Abstract

We determine the single mode condition and analyze the modes discrimination of 1.3 μm InP based photonic-crystal vertical-cavity surface-emitting diode laser. To this aim we apply the fully vectorial, three dimensional Plane Wave Admittance Method and analyze a broad range of photonic-crystal parameters such as hole etching depth, distance between the holes and their diameters.

© 2007 Optical Society of America

1. Introduction

The photonic-crystal (PC) vertical-cavity surface-emitting diode lasers (VCSELs) are very perspective light sources for telecommunication applications [1]. Therefore crucial is the determination of the laser parameters, which support the single mode operation. Several experimental reports proved the ability of single mode operation of PC VCSEL structures. A maximum output power of 3.1 mW and 1mW in the fundamental mode has been achieved for a single-defect PC VCSEL emitting at 0.85 μm [2] and 1.3 μm [3], respectively. The structure of the photonic-crystal is defined by three independent parameters: the distance between the PC holes (L), the diameter of the holes (a) and their etching depths. Hence all three parameters have to be analyzed to determine the PC parameters supporting single mode operation. The first attempts toward the determination of the single mode condition have been performed in Refs. [4, 5]. The first one has been based on 2D plane-wave expansion and a correction to the effective refractive index corresponding to finite depths of the holes and the second on the scalar, effective index method. Both of them however suffered the lack of fully vectorial, three dimensional, optical analysis, which is performed here.

The goal of this paper is to determine the PC parameters, which assure single mode operation of a 1.3-μm InAlGaAs/InP PC VCSEL. Additionally, we analyze the influence of the PC parameters on the mode discrimination. Our analysis is purely computational and therefore we do not impose any limitation on the aspect ratio of the PC holes (diameter to depth of the hole), since the technology shows constant improvement in that field. On the other hand, we realize that the etching process is limited presently to the aspect ratio close to about 15 [6], which also encourages investigating a broad range of PC parameters. The presented analysis concerns the complex eigen-wavelength. The real part of the eigen-wavelength corresponds to the modal wavelength of the emission while its imaginary part accounts for the modal gain (losses).

2. Laser structure

In previous considerations [7] a PC VCSEL emitting through the DBR free of PC structure (we call it hereafter bottom-emitting) has been proven as more effective than the one emitting through the PC DBR (refered as top-emitting). The advantage lies in the much better beam quality of the bottom-emitting concept. The presented study is based on 1.3 μm PC VCSEL structure of design shown in Fig. 1(a) and described in detail in Ref. [8]. The VCSEL cavity is bounded by Al0.9Ga0.1As/GaAs DBRs. The upper one consists of 35 pairs and the bottom one of 27 pairs. The considered here three-ring single-defect photonic crystal is determined by the parameters [as defined in Fig. 1(b)]: PC pitch (L) and hole diameter (a). The optical aperture, which determines also the active region aperture (2RA), is defined as a circle, touching the inner ring of holes. Hence, the relation between those parameters reads: R = L - 0.5a. The depth of the PC holes is determined as the distance, between the upper edge of the top DBR and the bottom of the hole. In the computational model, the refractive indices of the layers are complex and uniform, only the one of the active region is defined by a radially step-like function of the imaginary part. We consider the active region with a material gain equal to 2000 cm-1 for rRA and -1000 cm-1 for rRA.

 

Fig. 1. (a). Schematic view of the bottom-emitting 1300-nm InP-based AlGaInAs quantum well (QW) PC VCSEL design. The photonic-crystal (PC) is shown as straight air columns surrounding the central part of the VCSEL cavity and etched throughout the upper DBR, (b). the typical intensity distribution of the fundamental mode within the active region and the first ring of the photonic crystal. The diameter of the hole (a) the distance between the hole axes - the pitch (L) and the optical aperture (2 RA) are defined.

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3. Results

The fully vectorial, three-dimensional Plane Wave Admittance Method (PWAM) [9] applied here, combines two very effective approaches. In the plane of the active region the field is expanded in a basis of exponential functions and, using the Admittance Method, it is transformed through the layers determining an eigenvalue problem, which gives as a solution the complex eigen-wavelength and the field distribution of the mode within the VCSEL structure [8, 9]. The investigation presented here concerns the modes confined by a single defect of the PC structure. Other mechanisms such as spatial distribution of the gain or/and temperature can also contribute to the waveguiding, however this is beyond the scope of this work and only the modes guided by the PC will be investigated. Since there is no strict criterion distinguishing the modes confined within the PC, we consider only modes with 95% of the modal power confined within the hexagon determined by the third ring of the PC (in the active layer). The discrimination of the modes is defined as the difference of the real and imaginary eigen-wavelength of HE11 and HE21 modes. The analyzed PC parameters are varied within the limits and with the steps as follow RA: 1 μm – 6 μm, step 1 μm; a/L: 0.1 – 0.9, step 0.1; etching depth 0 – 13.8 μm and the step is the thickness of a single DBR pair.

 

Fig. 2. Region of single mode operation (gray fields) for six different optical apertures (from RA = 1 μm to RA = 6 μm) a) - f) mapped in the plane of etching depth and a/L ratio. The horizontal lines assign the PC parameters, which have been chosen in the analysis presented in Figs. 3 and 4.

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3.1 Single mode operation

Figure 2 illustrates the area of a single mode (s-m) operation region, which exists in a broad range of PC parameters. The s-m region is shown as gray area. From below it is limited by the cut-off etching depth, i.e. the depth for which well-confined mode in the PC defect appears [10]. From above it is limited by the cut-off etching depth for the first higher order mode. As it can be seen from Fig. 2 the s-m region is the largest for a/L ratios as in this case it is not limited from above by the appearance of first order mode. This “extended” s-m region is shifted toward smaller a/L ratios and is squeezed with the increase of RA: in the case of RA = 1 μm the s-m region is limited by 0.2 < a/L < 0.5 while for broad optical aperture (RA = 6 μm) by 0.1 < a/L< 0.2. For the remaining stripe of s-m region one can notice from Fig. 2 that its thickness is almost the same for different RA. It is approximately equal to 0.5 μm and does not depend on a/L. Only in the extreme case of RA = 1 μm it is somewhat thicker and approximately equals 1 μm. The position of the s-m region is shifted toward shallower etching depth with the increase of RA and a/L. However, the dependence on a/L is weak and for a/L > 0.6 the shift is insignificant. The described variation of the position and the area of s-m region are governed mostly by the change of the PC induced waveguiding. If it is stronger, the single mode region is shifted toward smaller a/L and shallower etching depths.

3.2. Mode discrimination

Figures 3(a) and 3(b) illustrates the influence of the etching depth on the wavelength of the emission for different configurations of PC parameters. The chosen characteristics correspond to the PC parameters assigned by the vertical lines in Figs. 2(b), 2(d)–2(f). The reduction of the aperture size contributes to a pronounced blue shift. Similar effect causes the increase of the a/L ratio however, not in the same extend. This can be explained since the reduction of the aperture as well as the broadening of the holes causes a spatial shrinkage of the mode as well as better discrimination of the modes (larger wavelength difference) as also observed for oxide-confined VCSELs [11]. Figures 3(c) and 3(d) present the discrimination of the modes with respect to the real part of eigen-wavelength. The largest discrimination is observed in the case of deep, broad holes and narrow apertures. In those case the impact of the PC holes is the strongest.

Figure 4 shows the modal gain difference between HE11 and HE21 modes. The largest difference is observed for PC parameters close to the single mode region. This can be explained since the modal gain is related to the mode overlap with the active region.

In the multimode region, but close to the single mode one, the first order mode is weakly confined by the PC, hence it suffers high losses, while the fundamental mode is relatively well localized into the active region (Fig. 5). The reduction of the mode discrimination is due to the improvement of the confinement for both HE11 and HE21 modes (Fig. 6). Therefore the largest discrimination of the modes is observed for narrow optical apertures as well as for narrow and/or shallow holes.

 

Fig. 3. Fundamental mode wavelength [(a) and (b)] and mode discrimination for the real eigen-wavelength [(c) and (d)] versus etching depth, under a change of the optical aperture (RA) [(a) and (c)] and the a/L ratio [(b) and (d)].

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Fig. 4. The difference of the modal gains αm0 and αm1 of the fundamental and the first order mode versus etching depth, under a change of (a) optical aperture and b) a/L ratio. Dots in (a) relates to the PC parameters used in Figs. 5 and 6.

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Fig. 5. Distribution of a) HE11 and b) HE21 mode within active region for 6.5 μm etching depth and PC parameters determined in Fig. 4(a) by the large black dots.

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Fig. 6. Distribution of a) HE11 and b) HE21 mode within active region for 12.4 μm etching depth and PC parameters determined in the Fig. 4(a) by the corresponding dot.

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4. Conclusions

In this paper we determine the regions of single mode operation and analyze the mode discrimination of a single-defect photonic-crystal vertical-cavity surface-emitting diode laser in a broad range of photonic crystal parameters. It has been found that the position of the single mode region depends mostly on the optical aperture (2 RA) and on the a/L ratio. For some combinations of RA and small a/L ratios, the first order mode does not appear in the structure regardless of the etching depth. For most of the PC parameters, this single mode region corresponds to the holes etched only in the top DBR, which significantly reduces the technological effort. The strongest emission wavelength discrimination, i.e. the largest difference in the wavelengths of the fundamental and the first order modes, is observed for PC parameters assuring strong confinement effect i.e.: deep, broad holes and narrow optical apertures. The discrimination with respect to the modal gain is better for shallow, narrow holes and narrow optical apertures.

Acknowledgment

This work was supported by Foreign Fellowships for Young PhD’s awarded by Foundation for Polish Science, grant No N515 004 32/0297 from the Polish Ministry of Science and Higher Education (MNiSzW), IAP Program of the Belgian government, COST P11, as well as GOA, FWO, and OZR of the VUB.

References and links

1. H. Li and K. Iga, Vertical-Cavity Surface-Emitting Laser Devices, (Berlin: Springer-Verlang, 2003).

2. A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325 – 326 (2005). [CrossRef]  

3. P. O. Leisher, A. J. Danner, and K. D. Choquette, “Single-Mode 1.3-μm Photonic Crystal Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. 18, 2156 – 2158 (2006). [CrossRef]  

4. N. Yokouchi, A. J. Danner, and K. D. Choquette “Etching depth dependence of the effective refractive index in two-dimensional photonic-crystal-patterned vertical-cavity surface-emitting laser structures,” Appl. Phys. Lett. 82, 1344 – 1346 (2003). [CrossRef]  

5. S. Ivanov, H. J. Unold, R. Michalzik, J. Maehnss, K. J. Ebeling, and I. A. Sukhoivanov, “Theoretical study of cold-cavity single-mode conditions in vertical-cavity surface-emitting lasers with incorporated two-dimensional photonic crystals,” J. Opt. Soc. Am. B 20, 2442 – 2447 (2003). [CrossRef]  

6. www.oxfordplasma.de/process/inp_phcr.htm

7. T. Czyszanowski, M. Dems, and K. Panajotov “Improvement of the beam quality in the long wavelength photonic crystal VCSEL,” submitted to Opt. Express

8. T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov “Optimal radii of Photonic Crystal holes within DBR mirrors in long wavelength VCSEL,” Opt. Express 15, 1301–1306 (2007). [CrossRef]   [PubMed]  

9. M. Dems, R. Kotynski, and K. Panajotov “Plane Wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196 – 3207 (2005). [CrossRef]   [PubMed]  

10. T. Czyszanowski, M. Dems, and K. Panajotov “Optimal parameters of Photonic-Crystal Vertical-Cavity Surface-Emitting Diode Lasers,” accepted by IEEE J. Lightwave Technol.

11. T. Czyszanowski and W. Nakwaski “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D: Appl. Phys. 39, 30 – 35 (2006). [CrossRef]  

References

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  1. H. Li and K. Iga, Vertical-Cavity Surface-Emitting Laser Devices, (Berlin: Springer-Verlang, 2003).
  2. A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325 – 326 (2005).
    [Crossref]
  3. P. O. Leisher, A. J. Danner, and K. D. Choquette, “Single-Mode 1.3-μm Photonic Crystal Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. 18, 2156 – 2158 (2006).
    [Crossref]
  4. N. Yokouchi, A. J. Danner, and K. D. Choquette “Etching depth dependence of the effective refractive index in two-dimensional photonic-crystal-patterned vertical-cavity surface-emitting laser structures,” Appl. Phys. Lett. 82, 1344 – 1346 (2003).
    [Crossref]
  5. S. Ivanov, H. J. Unold, R. Michalzik, J. Maehnss, K. J. Ebeling, and I. A. Sukhoivanov, “Theoretical study of cold-cavity single-mode conditions in vertical-cavity surface-emitting lasers with incorporated two-dimensional photonic crystals,” J. Opt. Soc. Am. B 20, 2442 – 2447 (2003).
    [Crossref]
  6. www.oxfordplasma.de/process/inp_phcr.htm
  7. T. Czyszanowski, M. Dems, and K. Panajotov “Improvement of the beam quality in the long wavelength photonic crystal VCSEL,” submitted to Opt. Express
  8. T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov “Optimal radii of Photonic Crystal holes within DBR mirrors in long wavelength VCSEL,” Opt. Express 15, 1301–1306 (2007).
    [Crossref] [PubMed]
  9. M. Dems, R. Kotynski, and K. Panajotov “Plane Wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196 – 3207 (2005).
    [Crossref] [PubMed]
  10. T. Czyszanowski, M. Dems, and K. Panajotov “Optimal parameters of Photonic-Crystal Vertical-Cavity Surface-Emitting Diode Lasers,” accepted by IEEE J. Lightwave Technol.
  11. T. Czyszanowski and W. Nakwaski “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D: Appl. Phys. 39, 30 – 35 (2006).
    [Crossref]

2007 (1)

2006 (2)

P. O. Leisher, A. J. Danner, and K. D. Choquette, “Single-Mode 1.3-μm Photonic Crystal Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. 18, 2156 – 2158 (2006).
[Crossref]

T. Czyszanowski and W. Nakwaski “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D: Appl. Phys. 39, 30 – 35 (2006).
[Crossref]

2005 (2)

A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325 – 326 (2005).
[Crossref]

M. Dems, R. Kotynski, and K. Panajotov “Plane Wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196 – 3207 (2005).
[Crossref] [PubMed]

2003 (2)

N. Yokouchi, A. J. Danner, and K. D. Choquette “Etching depth dependence of the effective refractive index in two-dimensional photonic-crystal-patterned vertical-cavity surface-emitting laser structures,” Appl. Phys. Lett. 82, 1344 – 1346 (2003).
[Crossref]

S. Ivanov, H. J. Unold, R. Michalzik, J. Maehnss, K. J. Ebeling, and I. A. Sukhoivanov, “Theoretical study of cold-cavity single-mode conditions in vertical-cavity surface-emitting lasers with incorporated two-dimensional photonic crystals,” J. Opt. Soc. Am. B 20, 2442 – 2447 (2003).
[Crossref]

Choquette, K. D.

P. O. Leisher, A. J. Danner, and K. D. Choquette, “Single-Mode 1.3-μm Photonic Crystal Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. 18, 2156 – 2158 (2006).
[Crossref]

A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325 – 326 (2005).
[Crossref]

N. Yokouchi, A. J. Danner, and K. D. Choquette “Etching depth dependence of the effective refractive index in two-dimensional photonic-crystal-patterned vertical-cavity surface-emitting laser structures,” Appl. Phys. Lett. 82, 1344 – 1346 (2003).
[Crossref]

Czyszanowski, T.

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov “Optimal radii of Photonic Crystal holes within DBR mirrors in long wavelength VCSEL,” Opt. Express 15, 1301–1306 (2007).
[Crossref] [PubMed]

T. Czyszanowski and W. Nakwaski “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D: Appl. Phys. 39, 30 – 35 (2006).
[Crossref]

T. Czyszanowski, M. Dems, and K. Panajotov “Improvement of the beam quality in the long wavelength photonic crystal VCSEL,” submitted to Opt. Express

T. Czyszanowski, M. Dems, and K. Panajotov “Optimal parameters of Photonic-Crystal Vertical-Cavity Surface-Emitting Diode Lasers,” accepted by IEEE J. Lightwave Technol.

Danner, A. J.

P. O. Leisher, A. J. Danner, and K. D. Choquette, “Single-Mode 1.3-μm Photonic Crystal Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. 18, 2156 – 2158 (2006).
[Crossref]

A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325 – 326 (2005).
[Crossref]

N. Yokouchi, A. J. Danner, and K. D. Choquette “Etching depth dependence of the effective refractive index in two-dimensional photonic-crystal-patterned vertical-cavity surface-emitting laser structures,” Appl. Phys. Lett. 82, 1344 – 1346 (2003).
[Crossref]

Dems, M.

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov “Optimal radii of Photonic Crystal holes within DBR mirrors in long wavelength VCSEL,” Opt. Express 15, 1301–1306 (2007).
[Crossref] [PubMed]

M. Dems, R. Kotynski, and K. Panajotov “Plane Wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196 – 3207 (2005).
[Crossref] [PubMed]

T. Czyszanowski, M. Dems, and K. Panajotov “Optimal parameters of Photonic-Crystal Vertical-Cavity Surface-Emitting Diode Lasers,” accepted by IEEE J. Lightwave Technol.

T. Czyszanowski, M. Dems, and K. Panajotov “Improvement of the beam quality in the long wavelength photonic crystal VCSEL,” submitted to Opt. Express

Ebeling, K. J.

Iga, K.

H. Li and K. Iga, Vertical-Cavity Surface-Emitting Laser Devices, (Berlin: Springer-Verlang, 2003).

Ivanov, S.

Kim, T. S.

A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325 – 326 (2005).
[Crossref]

Kotynski, R.

Leisher, P. O.

P. O. Leisher, A. J. Danner, and K. D. Choquette, “Single-Mode 1.3-μm Photonic Crystal Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. 18, 2156 – 2158 (2006).
[Crossref]

Li, H.

H. Li and K. Iga, Vertical-Cavity Surface-Emitting Laser Devices, (Berlin: Springer-Verlang, 2003).

Maehnss, J.

Michalzik, R.

Nakwaski, W.

T. Czyszanowski and W. Nakwaski “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D: Appl. Phys. 39, 30 – 35 (2006).
[Crossref]

Panajotov, K.

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov “Optimal radii of Photonic Crystal holes within DBR mirrors in long wavelength VCSEL,” Opt. Express 15, 1301–1306 (2007).
[Crossref] [PubMed]

M. Dems, R. Kotynski, and K. Panajotov “Plane Wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196 – 3207 (2005).
[Crossref] [PubMed]

T. Czyszanowski, M. Dems, and K. Panajotov “Improvement of the beam quality in the long wavelength photonic crystal VCSEL,” submitted to Opt. Express

T. Czyszanowski, M. Dems, and K. Panajotov “Optimal parameters of Photonic-Crystal Vertical-Cavity Surface-Emitting Diode Lasers,” accepted by IEEE J. Lightwave Technol.

Sukhoivanov, I. A.

Thienpont, H.

Unold, H. J.

Yokouchi, N.

N. Yokouchi, A. J. Danner, and K. D. Choquette “Etching depth dependence of the effective refractive index in two-dimensional photonic-crystal-patterned vertical-cavity surface-emitting laser structures,” Appl. Phys. Lett. 82, 1344 – 1346 (2003).
[Crossref]

Appl. Phys. Lett. (1)

N. Yokouchi, A. J. Danner, and K. D. Choquette “Etching depth dependence of the effective refractive index in two-dimensional photonic-crystal-patterned vertical-cavity surface-emitting laser structures,” Appl. Phys. Lett. 82, 1344 – 1346 (2003).
[Crossref]

Electron. Lett. (1)

A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325 – 326 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

P. O. Leisher, A. J. Danner, and K. D. Choquette, “Single-Mode 1.3-μm Photonic Crystal Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. 18, 2156 – 2158 (2006).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Phys. (1)

T. Czyszanowski and W. Nakwaski “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D: Appl. Phys. 39, 30 – 35 (2006).
[Crossref]

Opt. Express (2)

Other (4)

T. Czyszanowski, M. Dems, and K. Panajotov “Optimal parameters of Photonic-Crystal Vertical-Cavity Surface-Emitting Diode Lasers,” accepted by IEEE J. Lightwave Technol.

www.oxfordplasma.de/process/inp_phcr.htm

T. Czyszanowski, M. Dems, and K. Panajotov “Improvement of the beam quality in the long wavelength photonic crystal VCSEL,” submitted to Opt. Express

H. Li and K. Iga, Vertical-Cavity Surface-Emitting Laser Devices, (Berlin: Springer-Verlang, 2003).

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Figures (6)

Fig. 1.
Fig. 1.

(a). Schematic view of the bottom-emitting 1300-nm InP-based AlGaInAs quantum well (QW) PC VCSEL design. The photonic-crystal (PC) is shown as straight air columns surrounding the central part of the VCSEL cavity and etched throughout the upper DBR, (b). the typical intensity distribution of the fundamental mode within the active region and the first ring of the photonic crystal. The diameter of the hole (a) the distance between the hole axes - the pitch (L) and the optical aperture (2 RA ) are defined.

Fig. 2.
Fig. 2.

Region of single mode operation (gray fields) for six different optical apertures (from RA = 1 μm to RA = 6 μm) a) - f) mapped in the plane of etching depth and a/L ratio. The horizontal lines assign the PC parameters, which have been chosen in the analysis presented in Figs. 3 and 4.

Fig. 3.
Fig. 3.

Fundamental mode wavelength [(a) and (b)] and mode discrimination for the real eigen-wavelength [(c) and (d)] versus etching depth, under a change of the optical aperture (RA ) [(a) and (c)] and the a/L ratio [(b) and (d)].

Fig. 4.
Fig. 4.

The difference of the modal gains α m0 and α m1 of the fundamental and the first order mode versus etching depth, under a change of (a) optical aperture and b) a/L ratio. Dots in (a) relates to the PC parameters used in Figs. 5 and 6.

Fig. 5.
Fig. 5.

Distribution of a) HE11 and b) HE21 mode within active region for 6.5 μm etching depth and PC parameters determined in Fig. 4(a) by the large black dots.

Fig. 6.
Fig. 6.

Distribution of a) HE11 and b) HE21 mode within active region for 12.4 μm etching depth and PC parameters determined in the Fig. 4(a) by the corresponding dot.

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