have proposed newly designed PC-SELs which can directly emit circularly-polarized beam. We have investigated appropriate height and tilt angle of oblique-triangular-prism-shaped air holes in photonic-crystal layer to achieve high DOP. We have also revealed that high DOP can be achieved with larger cavity length. It was recently revealed that our proposed photonic-crystal structure can be fabricated by using angled-etch method [20] (see Appendix C for details). We continue further optimization toward the demonstration of PC-SELs with circularly polarized beam. We expect that our work can lead to the realization of chip-sized lens-free light source with circular polarization, contributing to improved laser applications.

Appendix A Results of rigorous coupled-wave analysis

In this section, we show the results of calculation of electromagnetic field with oblique-triangular-prism-shaped air holes using rigorous coupled-wave analysis (RCWA) [21].

Figure 8 shows the calculated absorptivity spectrum with left- and right-handed circularly polarized light incident. Incoming light was vertically irradiated. Lattice constant a was set as 295 nm, filling factor of air holes was set as 16%, height of air holes was set as 120 nm, and tilt angle θ was set as 26°. Absorption coefficient in MQW was set to be constant independently of the frequency. It is shown that absorptivity is drastically changed at the peak around 0.298 c/a. That indicate the strong resonance exists only in the left-circularly polarized light incident and this structure has circular dichroism at this frequency. It is expected that circularly polarized light emission can be realized by using this resonant mode. Figure 9 shows the calculated electric field distribution with left-circularly polarized light incident at the frequency of 0.2979 c/a. It is revealed that electric field distribution around air holes is not gradually sifted along the z direction although air holes are tilted.

 

Fig. 8 Calculated absorptivity spectrum with left- and right-handed circularly polarized light incident.

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Fig. 9 (a) Cross-sectional structure used in RCWA calculation. Distribution of (b) E_x and (c) E_y with left-circularly polarized light incident at the frequency of 0.2979 c/a. (d) In-plane electric field distribution around air holes in three different height indicated as I, II, and III in (a).

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Appendix B Calculation of far-field profile and polarization using three-dimensional coupled-wave theory

According to Ref [19], the far-field pattern and polarization of the output laser beam can be calculated using 3D-CWT. The complex radiation field just above the PC surface ΔE_j(x, y, z = d_PC/2) can be calculated, where d_PC is the height of the PC layer, and z = 0 is defined at the center of the PC layer (for further detail, see Ref [19].). The time-dependent amplitude of the far field F_x and F_y, functions of the deflection angle (θ_x, θ_y) from the surface normal, can be found by taking the Fourier transform of the radiation field and multiplying by an obliquity factor (cosθ_x+cosθ_y−1) [22] as below,

(1)$$\begin{array}{l}{F}_j\left({\theta }_x,{\theta }_y,t\right)\propto \left(\cos {\theta }_x+\cos {\theta }_y-1\right)\underset{0}{\overset{L}{{\displaystyle \iint }}}\Delta E_j\left(x,y,z=\frac{d_{\text{PC}}}{2}\right)e^{i\omega t}\\ ·e^{-i{k}_0\left(\text{tan}{\theta }_{x}x+\text{tan}{\theta }_{y}y\right)}dxdy,j=x,y\end{array}$$

Then, the time-averaged far-field profile is given as

(2)$$B({\theta }_x,{\theta }_y)=B_x({\theta }_x,{\theta }_y)+B_y\left({\theta }_x,{\theta }_y\right)$$

where $〈B({\theta }_x,{\theta }_y)〉$ and $〈B_y\left({\theta }_x,{\theta }_y\right)〉$ are defined as

(3)$$B_j\left({\theta }_x,{\theta }_y\right)=\frac{1}{T}\underset{0}{\overset{T}{{\displaystyle \int }}}|Re[F_j({\theta }_x,{\theta }_y,t)]{|}^{2}dt,j=x,y,T=2\pi /\omega$$

The polarization characteristics I(φ), function of polarization angle φ, is given as

(4)$$I\left(\phi \right)={{\displaystyle \iint }}^{\text{}}\left\{\frac{1}{T}\underset{0}{\overset{T}{{\displaystyle \int }}}{\left|Re\left[F_x\left({\theta }_x,{\theta }_y,t\right)\cos \phi +F_x\left({\theta }_x,{\theta }_y,t\right)\sin \phi \right]\right|}^{2}dt\right\}d{\theta }_{x}d{\theta }_y$$

We defined I_max and I_min as the maximum and minimum value of I(φ).

Stokes parameters S₀, S₁, S₂, and S₃ are also given as

(5)$$S_0=I\left(0°\right)+I\left(90°\right)$$

(6)$$S_1=I\left(0°\right)-I\left(90°\right)$$

(7)$$S_2=I\left(45°\right)-I\left(135°\right)$$

(8)$$S_3=I_{\lambda /4}\left(45°\right)-I_{\lambda /4}\left(135°\right)$$

where

(9)$$I_{\lambda /4}\left(\phi \right)={{\displaystyle \iint }}^{\text{}}\left\{\frac{1}{T}\underset{0}{\overset{T}{{\displaystyle \int }}}{\left|Re\left[F_x\left({\theta }_x,{\theta }_y,t\right)\cos \phi +i{F}_x\left({\theta }_x,{\theta }_y,t\right)\sin \phi \right]\right|}^{2}dt\right\}d{\theta }_{x}d{\theta }_y$$

Then degree of polarization (DOP) is given as $\sqrt{S_1{}^2+S_2{}^2+S_3{}^2}/S_0$.

Appendix C Demonstration of angled-etch method for realization of oblique-triangular-prism-shaped air holes

Figure 10 shows SEM images after etching. It was revealed that photonic crystal structure with oblique-triangular-prism-shaped air holes can be fabricated.

 

Fig. 10 (a) Top-view and (b) cross-sectional SEM images of photonic crystal structure with oblique-triangular-prism-shaped air holes after etching.

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Funding

This work was partially supported by the Accelerated Innovation Research Initiative Turning Top Science and Ideas into High-Impact Values (ACCEL) of the Japan Science and Technology (JST) Agency, and by the Consortium for Photon Science and Technology (C-PhoST) of the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). M. Nishimoto was supported by research fellowships of Japan Society for the Promotion of Science (JSPS).

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