Defect Design

From the previous sections, we determined several important parameters such as slab thickness (t =0.45a), hole radius (r=0.35a) and lattice constant (a=460nm).

In this section, we will design the shape of the defect. Only single defect is considered due to its small modal volume and simpleness.

Figure 3.2 Modified parameters d and r′

Modified parameters are shown in Figure 3.2. The radius of the nearest neighbor holes from the single defect is changed to r′ and they are pushed away from the defect by

d.

Using the PWE method, we found many defect shapes in which resonant modes (Figure 3.3) appear near 1550nm. Among them, three characteristic defect designs (Figure 3.4) are found by the FDTD simulation and only these three defect designs are

discussed in the next chapter.

(a) Degenerate dipole modes (b) Monopole mode

(c) Degenerate quadrupole modes (d) Hexapole mode Figure 3.3 Various fundamental resonant mode modes (d=0.1a, r′=0.25a )

Design A Design B Design C

Figure 3.4 Charateristic defect designs : Design A (r′ =0.25a, d =0.13a), Design B (r′ =0.25a, d =0), Design C (r′ =0.25a, d =0.17a)

References

[1] B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A.

Scherer, and Y. Qiu, “Liquid crystal electric tuning of a photonic crystal laser,” Appl.

Phys. Lett. 85, 360-362 (2004).

[2] S. T. Wu, C. S. Wu, M. Warenghem, and M. Ismaili, Opt. Eng. 32, 1775 (1993).

[3] S. T. Wu, Phys. Rev. A, 33, 1270 (1986).

[4] Hong-Gyu Park, Ki Hwang, Joon Huh, Han-Youl Ryu, Se-Heon Kim, Jeong-Soo Kim, and Yong-Hee Lee, “Characteristics of Modified Single-Defect Two-Dimensional Photonic Crystal Lasers”, IEEE JOURNAL OF QUANTUM ELECTRONICS 38, 1353 (2002).

[5] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press (1993).

[6] Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes”, IEEE JOURNAL OF QUANTUM ELECTRONICS, 39, 1081 (2003).

Chapter 4.

Characteristics of the Liquid Crystal Infiltrated Tunable Photonic Crystal Laser

In this chapter, we will analyze the simulation results of LC infiltrated tunable 2D PhC laser. Simulation conditions are introduced first and various characteristics in three defect designs, such as single mode laser wavelength shift and laser mode change, are analyzed.

4.1 Simulation Conditions

The simulation domain used in this research is illustrated in Figure 4.1. As mentioned in the previous chapter, InGaAsP quantum well material (ε =11.56 at λ=1550 nm) is employed as the slab material. Lattice constant (a) is 460 nm and slab thickness is

0.45a. The hole radius is 0.35a except the nearest neighbor holes from the defect.

Due to the lack of computing power, we used the simple structure which has 5 hole layers around the center defect. The domain size is 12.5a×11a×5a and the spatial and time steps are assumed to be ∆ =x 23 nm and ∆ =t 3.83 10× ⁻¹⁷s, respectively. At the boundaries of the simulation domain, uniaxial perfectly matched layers (UPML) are placed as an appropriate boundary condition.

LCs are placed at the white part in Figure 4.1. The LC alignment is defined by θ and φ in the right inset of Figure 4.1. At the interface of LC medium and the slab

Figure 4.1 The structure of Design A in the simulation domain whose size is 12.5a×11a×5a. The alignment of LCs are defined by θ and φ.

Figure 4.2 Isotropic interface region

x (Γ-K⁾

y (Γ-M⁾

x z

UPML

material, gradual dielectric change is imposed. This makes the diameter of the tunable LC holes decreased by ~ 100 nm as shown in Figure 4.2. This isotropic interface region could be regarded as LC anchoring region, yet it is quite rough approximation. In the LC medium excluding the isotropic interface region, LCs are assumed to be uniformly aligned.

The spontaneous dipole emission centered at 1550 nm with FWHM 20 nm is assumed as shown in Figure 4.3 (a), and these dipole sources with random phases are randomly distributed over the slab material in the center circle of diameter 5a (Figure 4.3 (b))

All the simulation conditions are same except the defect designs of 2D PhC slab.

0 200 400 600 800 1000

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Amplitude (a.u.)

Time (fs)

1500 1550 1600

-1.00E-009 0.00E+000 1.00E-009 2.00E-009 3.00E-009 4.00E-009 5.00E-009 6.00E-009 7.00E-009

Power (a.u.)

Wavelength (nm) FWHM 20nm

(a)

(b)

Figure 4.3 (a) The spontaneous dipole emission centered at 1550 nm with FWHM 20nm (b) Random dipole sources in the center circle of D=5a

Fourier Transform

4.2. Several Characteristics of the Liquid Crystal Infiltrated Photonic Crystal Laser

In this section, we will analyze the simulation results from the three different defect designs (Section 3.2.4) ; Design A (r′ =0.25a, d =0.13a), Design B (r′ =0.25a,

0

d = ), Design C (r′ =0.25a, d =0.17a).

4.2.1. Lasing Wavelength Shift of Single Mode (Design A)

In Design A, as shown in Figure 4.4, the lasing wavelength shifts toward shorter wavelength region (from 1558 nm to 1537 nm) as θ changes from 90° to 0°. It seems reasonable compared with the experimental results [1]. We also can notice that the lasing wavelength shift and Q-factor change are not linear to the change of θ . The theoretical Q-factor is 343 when θ =90^D,φ =0^D, and 539 when θ =0^D,φ =0^D. It is reasonable that Q-factor increases as θ changes from 90° to 0° since in-plane n_eff decreases as the LC director rotates from x-axis to z-axis.

From the Hz field profiles in Figure 4.5, we can find that the lasing mode is the hexapole mode at all the differentθ . However, there are differences between the modes.

In Figure 4.6, Ex field profile at θ =90^D is mainly (b), yet there is also another mode which can be known from the oscillation between (c) and (d), while E_x field profile at θ =0^D does not. This is attributed to the liquid crystal effect by which Ex-polarized light experiences n_eff = when LCs are aligned parallel to the x-axis (n_e θ =90^D,φ =0^D),

eff o

n = when LCs are aligned parallel to the z-axis (n θ =0^D). The same Ey field profiles at (e) and (f) confirm it again.

1500 1520 1540 1560 1580 1600

Power (a.u.)

Wavelength (nm)

θ=0^ο φ=0^ο θ=30^ο φ=0^ο

θ=60^ο φ=0^ο θ=90^ο φ=0^ο

0 1

Figure 4.4 Power spectrum (resolution limit : 2.5 nm)

(a) θ =90^D, φ =0^D (b) θ =60^D, φ =0^D

(c) θ =30^D, φ =0^D (d) θ =0^D, φ =0^D

Figure 4.5 Hz-field profiles

(a) Ex (θ =0^D, φ =0^D) (b) Ex (θ =90^D, φ =0^D)

(c) E_x (θ =90^D, φ =0^D) (d) E_x (θ =90^D, φ=0^D)

(e) Ey (θ =90^D, φ =0^D) (f) Ey (θ =0^D, φ =0^D)

(a) Top view (z=0) (b) Side view (x=0)

Figure 4.7 Electric field intensity profiles of the hexapole mode (θ =0^D, φ =0^D)

This LC effect can also be confirmed by comparing with the FDTD simulation which can handle only non-birefringent materials (Figure 4.8). In this conventional FDTD simulation, the lasing mode at 1537 nm and 1558 nm can be found when n_eff is

1.4986 and 1.5833, respectively. n_eff =1.5833 is the refractive index when the light propagating in-plane direction is mean angle 45°-polarized with respect to the LC director. Thus, we can conclude that the conventional FDTD simulation well-predicts the lasing wavelength. However, It fails to predict the oscillation in the E_x-polarized light mode as shown Figure 4.6 (b)-(d). Here, we can also notice that there exists the quadrupole mode at λ=1545 nm when n_eff =1.6841, which actually doesn’t occur.

1500 1520 1540 1560 1580 1600

Power (a.u.)

Wavelength (nm)

neff=1.4986 neff=1.5833 neff=1.6841

0 1

(a) Mean-polarized angle (b) Power spectrum (resolution limit : 2.5 nm)

n_eff =1.4986 n_eff =1.5833 n_eff =1.6841

(c) Hz field profiles

(d) E_x (left) and E_y (right) field profiles when n_eff =1.5833

Figure 4.8 Conventional FDTD simulation results

4.2.2. Degeneracy Splitting (Design B)

From the Design B, we can get the dipole mode as shown in Figure 4.9. Since the dipole mode is doubly degerated, two degenerate modes are expected to appear in the ordinary hole 2D PhC slab structure [2]. However, if LCs are infiltrated into the air-holes and LC director is aligned parallel to x-axis (θ =90^D φ =0^D) or y-axis (θ =90^D φ =90^D), the degeneracy disappears as shown in Figure 4.9 (b)-(c). This is attributed to the symmetry breaking caused by LC alignment.

From Figure 4.10, we can notice that this is due to the partial polarization of in-plane electric fields. If we suitably design the defect, we expect that the intrinsic polarization of the lasing mode can be changed by LC alignment change. The electric field intensity profiles for each case are shown in Figure 4.11.

(a) θ =0^D, φ =0^D (b) θ =90^D, φ =0^D (c) θ =90^D, φ =90^D

Figure 4.9 Hz field profiles

(a) Ex (θ =90^D, φ =0^D) (b) Ey (θ =90^D, φ =0^D)

(c) Ex (θ =90^D, φ =90^D) (d) Ey (θ =90^D, φ =90^D)

Figure 4.10 Transverse electric field profiles

(a) Top view (θ =90^D,φ =0^D) (b) Top view (θ =90^D,φ =90^D)

(c) Side view (θ =90^D,φ =0^D,x=0) (d) Side view (θ =90^D,φ=90^D,x=0)

(e) Side view (θ =90^D,φ =0^D,y=0) (f) Side view (θ =90^D,φ =90^D,y=0)

Figure 4.11 Electric field intensity profiles

4.2.3. Lasing Mode Change (Design C)

In Figure 4.12, we can see the lasing wavelength shift. As θ changes from 90° to 60°, the lasing wavelength shifts toward shorter wavelength region (from 1540 nm to 1534 nm). When θ changes from 60° to 30°, the lasing wavelength shifts toward longer wavelength region, which looks strange. However, when θ changes from 30°

to 0°, the lasing wavelength shifts from 1563 nm to 1558 nm, again toward shorter wavelength region .

1500 1520 1540 1560 1580 1600

Power (a.u.)

Wavelength (nm)

θ=0^o, φ=0^o θ=30^o, φ=0^o θ=60^o, φ=0^o θ=90^o, φ=0^o

0 1

Figure 4.12 Power spectrum (resolution limit : 2.5 nm)

Here, we can guess, when θ changes from 60° to 30°, the lasing mode is switched to another lasing mode. We can confirm it by analyzing the Hz field profiles at different θ .

From the Figure 4.13 (a) and (d), we can notice that the quadrupole mode of Q-factor 295 appears at 1540 nm when 90θ = ^D φ =0^D, and the hexapole mode of Q-factor

476 appears at 1558 nm when θ =0^D φ=0^D

When 60θ = ^D, 0φ = ^D, the quadrupole mode appears at 1534 nm, yet the mode profile slightly deforms like Figure 4.13-(b). When θ =30^D, 0φ= ^D, the deformed hexapole mode is dominant (Figure 4.13-(c)). These deformed modes are probably attributed to the multiple mode mixing.

(a) θ =90^D φ =0^D (b) θ =60^D, φ=0^D

(c) θ =30^D, φ =0^D (d) θ =0^D φ =0^D

Figure 4.13 Hz field profiles

(a) Top view (z=0) (b) Side view (x=0)

Figure 4.14 Electric field intensity profiles of the quadrupole mode (θ =90^D φ=0^D)

References

[1] B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A.

Scherer, and Y. Qiu, “Liquid crystal electric tuning of a photonic crystal laser,” Appl.

Phys. Lett. 85, 360-362 (2004).

[2] Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes”, IEEE JOURNAL OF QUANTUM ELECTRONICS, 39, 1081 (2003).

在文檔中 國 立 台 灣 大 學 物 理 學 研 究 所 (頁 36-56)