**2.1 Photonic crystal micro-cavity VCSEL model**

The active region of 980nm VCSEL was composed of three 8nm thick In0.2Ga0.8As quantum well layers with 10nm thick GaAs barrier layer. Al0.98Ga0.02As layer is incorporated between the P-type DBR and the active region to form lateral oxidation and provide both current and optical confinements. The reflectors were DBR mirrors with the reflectivity higher than 99%. In this work a periodic arrangement of air holes on the top DBR reflector was designed to form two-dimensional photonic crystal structure. Two kinds of lattice defect were produced to evaluate the Q factor of the microcavity. Schematic diagram of the structure was shown in Fig. 2.

Generally there were two types of two-dimensional periodic arrangement of photonic crystals: hexagonal lattice and square lattice. Under the similar lattice parameters of hole depth, diameter and distance, hexagonal lattice was suggested to obtain photonic band gap easily than the square lattice does. Once the photonic band gap was created, the band gap of hexagonal lattice was wider than that of square lattice. Therefore hexagonal lattice was often used in the design of PhC-VCSEL. When one or several holes were removed from the lattice, the periodicity of the lattice structure was destroyed. The simplest way is to remove one air hole from the center of the lattice. This created the H1 cavity, shown in Figure 3(a). The second photonic crystal defect structure, H2 microcavity, was to remove seven air holes from the center, as shown in Figure 3(b). In our simulation the air hole was etched through

980nm Photonic Microcavity Vertical Cavity Surface Emitting Laser 25

Where *a* is the lattice period, λ is the wavelength, *n0* is the refractive index of the cavity

According to photonic crystal theory, the following requirement of normalized frequency

Fig. 5. Normalized frequencies *Veff* of H1microcavity at different filling ratio

center, *neff* is the external refractive index of the photonic crystal cladding.

Fig. 4. Photonic crystal fiber

should be met to achieve single-mode operation.

2 2 1/2

<sup>0</sup> (2 / )( ) *V a nn eff eff* (1)

2.405 *Veff* (2)

the top DBR and stop above lateral oxidation layer. The period of air holes was chosen to be 5.5μm for the easiness of fabrication.

Fig. 2. Photonic crystal microcavity VCSEL

Fig. 3. Photonic crystal micro-cavities (a: H1 microcavity and b: H2 micro-cavity)

## **2.2 Analysis of single-mode condition**

Photonic crystal defect structure with several holes missing at the center was similar to photonic crystal fiber where the solid center was surrounded by periodic arrangement of air holes, as shown in Figure 4. The characteristics of microcavity was only determined by the arrangement of air holes and the configuration of defect. There is no active material in the PhC structure. Therefore, the theory of photonic crystal fiber was used to investigate the normalized frequency of PhC defect structure in this work.

In the theory of photonic crystal fiber, the normalized frequency was expressed as following:

$$N\_{\rm eff} = (2\,\text{\pi}\,a \,/\,\lambda)(n^2 \,\_0 - n^2 \,\_{\rm eff})^{1/2} \tag{1}$$

Where *a* is the lattice period, λ is the wavelength, *n0* is the refractive index of the cavity center, *neff* is the external refractive index of the photonic crystal cladding.

Fig. 4. Photonic crystal fiber

24 Photonic Crystals – Innovative Systems, Lasers and Waveguides

the top DBR and stop above lateral oxidation layer. The period of air holes was chosen to be

5.5μm for the easiness of fabrication.

Fig. 2. Photonic crystal microcavity VCSEL

**2.2 Analysis of single-mode condition** 

following:

(a) (b)

normalized frequency of PhC defect structure in this work.

Fig. 3. Photonic crystal micro-cavities (a: H1 microcavity and b: H2 micro-cavity)

Photonic crystal defect structure with several holes missing at the center was similar to photonic crystal fiber where the solid center was surrounded by periodic arrangement of air holes, as shown in Figure 4. The characteristics of microcavity was only determined by the arrangement of air holes and the configuration of defect. There is no active material in the PhC structure. Therefore, the theory of photonic crystal fiber was used to investigate the

In the theory of photonic crystal fiber, the normalized frequency was expressed as

According to photonic crystal theory, the following requirement of normalized frequency should be met to achieve single-mode operation.

$$V\_{\rm eff} < 2.405 \tag{2}$$

Fig. 5. Normalized frequencies *Veff* of H1microcavity at different filling ratio

980nm Photonic Microcavity Vertical Cavity Surface Emitting Laser 27

corresponds originally to multi-mode operation when the hole depth was set to be infinite as shown in Fig.5. This enables the fabrication of H1 microcavity much more easily while

Similar to the above analysis, H2 microcavity with seven holes in the center missing was

0 0 (2 3 / ) ( ) ) *V a nn n eff*

For single mode operation, the normalized frequency of H2 microcavity was smaller than that of H1 microcavity. At a filling ratio of 0.1, the output is single mode. This result might caused by relatively weak confinement of H2 microcavity compared with H1

1/2 2 2

(4)

calculated, as shown Fig.4. The normalized frequencies was as following

Fig. 7. Normalized frequencies *Veff* of H2 micro-cavity at different filling ratio

The cavity mode volume was reduced greatly in a photonic crystal microcavity, which improved the coupling of light field with the cavity mode. High quality factor of microcavity could be realized due to the light confinement provided by the photonic band

Cavity quality factor *Q* was an important parameter for evaluating photonic crystal VCSEL. Quality factor implies the ability of a microcavity to store energy. Obviously a photonic crystal microcavity with high Q was the purpose of an ideal design. *Q* was defined as:

**2.3 Quality factor** *Q* **of microcavity** 

gap in the lateral direction.

single mode operation was still maintained.

microcavity.

According to the analysis above, the normalized frequency of H1 photonic crystal microcavity was calculated. The filling ratio *r/a* of 0.1, 0.2, 0.4, 0.6 and 0.7 was suggested for the calculation of normalized frequency, as presented in Fig.5. It was shown that H1 photonic crystal micrcavity meets the requirement of single-mode operation when the filling ratio was less than 0.1. Obviously, smaller filling ratio was beneficial to single-mode operation. But too small filling ratio would cause additional difficulty in the fabrication process of photonic crystal structure. In the above calculation the hole depth was set to be infinite. However, the thickness of VCSEL chip is reasonably around 150μm like conventional edge emitting diode laser chip. It is very difficult, if required small filling ratio, to etch through the entire chip. And the mechanical strength of the device and the electrical properties would be deteriorated significantly. So the reliable hole depth was limited, which was not the case of identical photonic crystal fiber. Therefore the calculation above based on the theory of photonic crystal fiber should be modified as follows:

$$V\_{\rm eff} = \left(2\pi a \;/\,\lambda\right) \left[n\,^2\!\!/ \, -\left(n\_0 - \gamma \Delta n\right)^2\right]^{1/2} \tag{3}$$

Fig. 6. Normalized frequencies *Veff* of H1microcavity at different filling ratio based on modified calculation

The normalized frequencies *Veff* of H1microcavity were calculated based on the modified model of equation (3) for different filling ratio, as shown in Fig.6. The corresponding etching depth factor γ was 0.3. Comparing Fig.5 and Fig.6 carefully, it was observed that single mode operation was realized for a filling ratio of 0.4. Though this filling ratio

According to the analysis above, the normalized frequency of H1 photonic crystal microcavity was calculated. The filling ratio *r/a* of 0.1, 0.2, 0.4, 0.6 and 0.7 was suggested for the calculation of normalized frequency, as presented in Fig.5. It was shown that H1 photonic crystal micrcavity meets the requirement of single-mode operation when the filling ratio was less than 0.1. Obviously, smaller filling ratio was beneficial to single-mode operation. But too small filling ratio would cause additional difficulty in the fabrication process of photonic crystal structure. In the above calculation the hole depth was set to be infinite. However, the thickness of VCSEL chip is reasonably around 150μm like conventional edge emitting diode laser chip. It is very difficult, if required small filling ratio, to etch through the entire chip. And the mechanical strength of the device and the electrical properties would be deteriorated significantly. So the reliable hole depth was limited, which was not the case of identical photonic crystal fiber. Therefore the calculation above based on

1/2 2 2

0 0 (2 / ) ( ) ) *V a nn n eff* (3)

the theory of photonic crystal fiber should be modified as follows:

Fig. 6. Normalized frequencies *Veff* of H1microcavity at different filling ratio based on

The normalized frequencies *Veff* of H1microcavity were calculated based on the modified model of equation (3) for different filling ratio, as shown in Fig.6. The corresponding etching depth factor γ was 0.3. Comparing Fig.5 and Fig.6 carefully, it was observed that single mode operation was realized for a filling ratio of 0.4. Though this filling ratio

modified calculation

corresponds originally to multi-mode operation when the hole depth was set to be infinite as shown in Fig.5. This enables the fabrication of H1 microcavity much more easily while single mode operation was still maintained.

Similar to the above analysis, H2 microcavity with seven holes in the center missing was calculated, as shown Fig.4. The normalized frequencies was as following

$$V\_{eff} = \left(2\pi\sqrt{3}a \;/\,\lambda\right) \left[n^2 \; \_0 - \left(n\_0 - \gamma\Delta n\right)^2\right]^{1/2} \tag{4}$$

For single mode operation, the normalized frequency of H2 microcavity was smaller than that of H1 microcavity. At a filling ratio of 0.1, the output is single mode. This result might caused by relatively weak confinement of H2 microcavity compared with H1 microcavity.

Fig. 7. Normalized frequencies *Veff* of H2 micro-cavity at different filling ratio

#### **2.3 Quality factor** *Q* **of microcavity**

The cavity mode volume was reduced greatly in a photonic crystal microcavity, which improved the coupling of light field with the cavity mode. High quality factor of microcavity could be realized due to the light confinement provided by the photonic band gap in the lateral direction.

Cavity quality factor *Q* was an important parameter for evaluating photonic crystal VCSEL. Quality factor implies the ability of a microcavity to store energy. Obviously a photonic crystal microcavity with high Q was the purpose of an ideal design. *Q* was defined as:

980nm Photonic Microcavity Vertical Cavity Surface Emitting Laser 29

 

Where *q* is the total number of photon in microcavity, *h* is Planck constant, is the resonant frequency. Now the quality factors *Q* of H1 and H2 microcavity were calculated for different

It was shown that the Q value reach a maximum of 4832(H1) and 3931(H2) when the filling

Photonic crystal micro-cavity VCSEL with hexagonal lattice of air holes was discussed in regarding the quality factor and the requirement of single mode operation. The normalized frequencies of two types of microcavities (H1 and H2) were calculated based on modified theory of photonic crystal fiber. A filling ratio of 0.4 for H1 microcavity was considered to be a good choice for single mode operation when the etching depth factor was 0.3. For H2 microcavity, the filling ratio less than 0.1 were necessary for single mode operation. The difference between the filling ratios for H1 and H2 microcavities might suggest weak confinement of H2 microcavity. Quality factors *Q* of two microcavities were

H Soda, K Iga, C Kitahara*,et al*.GaInAsP/InP Surface Emitting Injection Lasers[J], *Jpn J Appl* 

R.S.Geels, S.W.Corzine, L.A.Coldren. InGaAs Vertical-Cavity Surface-Emitting Lasers, *IEEE* 

Kenichi Iga, Vertical-Cavity Surface-Emitting Laser:Its Conception and Evolution[J],*Japanese* 

Li-Gao Zei, Stephan Ebers, Joerg-Reindhardt Kropp, *el a1.* Noise Performance of Multimode

Maria Susana Torre, Cristina Masoller, K. Alan Shore, Synchronization of unidirectionally

D. S. Song, S. H. Kim, H. G. Park, c. K. Kim, and Y. H. Lee, "Single-fundamental-mode

T. S. Kim, A. J. Danner, D. M. Grasso, E. W. Young, and K. D. Choquette, "Single

D. S. Song, Y. J. Lee, H. W. Choi, and Y. H. Lee, "Polarization-controlled, single-transverse-

K.D. Choquette, R.P. Schneider, K.L. Lear, R.E. Leibenguth, IEEE J. Sel. Top. Quantum

coupled multi-transverse-mode vertical-cavity surface-emitting lasers[J], *Journal of* 

photonic-crystal verticalcavity surface-emitting lasers," Appl. Phys. Lett. 80,

fundamental mode photonic crystal vertical cavity surface emitting laser with 9

mode, photoniccrystal, vertical-cavity, surface-emitting lasers," Appl. Phys. Lett.

VCSELs[J]. *Journal of Lightwave Technology*.2001,19(6):884-887.

*the Optical Society of America B*.2004,21(10):1772-1880.

GHz bandwidth," Electron. Lett. 40, 1340–1341 (2004).

*dh <sup>h</sup> dt*

(5)

Q 2 *qh*

filling ratio, as shown in Fig.8.

calculated to be 4832(H1) and 3931(H2) respectively.

*J. Quantum Electron.*, 1991,27:1359-1362.

*Journal of Applied Physics*,2008, 47(1):1-10.

*Phys*,1979,18(12):2329-2330.

3901–3903 (2002).

82, 3182–3184 (2003).

Electron. 1 (1995) 661.

ratio was 0.3.

**3. Conclusion** 

**4. References** 

(a)

(b)

Fig. 8. Quality factor of microcavity at different filling ratio (a: H1 microcavity, and b: H2 microcavity)

$$\mathbf{Q} = 2\pi \frac{q \text{hv}}{\left(-\frac{dh}{dt}\right) \text{h}} \tag{5}$$

Where *q* is the total number of photon in microcavity, *h* is Planck constant, is the resonant frequency. Now the quality factors *Q* of H1 and H2 microcavity were calculated for different filling ratio, as shown in Fig.8.

It was shown that the Q value reach a maximum of 4832(H1) and 3931(H2) when the filling ratio was 0.3.

### **3. Conclusion**

28 Photonic Crystals – Innovative Systems, Lasers and Waveguides

0.15 0.20 0.25 0.30 0.35 0.40 0.45

r/a

0.15 0.20 0.25 0.30 0.35 0.40 0.45

r/a

(b)

Fig. 8. Quality factor of microcavity at different filling ratio (a: H1 microcavity, and b: H2

(a)

500

1000

1500

2000

2500

Q

microcavity)

3000

3500

4000

Q

Photonic crystal micro-cavity VCSEL with hexagonal lattice of air holes was discussed in regarding the quality factor and the requirement of single mode operation. The normalized frequencies of two types of microcavities (H1 and H2) were calculated based on modified theory of photonic crystal fiber. A filling ratio of 0.4 for H1 microcavity was considered to be a good choice for single mode operation when the etching depth factor was 0.3. For H2 microcavity, the filling ratio less than 0.1 were necessary for single mode operation. The difference between the filling ratios for H1 and H2 microcavities might suggest weak confinement of H2 microcavity. Quality factors *Q* of two microcavities were calculated to be 4832(H1) and 3931(H2) respectively.
