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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 25 — Dec. 8, 2008
  • pp: 20789–20802
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Diffraction loss in long-wavelength buried tunnel junction VCSELs analyzed with a hybrid coupled-cavity transfer-matrix model

Jörgen Bengtsson, Johan Gustavsson, Å sa Haglund, Anders Larsson, Alexander Bachmann, Kaveh Kashani-Shirazi, and Markus-Christian Amann  »View Author Affiliations


Optics Express, Vol. 16, Issue 25, pp. 20789-20802 (2008)
http://dx.doi.org/10.1364/OE.16.020789


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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

Currently, much effort is devoted to developing electrically pumped vertical-cavity surface-emitting lasers (VCSELs) for longer wavelengths so that the mid-infrared spectral region can be reached. This will enable new applications such as laser absorption spectroscopy to detect trace amounts of specific gases, for which today the distributed feedback (DFB) edge emitting laser is the standard laser source. Replacing the DFB laser with a long-wavelength VCSEL may enable gas analyzers with a lower cost, lower power consumption and the ability to rapidly scan the emission wavelength by several nanometers without mode hopping.

Several gases for which highly sensitive low cost analyzers would be of particular practical importance, such as CO, HF, and CH4, have strong absorption lines at wavelengths above 2 µm, 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 µm, 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]

]. Naturally, other material systems have been investigated, and particularly those based on GaSb have been identified as promising candidates [2

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

].

In the struggle towards longer wavelengths, the diffraction loss for the intra-cavity field in the VCSEL has emerged as an important point for concern. There are at least two reasons for this. First, the considerably longer wavelength of these devices, compared to conventional VCSELs, is not entirely accompanied by a corresponding up-scaling of the device size, which means that the typical intra-cavity feature size, measured in units of the emission wavelength, is smaller for these new VCSELs. And since diffraction is more pronounced for smaller features, it is quite natural to suspect that diffraction loss would be more prominent in long-wavelength VCSELs. Second, in the design of these devices one often introduces non-conventional features into the cavity for improved performance, such as the buried tunnel junction (BTJ). In addition to its desired effect, this may also introduce strong phase modulation of the propagating intra-cavity field, which can lead to an increased amount of light being lost by diffraction, resulting in an increased and possibly even detrimental diffraction loss.

In this work we model a recently experimentally demonstrated electrically pumped GaSbbased VCSEL [3

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]

] with room temperature emission at 2.3 µm, 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 µm BTJ diameter, which corresponds to a threshold current density of 2.6 kA/cm2, the maximum output power was 87 µW, 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.

Wd=WtotWmirrors,normal;ηd=WdWtot
(1)

where W tot is the total decrease of the energy (e.g., per unit time or per cavity round-trip) of the field in the cavity and W mirrors,normal 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, W mirrors,normal 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 η d, the diffraction loss relative to the total cavity loss.

Since there is no entirely evident definition of diffraction loss, different researchers use slightly different definitions, mostly dependent on what quantities are most readily extracted from their respective models. However, this should not be a serious problem since the purpose of the analysis is generally to establish whether diffraction loss is significant rather than producing very precise values. The important thing, and the challenge for the numerical method, is to calculate a truthful intra-cavity field and from it obtain the value for the total cavity loss, because, as mentioned, the non-axial propagation and the increasingly detailed interior structure of modern VCSEL designs require careful modeling.

Although the intra-cavity diffraction loss is not directly measurable, it is at least experimentally evident already from the far field patterns that diffraction does occur because of intracavity structures. In [4

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]

] the far field diffraction from VCSELs with rectangular oxide apertures was studied, and just as expected the diffraction was more pronounced for the smaller apertures (which were still fairly large compared to the wavelength). It is further possible to obtain indirectly a quantitative estimation for the diffraction loss from, e.g., differential efficiency measurements [5

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]

], but this still requires some sort of underlying model and assumptions.

To estimate the diffraction loss theoretically, one calculates the intra-cavity field by simplifying the structure and/or restricting the intra-cavity field. The structural simplification can be direct [6

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).

], such as assuming that the central region between the mirrors is a geometrically simple waveguide [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]

], or occur in processing the structure before solving for the intra-cavity fields by an effective index or weighting index approach [6

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).

], [8

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

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]

]. The field restriction can be either that the field is assumed to be a superposition of certain (eigen)modes of some (generally simplified) cavity geometry [11

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

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]

], or an explicit assumption of the spatial dependency, e.g., separability in one of the coordinates [8

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

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]

]. To avoid making assumptions about the intracavity field, one can use an iterative field-finding approach based on the Fox-Li method [13

13. A. G. Fox and T. Li, “Resonant modes in maser interferometer,” Bell System Tech. J. 40, 453–488 (1961).

]. It was used in [5

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]

] to estimate the diffraction loss in oxide aperture VCSELs. Compared to the original Fox-Li scheme, the method was modified by including DBR mirrors with an angle-dependent reflectivity by a transfer matrix method (TMM) [14

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]

], as well as intra-cavity elements, most importantly the oxide aperture. Since the Fox-Li analysis does not account for standing wave effects it was necessary to introduce a position dependent effective phase modulation of the oxide aperture. In our modeling, we use the same geometrical dimensionality as in [5

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]

], with a fully 3D central region (which in our case also includes a number of layers of the top mirror where the BTJ has caused a structural change) terminated with layered (1D) structures representing the top and bottom mirrors; the fields are 3D everywhere. However, to accurately simulate the complex intra-cavity geometry the central region is further divided into coupled sub-cavities, which means that the total standing-wave field can now be accounted for. This mimics the field propagation in a real cavity and thus, as an example, the position dependent guiding effect of an oxide aperture would result naturally from the field propagation.

2. Numerical model

In the simulations, the central VCSEL structure is made up of a number of layers, or sub-cavities, in the longitudinal (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.

Fig. 1. Illustration of the relation between the actual VCSEL structure and the numerical model. The numerical model for the central VCSEL region is built up by coupled layers, or coupled cavities, where each may have a refractive index with an arbitrary lateral variation. Also indicated are the four optical fields in each layer, and their relations via transmission, reflection and propagation.

Fig. 2. Amplitude reflectance (top) and phase shift upon reflection (bottom), here calculated for the starting wavelength of 2340 nm, of a plane wave as a function of the incidence angle from the x- and y-axis, respectively, for the top DBR structure used in the simulation.

In the 3D central part of the VCSEL, the layers can have an arbitrary lateral refractive index distribution 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

E(x,y)m,r,l=t(x,y)m1,r,rE(x,y)m1,r,r+r(x,y)m,l,lE(x,y)m,l,l
E(x,y)m,l,r=t(x,y)m+1,l,lE(x,y)m+1,l,l+r(x,y)m,r,rE(x,y)m,r,r
(2)

except at the two boundaries facing the top and bottom DBR structures, where there is only a reflected but no transmitted wave, calculated as described above. The transmission and reflection functions 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.

After the starting fields have been calculated, the propagated fields are calculated - 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) m [16

16. J. W. Goodman, Introduction to Fourier optics, (McGraw-Hill1996).

].

At a sufficiently large distance from the optical axis not to influence the propagation of the guided cavity field is a numerically absorbing material that attenuates any field approaching the numerical window of the calculations, thus avoiding numerical spurious reflections of the outward-propagating field. The absorption coefficient is very small and increases parabolically away from the surface of the absorbing material so as to minimize any reflections from the absorbing material itself. Because of the absorbing material the propagated field suffers a very slight loss, which is one of the contributions to the overall diffraction loss: the lateral walk-off of the non-axial field. The other loss contribution, the excess loss at the DBR mirrors, is obtained by simply taking the difference between the field power before and after reflection in the mirrors, and subtracting the corresponding difference using DBR reflectance values for normal incidence.

3. Physical and model structure

Figure 4 shows a schematic illustration of the studied 2.3 µm BTJ VCSEL. It consists of a GaSb substrate, a 24-pair n-doped AlAs0.09Sb0.91/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 Ga0.63In0.37As0.03Sb0.97 quantum wells separated by 8-nm-thick Al0.33Ga0.67As0.03Sb0.97 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 J, 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.

Fig. 3. Flowchart of the numerical algorithm to simulate the VCSEL cavity field and calculate the diffraction loss. The variables N p and Δλ p are decision parameters set by the user.

Fig. 4. Schematic of the BTJ VCSEL cavity structure modeled in this study.

As an optional feature the model can simulate a gradual, rather than abrupt, change of the longitudinal position of each layer from the area on top of the BTJ pillar to that outside. This is modeled as a smoothing of the lateral refractive index step by a convolution of the abrupt index distribution with a Gaussian function with a certain diameter, 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,m=1=1 µm. Also note the different scales in the longitudinal (z) and lateral (x,y) directions in this figure.

4. Results from the simulations

The simulations were performed for different diameters of the BTJ pillar. An abrupt transition was assumed in all layers, i.e., 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.

Fig. 5. The on-axis refractive index variation in the VCSEL structure (top), and the numerical representation of the coupled-cavity section (bottom). Case A is for a perfect reproduction of the steep walls of the BTJ throughout the top layers, i.e., D trans,m=0, while Case B is for a transition region increasing linearly with the distance from the BTJ (to the right) having a maximum value D trans,m =1=1 µm.
Fig. 6. Evolution towards convergence of the iterative simulation for a VCSEL with D BTJ=6 µm and D trans,m=0. The inset shows the laterally spatially averaged phase value for the field in a fixed longitudinal position as a function of the number of iterations (cavity round-trips); the sudden phase jumps result from the field averaging in the simulation method. The main figure shows the calculation wavelength, which is dynamically adjusted during the iterations to enforce true repetition of the field, so that the field phase does not change between the round-trips; the inset shows that this goal is accomplished after ~1000 iterations.

From the calculated fields the layer propagation losses - the walk-off losses to the numerically absorbing material near the boundary of the calculation window - were obtained as described, as well as the total losses in the reflections from the top and bottom DBRs. Together, all these losses constitute the total round-trip loss, and subtracting from it the mirror losses for normal incidence we obtained the so defined diffraction loss. The fraction of the total diffraction loss caused by walk-off and mirrors excess loss, respectively, varied slightly for different values of D BTJ, 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 D trans,m=0. As expected, the general trend is that the diffraction loss increases with decreasing diameter of the BTJ, particularly below ~6 µm. 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]

], and diffraction losses due only to lateral cavity walk-off [6

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).

].

Fig. 7. Cross section in the yz-plane of the absolute value of the standing-wave field in the coupled-cavity section for the VCSEL with D BTJ=6 µm and D trans,m=0, calculated from the fields E(x,y)m,d,p obtained in the final iteration. The inset shows the amplitude profiles of the left-propagating field at the left boundary in all the 15 layers.

It is perhaps even more informative to relate the diffraction loss to the total cavity loss, i.e. to calculate η d, 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 BTJ=6 µm. 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 µm 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 BTJ 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 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

To improve the laser performance and reach new emission wavelengths, novel intra-cavity structures such as multiple oxide apertures or BTJs are increasingly employed in modern VC- SEL designs. Often this leads to quite an abrupt spatial phase modulation of the intra-cavity field, which increases the risk of excessive diffraction loss. We have developed a numerical model that handles non-axial propagation of a 3D field in a 3D central VCSEL structure terminated by two 1D DBR structures and applied it to an experimentally recently realized GaSb-based VCSEL with 2.3 µm 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 µm, and it was found that if the height step occurred over a distance of ~2 µm, 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.

Fig. 8. Diffraction loss per round-trip and resonance wavelength as functions of the diameter of the BTJ, for D trans,m=0.

Fig. 9. Diffraction loss normalized to the total round-trip loss as a function of the BTJ diameter, for D trans,m=0. The inset shows the normalized diffraction loss as a function of the transition distance between the high and the low levels of the topmost layer in the dielectric top mirror, for D BTJ=6 µm. A non-zero value for D trans,m =1 implies that a gradual transition is assumed to occur also in all other layers in the physical top DBR. Also shown is the resonance wavelength for these cases.

Acknowledgment

This work was partly funded by the project NEMIS of the European Union (contr.: 031845) and the project NOSE of the Federal Ministry of Education and Research of Germany (contr.: 13N8772).

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. 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]

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]

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. 32, 607–616 (1996). [CrossRef]

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 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]

13.

A. G. Fox and T. Li, “Resonant modes in maser interferometer,” Bell System Tech. J. 40, 453–488 (1961).

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]

15.

P. Yeh, Optical waves in layered media, (Wiley2005).

16.

J. W. Goodman, Introduction to Fourier optics, (McGraw-Hill1996).

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

  1. 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]
  2. A. Krier, Mid-infrared semiconductor optoelectronics, (Springer-Verlag 2006). [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]
  4. 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]
  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]
  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. 32, 607-616 (1996). [CrossRef]
  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 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]
  13. A. G. Fox and T. Li, "Resonant modes in maser interferometer," Bell System Tech. J. 40, 453-488 (1961).
  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]
  15. P. Yeh, Optical waves in layered media, (Wiley 2005).
  16. J. W. Goodman, Introduction to Fourier optics, (McGraw-Hill 1996).

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