## Diffraction loss in long-wavelength buried tunnel junction VCSELs analyzed with a hybrid coupled-cavity transfer-matrix model

Optics Express, Vol. 16, Issue 25, pp. 20789-20802 (2008)

http://dx.doi.org/10.1364/OE.16.020789

Acrobat PDF (774 KB)

### Abstract

Intra-cavity diffraction in VCSELs is a loss mechanism that potentially can cause a significant decrease in efficiency and a rise in the threshold current, particularly in cavities with small lateral features with a high index contrast. One such VCSEL type is the 2.3 µm GaSb-based buried tunnel junction (BTJ) VCSEL studied in this work, where the BTJ induced topology of the top layers gives rise to excess loss through diffraction. Diffraction loss is difficult to measure, and also the numerical estimation must be done with care because of the non-axial propagation of the diffracted fields. We present a simulation method with spatially varying dimensionality, such that the field is three-dimensional (3D) in the entire cavity, whereas the material structure of the cavity is modelled in 3D near the BTJ and the layers with a varying topology, but elsewhere is assumed to be 1D like in a regular DBR structure. We find that the diffraction loss displays a non-monotonic behaviour as a function of the BTJ diameter, but as expected it rapidly increases below a certain diameter of the BTJ and may even become the dominant cause of loss in some device designs. We also show that the diffraction loss can be much reduced if the layers above the BTJ can be deposited such that the surface profile becomes smoother with increasing distance from the BTJ.

© 2008 Optical Society of America

## 1. Introduction

_{4}, have strong absorption lines at wavelengths above 2

*µ*m, which cannot be conveniently reached with conventional VCSELs based on GaAs or InP. The longest wavelength ever reported for an InP-based VCSEL is 2.3

*µ*m, and this value is considered to be something of a fundamental limit for this material system [1

1. M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. Rönneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, “Electrically pumped room temperature CW VCSELs with 2.3 µm emission wavelength,” Electron. Lett. **42**, 640–641 (2006). [CrossRef]

2. A. Krier, *Mid-infrared semiconductor optoelectronics*, (Springer-Verlag2006). [CrossRef]

3. A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, “Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 µm,” Electron. Lett. **44**, 202–203 (2008). [CrossRef]

*µ*m, and calculate the intra-cavity diffraction loss. The device has a BTJ, which in effect replaces hole currents by electron currents, in order to minimize the use of p-doped materials in the structure. In this way the resistive heating is reduced as well as the optical absorption loss, since the n-doped material has a higher electrical conductivity and a lower free carrier absorption of the laser radiation. In spite of these advancements, the measured VCSEL performance was not too impressive; the threshold current was 3.3 mA for a device with 9

*µ*m BTJ diameter, which corresponds to a threshold current density of 2.6 kA/cm

^{2}, the maximum output power was 87

*µ*W, and the differential efficiency was merely 2%. Of course, a multitude of effects might contribute, e.g. a misalignment of the top of the cavity to the BTJ or a poor match between the gain spectrum and the cavity resonance wavelength. In light of the above discussion it is also natural to ask whether diffraction loss might play a decisive role in degrading the performance. In the remainder of this Introduction we will give a convenient definition of the diffraction loss, and briefly review some attempts to model this quantity in VCSELs. In Section 2 we describe the numerical method used in the work presented here, and Section 3 contains a description of the physical structure of the VCSEL and how it is represented in the numerical model. Finally, Section 4 contains the simulation results. The results indicate that the varying topology of the top DBR mirror, which is caused by the introduction of the BTJ, gives rise to diffraction loss which for a small diameter of the BTJ, and a top mirror topology that closely follows the shape of the etched BTJ, can even be the dominant loss mechanism.

*W*

*is the total decrease of the energy (e.g., per unit time or per cavity round-trip) of the field in the cavity and*

_{tot}*W*

*,*

_{mirrors}*is the energy decrease by transmission through the top and bottom mirrors had the field in the cavity been perfectly on-axis, i.e. normal to the DBR mirror layers. Since we assume that the cavity is passive (no gain) and neglect absorption losses,*

_{normal}*W*

*,*

_{mirrors}*would be the only loss term in a diffraction-free cavity. To assess the importance of the diffraction loss it is convenient to characterize it with the parameter*

_{normal}*η*

*, the diffraction loss relative to the total cavity loss.*

_{d}4. P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear, “Diffraction from oxide confinement apertures in vertical-cavity lasers,” Appl. Phys. Lett. **75**, 754–756 (1999). [CrossRef]

5. E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in vertical-cavity lasers,” J. Sel. Top. Quantum Electron. **3**, 379–389 (1997). [CrossRef]

7. R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, “VCSEL diffraction-loss theory,” IEE Proc. Optoelectron.142, 77–81 (1995). [CrossRef]

8. G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” J. Quantum Electron. **32**, 607–616 (1996). [CrossRef]

10. P. Mackowiak, R. P. Sarzala, M. Wasiak, and W. Nakwaski, “Radial optical confinement in nitride VCSELs,” J. Phys. D **36**, 2041–2045 (2003). [CrossRef]

11. S. Riyopoulos and D. Dialetis, “Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure,” J. Opt. Soc. Am. B **18**, 1497–1511 (2001). [CrossRef]

12. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. **33**, 327–341 (2001). [CrossRef]

8. G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” J. Quantum Electron. **32**, 607–616 (1996). [CrossRef]

10. P. Mackowiak, R. P. Sarzala, M. Wasiak, and W. Nakwaski, “Radial optical confinement in nitride VCSELs,” J. Phys. D **36**, 2041–2045 (2003). [CrossRef]

5. E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in vertical-cavity lasers,” J. Sel. Top. Quantum Electron. **3**, 379–389 (1997). [CrossRef]

14. D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” J. Quantum Electron. **29**, 1950–1962 (1993). [CrossRef]

5. E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in vertical-cavity lasers,” J. Sel. Top. Quantum Electron. **3**, 379–389 (1997). [CrossRef]

## 2. Numerical model

*z*) direction, see Fig. 1, which need not necessarily correspond to physical layers in the actual structure. However, within a layer the refractive index

*n*of the structure is assumed to be constant in the longitudinal direction; if there actually is a slight

*n*(

*z*) variation within the layer, but one does not want to increase the computational cost by introducing additional layers, one uses a value for

*n*such that the optical path length in the z-direction is preserved. Thus, as indicated in the figure, each layer is characterized by a, possibly laterally varying, longitudinally weighted refractive index within the layer.

*x*- or

*y*-direction. Further, we see that the reflection even at normal incidence is quite low since a number of mirror pairs were rather included in the 3D structure in the central part of the VCSEL, as we shall see, thus leaving only a few pairs to represent the peripheral 1D DBR top mirror in the TMMcalculation, giving a low reflectivity for this TMM section.

*n*(

*x*,

*y*)

*m*, where

*m*is the layer, or sub-cavity, number. In each layer the total optical field is represented by one right- and one left-propagating field, calculated at both the left and right boundaries of the layer, so that we have a set of fields

*E*(

*x*,

*y*)

_{m,d,p}, where

*d*is the direction (either

*r*or

*l*, for right- or left-propagating), and

*p*is the position (either

*r*or

*l*, for right or left boundary). In one iterative step, the “starting fields”

*E*(

*x*,

*y*)

_{m,r,l}and

*E*(

*x*,

*y*)

_{m,l,r}, propagating into the layer, are calculated first, from the “propagated fields” in the layer and its nearest neighboring layers, according to

*t*(

*x*,

*y*)

_{m,l,l}and

*r*(

*x*,

*y*)

_{m,l,l}are the Fresnel equation expressions for normal incidence at the left boundary, with the incident medium being that of layer

*m*and with layer

*m*-1 on the other side, and likewise for

*t*(

*x*,

*y*)

_{m,r,r}and

*r*(

*x*,

*y*)

_{m,r,r}but for the right boundary with layer

*m*+1 on the other side. The refractive indices used in the Fresnel equations are the laterally varying functions

*n*(

*x*,

*y*)

*m*and

*n*(

*x*,

*y*)

_{m±1}so the reflectance and transmittance of each layer are also laterally varying, as explicitly indicated by their (

*x*,

*y*)-dependency.

*E*(

*x*,

*y*)

_{m,r,r}from

*E*(

*x*,

*y*)

_{m,r,l}and

*E*(

*x*,

*y*)

_{m,l,l}from

*E*(

*x*,

*y*)

_{m,l,r}- by a beam propagation method (BPM), which is a stepwise application of the non-paraxial free-space angular spectrum propagation method, in each step corrected for the laterally varying refractive index

*n*(

*x*,

*y*)

*[16].*

_{m}## 3. Physical and model structure

*µ*m BTJ VCSEL. It consists of a GaSb substrate, a 24-pair n-doped AlAs

_{0.09}Sb

_{0.91}/GaSb bottom DBR and a four-pair

*α*- Si/SiO2 dielectric top mirror, of which some pairs are included in the central coupled cavity region in the model rather than in the top DBR, as we shall see. The cavity has an active region with five 11-nm-thick Ga

_{0.63}In

_{0.37}As

_{0.03}Sb

_{0.97}quantum wells separated by 8-nm-thick Al

_{0.33}Ga

_{0.67}As

_{0.03}Sb

_{0.97}barriers. On top of the active region a highly doped p

^{+}-GaSb/n

^{+}- InAsSb BTJ is positioned at a node of the optical field intensity. The BTJ serves as a current aperture and reduces optical losses by replacing absorbing p-doped material by n-type material. Because the n

^{+}-doped layer of the BTJ is etched laterally to confine the current it has the shape of a cylindrical pillar with a certain diameter,

*D*

_{BT}*, which will be varied in the simulation to study its effects on the diffraction loss, and a height of 72 nm. This pillar will affect the topology of all the layers above it, creating a longitudinal offset of approximately the pillar height in the peripheral lateral parts of each layer. This is thus the case also for the dielectric layers in the top DBR mirror, although for simplicity they are drawn completely flat in the figure. It is this change of the topology of the top DBR mirror that makes the cavity index guided and causes the increased diffraction loss.*

_{J}*D*

_{trans,m}, in layer number

*m*, which then is a measure of the distance over which the gradual change takes place. It is further natural to assume that this change becomes more gradual the farther away the layer is from the BTJ. Therefore, the diameter of the Gaussian was set to increase with each layer, being simply proportional to the distance from the BTJ; this is shown in the lower part of the composite structure shown in Fig. 5, labeled Case B, which shows the case

*D*

*,*

_{trans}*=1=1*

_{m}*µ*m. Also note the different scales in the longitudinal (

*z*) and lateral (

*x*,

*y*) directions in this figure.

## 4. Results from the simulations

*D*

_{trans,m}=0, when studying the effects of changing the BTJ diameter, but in a subsequent set of simulations

*D*

_{trans,m}was varied for a fixed value of the BTJ diameter. The evolution of the optical field towards a stable, self-repetitive solution in space and frequency is most conveniently monitored by observing the phase of the field (the intensity-weighted spatial average of the phase of the field in one of the planes where the field is calculated) and the corresponding dynamic wavelength which is changed in accordance with the round-trip phase change. These quantities are plotted in Fig. 6 as functions of the number of iterations, where an iteration is one update of all the fields

*E*(

*x*,

*y*)

_{m,d,p}and thus is essentially a cavity round-trip. The total number of fields

*E*(

*x*,

*y*)

_{m,d,p}was 60, since we have left- and right-propagating fields at the left and right boundaries in the 15 layers representing the central coupled-cavity structure. In all simulations we used ~15000 iterations, which was always more than sufficient for convergence. One iteration took less than five seconds on an ordinary personal computer (2.4 GHz, 2GB memory) for this system with 15 coupled cavities, each field

*E*(

*x*,

*y*)

_{m,d,p}being sampled in a 256×256 element matrix. The finally obtained intra-cavity field distribution is displayed in Fig. 7 where the inset shows an amplitude trace along the y-axis for the left-propagating fields at the left boundary in each layer, i.e., for

*E*(

*x*=0,

*y*)

_{m,l,l},

*m*=1,…,15; the BTJ-induced topology change obviously disturbs the field profile somewhat. The main figure shows the total intra-cavity standing-wave field, as a cross section in the (

*y*,

*z*)-plane. The field was obtained by calculating the right- and left-propagating fields in a large number of lon- gitudinal positions, based on their known values at the layer boundaries, and then adding them in each position. The longer wavelength in the low-index layers of the dielectric top mirror is evident as well as how rapidly the field decays in the mirror; if we had plotted the energy density (intensity) rather than the field amplitude of the standing wave the decay would appear to occur over an even shorter distance. We can also observe that the BTJ, in the rightmost part of the figure, is indeed at a field node, as it was specified to be.

*D*

*, but the two loss types were always of similar magnitude, showing that neither should be neglected in a thorough analysis. The results for the diffraction loss are shown in Fig. 8; the diffraction loss per round-trip and the resonance wavelength are shown for VCSELs with abrupt transition, i.e., with*

_{BTJ}*D*

_{trans,m}=0. As expected, the general trend is that the diffraction loss increases with decreasing diameter of the BTJ, particularly below ~6

*µ*m. Interestingly though, the loss curve shows an oscillatory tendency, and is not even monotonic. Similar oscillatory behavior has been reported previously in simplified analyzes, both for diffraction losses due only to non-axial DBR reflection [7

7. R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, “VCSEL diffraction-loss theory,” IEE Proc. Optoelectron.142, 77–81 (1995). [CrossRef]

*η*

*, in order to judge how severe the diffraction effects actually are; this is shown in Fig. 9 where the diffraction loss is expressed in percent of the total loss. In the inset results from simulations with a laterally gradually changing layer topology are shown, calculated for a VCSEL with*

_{d}*D*

*=6*

_{BTJ}*µ*m. First, it is clear from the figure that diffraction losses may be significant, and even dominant, among the loss mechanisms for structures with a small BTJ diameter. On the other hand, it is also clear that if the growth of the layers on top of the BTJ can be performed under conditions that planarize the surface, in effect increasing the

*D*

_{trans,m}with each new layer, this could lead to a marked loss reduction, at least if one can obtain transition distances exceeding ~1

*µ*m in the outermost layers of the dielectric mirror. It should be noted that although we use

*D*

_{trans,m}

_{=1}as the parameter that describes the degree of planarization, this planarizing effect is assumed to occur in all layers above the BTJ, and contributes most to the reduced diffraction loss closer to the BTJ where the optical field is stronger. Note also the difference in resonance wavelength behavior for the two methods of reducing the diffraction loss: when increasing

*D*

*the resonance (vacuum) wavelength also increases, since the field is more confined to the increasingly large higher-index pillar region near the optical axis; increasing*

_{BTJ}*D*

_{trans,m}on the other hand lowers the confinement so that the field experiences more of the peripheral part of the lateral structure, which has a lower effective index that leads to a reduced resonance wavelength.

## 5. Summary and concluding remarks

*µ*m emission. We found that the topological change in the dielectric top mirror caused by the etched BTJ may give rise to substantial diffraction loss, constituting as much as ~20–60% of the total cavity loss for BTJ diameters considered for practical VCSELs. To reduce the diffraction loss without increasing the BTJ diameter two strategies can be used; either to make the spatial phase modulation less abrupt by using gradual rather than sudden changes in the intra-cavity structure, or to position the structure in a place in the cavity where the field amplitude is low. However, these measures are usually accompanied by a change in the degree of index guiding of the intra-cavity field and a change in the resonance wavelength. The developed model accounts for guiding and standing-wave effects, and enforces true cavity field round-trip repetition by dynamic adjustment of the wavelength, and is therefore well suited to assess the advantages and drawbacks of any suggested measure to reduce the cavity diffraction loss. The perhaps simplest measure is to increase the planarizing effect when depositing the layers on top of the BTJ, thus making the height step of a layer smoother. The effects of this measure were simulated for a BTJ diameter of 6

*µ*m, and it was found that if the height step occurred over a distance of ~2

*µ*m, rather than abruptly, the diffraction loss was reduced by as much as ~70%. As evidenced by the reduced resonance wavelength, by ~2 nm, the degree of index guiding is also lowered (though the field is still guided); this may also help discriminating against higher order modes since they generally suffer more from a reduced guiding.

## Acknowledgment

## References and links

1. | M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. Rönneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, “Electrically pumped room temperature CW VCSELs with 2.3 µm emission wavelength,” Electron. Lett. |

2. | A. Krier, |

3. | A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, “Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 µm,” Electron. Lett. |

4. | P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear, “Diffraction from oxide confinement apertures in vertical-cavity lasers,” Appl. Phys. Lett. |

5. | E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in vertical-cavity lasers,” J. Sel. Top. Quantum Electron. |

6. | G. R. Hadley and K. L. Lear, “Diffraction loss of confined modes in microcavities,” 1997 Digest of the IEEE/LEOS Summer Topical Meetings (Cat. No. 97TH8276), 65–66 (1997). |

7. | R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, “VCSEL diffraction-loss theory,” IEE Proc. Optoelectron.142, 77–81 (1995). [CrossRef] |

8. | G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” J. Quantum Electron. |

9. | M. J. Noble, J. P. Loehr, and J. A. Lott, “Semi-analytic calculation of diffraction losses and threshold currents in microcavity VCSELs,” Proc. LEOS’98. IEEE Lasers and Electro-Optics Society 1998 Annual Meeting (Cat. No. 98CH36243) 1, 212–213 (1998). |

10. | P. Mackowiak, R. P. Sarzala, M. Wasiak, and W. Nakwaski, “Radial optical confinement in nitride VCSELs,” J. Phys. D |

11. | S. Riyopoulos and D. Dialetis, “Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure,” J. Opt. Soc. Am. B |

12. | P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. |

13. | A. G. Fox and T. Li, “Resonant modes in maser interferometer,” Bell System Tech. J. |

14. | D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” J. Quantum Electron. |

15. | P. Yeh, |

16. | J. W. Goodman, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(140.3070) Lasers and laser optics : Infrared and far-infrared lasers

(140.3410) Lasers and laser optics : Laser resonators

(140.5960) Lasers and laser optics : Semiconductor lasers

(230.0250) Optical devices : Optoelectronics

(140.7260) Lasers and laser optics : Vertical cavity surface emitting lasers

**ToC Category:**

Optical Devices

**History**

Original Manuscript: October 20, 2008

Revised Manuscript: November 25, 2008

Manuscript Accepted: November 25, 2008

Published: December 1, 2008

**Citation**

Jörgen Bengtsson, Johan Gustavsson, Åsa Haglund, Anders Larsson, Alexander Bachmann, Kaveh Kashani-Shirazi, and Markus-Christian Amann, "Diffraction loss in long-wavelength buried tunnel junction VCSELs analyzed with a hybrid coupled-cavity transfer-matrix model," Opt. Express **16**, 20789-20802 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20789

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### References

- M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006). [CrossRef]
- A. Krier, Mid-infrared semiconductor optoelectronics, (Springer-Verlag 2006). [CrossRef]
- A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008). [CrossRef]
- P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999). [CrossRef]
- E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997). [CrossRef]
- G. R. Hadley and K. L. Lear, "Diffraction loss of confined modes in microcavities," 1997 Digest of the IEEE/LEOS Summer Topical Meetings (Cat. No. 97TH8276), 65-66 (1997).
- R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995). [CrossRef]
- G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996). [CrossRef]
- M. J. Noble, J. P. Loehr, and J. A. Lott, "Semi-analytic calculation of diffraction losses and threshold currents in microcavity VCSELs," Proc. LEOS’98. IEEE Lasers and Electro-Optics Society 1998 Annual Meeting (Cat. No. 98CH36243) 1, 212-213 (1998).
- P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003). [CrossRef]
- S. Riyopoulos and D. Dialetis, "Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure," J. Opt. Soc. Am. B 18, 1497-1511 (2001). [CrossRef]
- P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001). [CrossRef]
- A. G. Fox and T. Li, "Resonant modes in maser interferometer," Bell System Tech. J. 40, 453-488 (1961).
- D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993). [CrossRef]
- P. Yeh, Optical waves in layered media, (Wiley 2005).
- J. W. Goodman, Introduction to Fourier optics, (McGraw-Hill 1996).

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