November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
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Spatially resolved Stokes parameters of small-area vertical-cavity surface-emitting lasers: experiment and simulation Andreas Molitor,1,* Pierluigi Debernardi,2 Sébastien Hartmann,1 and Wolfgang Elsäßer1,3 1
Institute of Applied Physics, Technical University Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany 2
3
Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni, 10129 Torino, Italy
Also with Center of Smart Interfaces, Technical University Darmstadt, Petersenstr. 32, 64287 Darmstadt, Germany *Corresponding author: [email protected]‑darmstadt.de Received September 6, 2013; revised October 10, 2013; accepted October 15, 2013; posted October 16, 2013 (Doc. ID 197132); published November 12, 2013 We present experimental investigations on spatially resolved Stokes parameters of vertical-cavity surface-emitting lasers (VCSELs) with a small aperture diameter of 3 μm and a monolithically integrated surface grating on top of the structure to technologically control the polarization. As expected, the grating fixes the state of polarization, but still shows both a spatially nonuniform linear polarization distribution of the fundamental transverse mode as well as an interesting eight-lobe pattern of circular polarization in terms of change of sign. These experimental findings are reproduced by numerical simulations using a fully vectorial three-dimensional model. © 2013 Optical Society of America OCIS codes: (120.2130) Ellipsometry and polarimetry; (250.7260) Vertical cavity surface emitting lasers; (260.5430) Polarization. http://dx.doi.org/10.1364/OL.38.004777
The polarization state of light is a fascinating phenomenon because of its versatility in many optical applications like ellipsometry [1], microscopy [2], and three-dimensional (3D) imaging [3]. Furthermore, polarization can be spatially dependent, namely in azimuthally or radially polarized beams of light that are used for high-resolution microscopy [4] or as an optical tweezer to manipulate objects on a microscale [5]. Besides these optical applications of polarized light, the polarization behavior of light directly emitted by optoelectronic devices has been investigated very recently, especially for many different types of vertical-cavity surface-emitting lasers (VCSELs) [6–11] to understand fundamental physical mechanisms, resulting in complex polarization behavior, such as polarization competition, switches, and hysteresis. Even a spatial inhomogeneous polarization of the fundamental transverse mode has been reported [12]. In [12], the polarization analysis of the spatially resolved transverse mode has been performed on a linear polarization projection basis only, and uncovered a weak four-lobe structure beside the dominant Gaussian-shaped mode, in good agreement with theory. We have shown very recently that the Stokes formalism is mandatory to reveal the full information of the state of polarization of the VCSEL’s emitted light [13]. Using an extended experimental technique, we demonstrated for what we believe is the first time spatially resolved Stokes parameters of the fundamental transverse mode of a small-area surface-grating VCSEL, showing both a spatially nonuniform linear polarization distribution of the fundamental transverse mode as well as an interesting pattern of circular polarization. These experimental findings are explained by the combination of the two orthogonal field distributions with complex values, which represent the solution of Maxwell’s equations in the VCSEL and are confirmed by comparing to numerical simulations using the 3D 0146-9592/13/224777-04$15.00/0
VCSEL ELectroMagnetic (VELM) code [14], which has already been successfully applied to compute the spatially resolved modal polarization features of different devices [15]. Figure 1(a) shows a schematic drawing of a VCSEL that consists of a quantum-well active layer (red), an injection current confining oxide layer (black) sandwiched between the top and the bottom distributed Bragg reflector mirrors framed by an electrical contact (yellow), and finally the monolithically integrated surface grating on the top layer of the VCSEL. The investigated devices (ULM Photonics Philips) emit at a central wavelength of 850 nm, and have been selected with respect to both the diameter of their oxide aperture of 3 μm (black) and their surface-grating parameters, namely an etching depth of 52 nm (i) and a grating period of 0.9 μm (ii), therefore providing a reliable polarization control, such that neither a polarization switching nor a change of the orientation of the linear polarization is observable. The orientation of the equidistant grating grooves has been thoughtfully selected to align with the crystal axis [110] of the semiconductor device, depicted in the photographic picture [Fig. 1(b)] showing a top view of the VCSEL with its crystal axes and the surface grating on top. This selected device is appropriate because of its small oxide aperture diameter: It emits transverse singlemode, thus the resulting weaker four-lobe component can be analyzed experimentally via Stokes formalism, although its intensity is reduced by a factor in the range between 104 –105 with respect to the intensity of the Gaussian-shaped component as it has been reported in [12]. These experimentally obtained spatially resolved Stokes parameters are compared with their numerically simulated counterparts generated by VELM code [14] to understand the underlying mechanisms. The single-mode operation of a small oxide aperture diameter VCSEL © 2013 Optical Society of America
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Fig. 1. (a) Schematic drawing of the VCSEL structure showing the active layer (red), the oxide aperture (black), the high reflecting DBR mirrors framed by the electrical contacts (yellow) and the surface grating with its parameters etching depth (i) and grating period (ii) on top. (b) A photographical picture of the top view of the VCSEL, showing the orientation of the grating grooves with respect to the crystal axis [110] and [−110] here depicted schematically with a red and a blue arrow, respectively. (c) Stokes parameters as a function of pump current measured using a photo detector. The polarization control due to the surface grating is clearly visible regarding the constant state of polarization represented by a dominant contribution of the negative value of S 1 (red plus) and small but nonzero contributions of S 3 (green circle) and S 2 (blue cross). Due to the single-mode operation the VCSEL’s total emission is fully polarized reflected in the DOP value close to one above laser threshold at 2 mA.
allows us to neglect transverse mode competition in the numerical simulations. Therefore, a simpler coldcavity approach is appropriate for comparing obtained results of the spatially resolved Stokes parameters experimentally and numerically. In Fig. 1(c), the normalized Stokes parameters S 1 , S 2 and S 3 as well as the degree of polarization (DOP) of the total emission are plotted as a function of the pump current. The parameter S 0 describes the total intensity of the light beam and is determined to normalize all the other Stokes parameters. Thus, they will only take values between −1 and 1, allowing directly comparable measurements at different pump currents. The parameter S 1 describes the linear horizontal (crystal axis [−110], S 1 > 0), or vertical (crystal axis [110], S 1 < 0) polarization. The parameter S 2 reflects the amount of linear polarization with an orientation of 45° (S 2 > 0) or −45° (S 2 < 0) and the parameter S 3 specifies the amount of right or left circular polarization within the investigated light beam. The degree of polarization (DOP) term:
s S 21 S 22 S 23 DOP ; S 20
(1)
therefore quantifies the ratio of the intensity of the fully polarized part to the total intensity (fully polarized DOP 1 and fully unpolarized DOP 0). Below the laser threshold at 2.0 mA, all these values are very close to zero, representing an unpolarized state of light due to the spontaneous emission character of the emitted light. Above the laser threshold, a sharp increase of the DOP (magenta open square) from 0 to 1 and a sharp decrease of S 1 (red plus) from 0 to −1 are observable. The state of polarization of the fully polarized light (DOP 1) does not change with increasing pump current and is reflected in the dominant value of S 1 ≈ −1 (red plus) representing a linearly polarized state of light oriented parallel to the crystal axis [110] and the very small, negative and constant contributions of S 2 and S 3 (blue cross and green circle) demonstrating the polarization controlling functionality of the selected surface grating. To unveil the spatial distribution of the Stokes vector Sx; y, we extended our experimental setup as it is depicted in Fig. 2. This schematic drawing contains the VCSEL and its emitted light, after collimation by a lens, passes a combination of a revolvable quarter-wave plate and a linear polarizer, which has its transmission axis fixed to the horizontal axis (crystal axis [−110]). The spatial distribution of the intensity of the near field of the VCSEL is detected by a charge-coupled device (CCD) camera connected to a computer that records the twodimensional intensity distributions as a function of the angle β in units of radians (see Fig. 2) between the horizontal axis aligned with the crystal axis [−110] and the fast axis of the quarter-wave plate [16,17]. The resulting Stokes vector of the detected light S0 x; y at the CCD camera can be calculated considering
Fig. 2. Schematic of the experimental setup showing the VCSEL with the crystal axes [110] and [−110]. The emitted light is collimated by a lens, passes a combination of a quarter-wave plate with revolvable fast axis and a polarizer with fixed transmission axis and finally the spatial intensity distribution is detected by a CCD camera. In the lower part, a representative set of the intensity of three selected pixels (1—green, 2—blue, 3 —red) of the near-field image (top left) is shown as a function of the of the angle β together with a fit to the measured data (lines) to extract the Stokes parameters. The 2D Stokes parameter distribution is calculated for each pixel using this procedure.
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
the combined effect of around an angle β, and vector of the VCSEL’s to the Müllers matrix pressed by
a quarter-wave plate revolvable a linear polarizer on the Stokes emitted light Sx; y according formalism, mathematically ex-
S0 x; y M ̳ LP M ̳ WP β Sx; y:
(2)
Here, M ̳ LP and M ̳ WP β represent the 4 × 4 Müller matrices for a linear polarizer and a revolvable quarterwave plate, respectively. The first Stokes parameter of the detected light S 00 β; x; y can be simplified to the following mathematical expression:
Figure 3 shows both these experimentally obtained spatially resolved normalized Stokes parameters S i exp x; y with i 0; 1; 2; 3 as well as their numerically simulated counterparts S i sim x; y with i 0; 1; 2; 3 for a small-area, surface-grating VCSEL at a pump current of 3 mA. The numerically simulated Stokes parameters S i sim x; y have been calculated using the following mathematical equations showing the definition of the Stokes parameters [17] in terms of two Cartesian components E x and E y of the mode profile computed by VELM (the asterisk indicates the complex conjugate):
1 S 00 β; x; y S 0 x; y S 1 x; ycos2 2β 2 S 2 x; y cos2β sin2β S 3 x; y sin2β:
4779
S 0 E x E x E y E y ;
(4a)
S 1 E x E x − E y Ey ;
(4b)
S 2 E x E y E x E y ;
(4c)
S 3 iE x E y − E x E y :
(4d)
(3) and S 00 xi ; yi
of The intensity of each corresponding pixel the recorded images obtained for different angles β is plotted as a function of β and a multiparameter fit to these obtained data as shown for three pixels in the lower part of Fig. 2 allows us to derive the spatial distribution of the Stokes vector Sx; y as a function of the two orthogonal coordinate axes x and y.
In the left part of Fig. 3, the first Stokes parameter S 0 exp x; y represents the total intensity of the incident light and is in good agreement with the near-field image recorded without the combination of a linear polarizer and a quarter-wave plate (not depicted here). The minimal value of S 1 exp x; y of −1 reflects the linear vertical polarization of the dominant mode and coincides with the results depicted in Fig. 1 (red plus). A bracket shape framing the Gaussian-shaped central spot also shows up, like what had been reported earlier in [18] and which, in the present case, is attributed to an oblong shape of the oxide aperture, originating from manufacturing processes [14,19]. In a standard VCSEL without a surface grating, an elliptically polarized beam of light is in general generated as we have shown in [13]. In the present case, the surfacegrating on top of our investigated devices fixes the state of polarization such that a dominant linear polarization S 1 exp x; y with an orientation parallel to the grating grooves due to the induced dichroism [6] is observable. However a weak four-lobe pattern still shows up in S 2 exp x; y (with a change of sign) and in S 3 exp x; y one even observes an eight-lobe distribution. Both S 2 exp x; y and S 3 exp x; y are showing an inner part with the same sign pattern. Additionally an outer part with a complementary sign pattern is observable for S 3 exp x; y. This clearly reflects the four-lobe symmetry of the weak field component I x exp x; y depicted in Fig. 4 together with the strong field component I y exp x; y and their simulated counterparts. These two-dimensional intensity distributions on a linear projection basis only have been calculated from the experimentally obtained spatially resolved Stokes parameters S 0 exp x; y and S 1 exp x; y using the following mathematical expression: 1 I x x; y S 0 x; y S 1 x; y 2
Fig. 3. Spatially resolved Stokes parameters: experimental (left) and numerically simulated (right) results.
and
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Fig. 4. Spatially resolved I x x; y and I y x; y calculated from the experimentally obtained Stokes parameters (left) in comparison with their numerically simulated counterparts (right).
In conclusion, we demonstrated what we believe is a new technique to measure spatially resolved Stokes parameters of VCSELs. Furthermore, we successfully compared our experimentally obtained results of spatially resolved Stokes parameters of a small-area surface-grating VCSEL with their numerically simulated counterparts. This also confirms the rigorous functionality of the VELM code with respect to the spatial distribution of polarization in terms of the Stokes formalism and demonstrates that the measured Sx; y patterns are consequently caused by the modal features of the small-sized VCSEL. This comparison and further investigations on spatiospectrally resolved Stokes parameter measurements also of multitransverse mode VCSELs will offer insight into the complex polarization behavior of VCSELs. This knowledge combined with the powerful VELM simulation tool can then be used to design VCSELs with particular polarization properties for special applications.
(5b)
We thank Dr. Johannes Michael Ostermann and Dr. Rainer Michalzik for providing excellent VCSEL structures.
The calculations were performed on the one hand to visualize the four-lobe intensity distribution in the x direction as it has been reported already in [12] and on the other hand to determine the ratio between the intensity of the dominant and the weak component to be 15 10−4 , to be comparable with the theoretical findings. The maximal and minimal values of the simulated results (maximal value of S 0 sim x; y 1; minimal value of S 1 sim x; y −1) are in excellent agreement with the experimentally obtained results, confirming the dominant linear vertical polarization character of the emission. The patterns of S 2 exp x; y and S 3 exp x; y are also fully reproduced by the simulations. Also, the maximal and minimal values of S 2 sim x; y and S 3 sim x; y compare very well with their experimental counterparts, being in the range between −0.02 and 0.02 for both S 2 x; y and S 3 x; y. As a consequence, also the weak mode component I x exp x; y and I x sim x; y compare very well. By comparing the experimentally obtained results and the numerically simulated results generated by VELM code, we were able prove that, in spite the fact that we have a polarization controlling surface grating, a nonzero amount of (i) linear polarization S 2 x; y and (ii) circular polarization component S 3 x; y show up. The amount of S 2 x; y and S 3 x; y and their complex four-lobe and eight-lobe pattern in terms of change of sign, respectively, can be explained by the superposition of the two complex orthogonal linearly polarized electromagnetic field components, originating from the solution of Maxwell’s equations in our VCSEL structure [12]. Indeed, the VELM code applied to a small-area, surfacegrating VCSEL with a slightly oblong oxide aperture (whose major axis is perpendicular to the field brackets), excellently reproduces the bracket structure, the fourand eight-lobe pattern as well as their normalized values.
References 1. W. E. J. Neal and R. W. Fane, J. Phys. E 6, 409 (1973). 2. S. Inoué, J. Cell Biol. 89, 346 (1981). 3. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, Opt. Lett. 32, 481 (2007). 4. R. Dom, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003). 5. T. A. Nieminen, N. R. Hechenburg, and H. RubinszteinDunlop, Opt. Lett. 33, 122 (2008). 6. P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005). 7. M. S. Torre, A. Valle, and L. Pesquera, Opt. Quantum Electron. 38, 445 (2006). 8. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997). 9. M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 84, 4337 (2000). 10. M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995). 11. R. Michalzik, VCSELs—Fundamentals, Technology and Applications of VCSELs (Springer, 2013). 12. L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001). 13. A. Molitor, S. Hartmann, and W. Elsässer, Opt. Lett. 37, 4799 (2012). 14. P. Debernardi, IEEE J. Quantum Electron. 45, 979 (2009). 15. P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003). 16. H. G. Berry, G. Gabrielse, and A. E. Livingston, Appl. Opt. 16, 3200 (1977). 17. B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007). 18. P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013). 19. P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
1 I y x; y S 0 x; y − S 1 x; y: 2
4777
Spatially resolved Stokes parameters of small-area vertical-cavity surface-emitting lasers: experiment and simulation Andreas Molitor,1,* Pierluigi Debernardi,2 Sébastien Hartmann,1 and Wolfgang Elsäßer1,3 1
Institute of Applied Physics, Technical University Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany 2
3
Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni, 10129 Torino, Italy
Also with Center of Smart Interfaces, Technical University Darmstadt, Petersenstr. 32, 64287 Darmstadt, Germany *Corresponding author: [email protected]‑darmstadt.de Received September 6, 2013; revised October 10, 2013; accepted October 15, 2013; posted October 16, 2013 (Doc. ID 197132); published November 12, 2013 We present experimental investigations on spatially resolved Stokes parameters of vertical-cavity surface-emitting lasers (VCSELs) with a small aperture diameter of 3 μm and a monolithically integrated surface grating on top of the structure to technologically control the polarization. As expected, the grating fixes the state of polarization, but still shows both a spatially nonuniform linear polarization distribution of the fundamental transverse mode as well as an interesting eight-lobe pattern of circular polarization in terms of change of sign. These experimental findings are reproduced by numerical simulations using a fully vectorial three-dimensional model. © 2013 Optical Society of America OCIS codes: (120.2130) Ellipsometry and polarimetry; (250.7260) Vertical cavity surface emitting lasers; (260.5430) Polarization. http://dx.doi.org/10.1364/OL.38.004777
The polarization state of light is a fascinating phenomenon because of its versatility in many optical applications like ellipsometry [1], microscopy [2], and three-dimensional (3D) imaging [3]. Furthermore, polarization can be spatially dependent, namely in azimuthally or radially polarized beams of light that are used for high-resolution microscopy [4] or as an optical tweezer to manipulate objects on a microscale [5]. Besides these optical applications of polarized light, the polarization behavior of light directly emitted by optoelectronic devices has been investigated very recently, especially for many different types of vertical-cavity surface-emitting lasers (VCSELs) [6–11] to understand fundamental physical mechanisms, resulting in complex polarization behavior, such as polarization competition, switches, and hysteresis. Even a spatial inhomogeneous polarization of the fundamental transverse mode has been reported [12]. In [12], the polarization analysis of the spatially resolved transverse mode has been performed on a linear polarization projection basis only, and uncovered a weak four-lobe structure beside the dominant Gaussian-shaped mode, in good agreement with theory. We have shown very recently that the Stokes formalism is mandatory to reveal the full information of the state of polarization of the VCSEL’s emitted light [13]. Using an extended experimental technique, we demonstrated for what we believe is the first time spatially resolved Stokes parameters of the fundamental transverse mode of a small-area surface-grating VCSEL, showing both a spatially nonuniform linear polarization distribution of the fundamental transverse mode as well as an interesting pattern of circular polarization. These experimental findings are explained by the combination of the two orthogonal field distributions with complex values, which represent the solution of Maxwell’s equations in the VCSEL and are confirmed by comparing to numerical simulations using the 3D 0146-9592/13/224777-04$15.00/0
VCSEL ELectroMagnetic (VELM) code [14], which has already been successfully applied to compute the spatially resolved modal polarization features of different devices [15]. Figure 1(a) shows a schematic drawing of a VCSEL that consists of a quantum-well active layer (red), an injection current confining oxide layer (black) sandwiched between the top and the bottom distributed Bragg reflector mirrors framed by an electrical contact (yellow), and finally the monolithically integrated surface grating on the top layer of the VCSEL. The investigated devices (ULM Photonics Philips) emit at a central wavelength of 850 nm, and have been selected with respect to both the diameter of their oxide aperture of 3 μm (black) and their surface-grating parameters, namely an etching depth of 52 nm (i) and a grating period of 0.9 μm (ii), therefore providing a reliable polarization control, such that neither a polarization switching nor a change of the orientation of the linear polarization is observable. The orientation of the equidistant grating grooves has been thoughtfully selected to align with the crystal axis [110] of the semiconductor device, depicted in the photographic picture [Fig. 1(b)] showing a top view of the VCSEL with its crystal axes and the surface grating on top. This selected device is appropriate because of its small oxide aperture diameter: It emits transverse singlemode, thus the resulting weaker four-lobe component can be analyzed experimentally via Stokes formalism, although its intensity is reduced by a factor in the range between 104 –105 with respect to the intensity of the Gaussian-shaped component as it has been reported in [12]. These experimentally obtained spatially resolved Stokes parameters are compared with their numerically simulated counterparts generated by VELM code [14] to understand the underlying mechanisms. The single-mode operation of a small oxide aperture diameter VCSEL © 2013 Optical Society of America
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Fig. 1. (a) Schematic drawing of the VCSEL structure showing the active layer (red), the oxide aperture (black), the high reflecting DBR mirrors framed by the electrical contacts (yellow) and the surface grating with its parameters etching depth (i) and grating period (ii) on top. (b) A photographical picture of the top view of the VCSEL, showing the orientation of the grating grooves with respect to the crystal axis [110] and [−110] here depicted schematically with a red and a blue arrow, respectively. (c) Stokes parameters as a function of pump current measured using a photo detector. The polarization control due to the surface grating is clearly visible regarding the constant state of polarization represented by a dominant contribution of the negative value of S 1 (red plus) and small but nonzero contributions of S 3 (green circle) and S 2 (blue cross). Due to the single-mode operation the VCSEL’s total emission is fully polarized reflected in the DOP value close to one above laser threshold at 2 mA.
allows us to neglect transverse mode competition in the numerical simulations. Therefore, a simpler coldcavity approach is appropriate for comparing obtained results of the spatially resolved Stokes parameters experimentally and numerically. In Fig. 1(c), the normalized Stokes parameters S 1 , S 2 and S 3 as well as the degree of polarization (DOP) of the total emission are plotted as a function of the pump current. The parameter S 0 describes the total intensity of the light beam and is determined to normalize all the other Stokes parameters. Thus, they will only take values between −1 and 1, allowing directly comparable measurements at different pump currents. The parameter S 1 describes the linear horizontal (crystal axis [−110], S 1 > 0), or vertical (crystal axis [110], S 1 < 0) polarization. The parameter S 2 reflects the amount of linear polarization with an orientation of 45° (S 2 > 0) or −45° (S 2 < 0) and the parameter S 3 specifies the amount of right or left circular polarization within the investigated light beam. The degree of polarization (DOP) term:
s S 21 S 22 S 23 DOP ; S 20
(1)
therefore quantifies the ratio of the intensity of the fully polarized part to the total intensity (fully polarized DOP 1 and fully unpolarized DOP 0). Below the laser threshold at 2.0 mA, all these values are very close to zero, representing an unpolarized state of light due to the spontaneous emission character of the emitted light. Above the laser threshold, a sharp increase of the DOP (magenta open square) from 0 to 1 and a sharp decrease of S 1 (red plus) from 0 to −1 are observable. The state of polarization of the fully polarized light (DOP 1) does not change with increasing pump current and is reflected in the dominant value of S 1 ≈ −1 (red plus) representing a linearly polarized state of light oriented parallel to the crystal axis [110] and the very small, negative and constant contributions of S 2 and S 3 (blue cross and green circle) demonstrating the polarization controlling functionality of the selected surface grating. To unveil the spatial distribution of the Stokes vector Sx; y, we extended our experimental setup as it is depicted in Fig. 2. This schematic drawing contains the VCSEL and its emitted light, after collimation by a lens, passes a combination of a revolvable quarter-wave plate and a linear polarizer, which has its transmission axis fixed to the horizontal axis (crystal axis [−110]). The spatial distribution of the intensity of the near field of the VCSEL is detected by a charge-coupled device (CCD) camera connected to a computer that records the twodimensional intensity distributions as a function of the angle β in units of radians (see Fig. 2) between the horizontal axis aligned with the crystal axis [−110] and the fast axis of the quarter-wave plate [16,17]. The resulting Stokes vector of the detected light S0 x; y at the CCD camera can be calculated considering
Fig. 2. Schematic of the experimental setup showing the VCSEL with the crystal axes [110] and [−110]. The emitted light is collimated by a lens, passes a combination of a quarter-wave plate with revolvable fast axis and a polarizer with fixed transmission axis and finally the spatial intensity distribution is detected by a CCD camera. In the lower part, a representative set of the intensity of three selected pixels (1—green, 2—blue, 3 —red) of the near-field image (top left) is shown as a function of the of the angle β together with a fit to the measured data (lines) to extract the Stokes parameters. The 2D Stokes parameter distribution is calculated for each pixel using this procedure.
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
the combined effect of around an angle β, and vector of the VCSEL’s to the Müllers matrix pressed by
a quarter-wave plate revolvable a linear polarizer on the Stokes emitted light Sx; y according formalism, mathematically ex-
S0 x; y M ̳ LP M ̳ WP β Sx; y:
(2)
Here, M ̳ LP and M ̳ WP β represent the 4 × 4 Müller matrices for a linear polarizer and a revolvable quarterwave plate, respectively. The first Stokes parameter of the detected light S 00 β; x; y can be simplified to the following mathematical expression:
Figure 3 shows both these experimentally obtained spatially resolved normalized Stokes parameters S i exp x; y with i 0; 1; 2; 3 as well as their numerically simulated counterparts S i sim x; y with i 0; 1; 2; 3 for a small-area, surface-grating VCSEL at a pump current of 3 mA. The numerically simulated Stokes parameters S i sim x; y have been calculated using the following mathematical equations showing the definition of the Stokes parameters [17] in terms of two Cartesian components E x and E y of the mode profile computed by VELM (the asterisk indicates the complex conjugate):
1 S 00 β; x; y S 0 x; y S 1 x; ycos2 2β 2 S 2 x; y cos2β sin2β S 3 x; y sin2β:
4779
S 0 E x E x E y E y ;
(4a)
S 1 E x E x − E y Ey ;
(4b)
S 2 E x E y E x E y ;
(4c)
S 3 iE x E y − E x E y :
(4d)
(3) and S 00 xi ; yi
of The intensity of each corresponding pixel the recorded images obtained for different angles β is plotted as a function of β and a multiparameter fit to these obtained data as shown for three pixels in the lower part of Fig. 2 allows us to derive the spatial distribution of the Stokes vector Sx; y as a function of the two orthogonal coordinate axes x and y.
In the left part of Fig. 3, the first Stokes parameter S 0 exp x; y represents the total intensity of the incident light and is in good agreement with the near-field image recorded without the combination of a linear polarizer and a quarter-wave plate (not depicted here). The minimal value of S 1 exp x; y of −1 reflects the linear vertical polarization of the dominant mode and coincides with the results depicted in Fig. 1 (red plus). A bracket shape framing the Gaussian-shaped central spot also shows up, like what had been reported earlier in [18] and which, in the present case, is attributed to an oblong shape of the oxide aperture, originating from manufacturing processes [14,19]. In a standard VCSEL without a surface grating, an elliptically polarized beam of light is in general generated as we have shown in [13]. In the present case, the surfacegrating on top of our investigated devices fixes the state of polarization such that a dominant linear polarization S 1 exp x; y with an orientation parallel to the grating grooves due to the induced dichroism [6] is observable. However a weak four-lobe pattern still shows up in S 2 exp x; y (with a change of sign) and in S 3 exp x; y one even observes an eight-lobe distribution. Both S 2 exp x; y and S 3 exp x; y are showing an inner part with the same sign pattern. Additionally an outer part with a complementary sign pattern is observable for S 3 exp x; y. This clearly reflects the four-lobe symmetry of the weak field component I x exp x; y depicted in Fig. 4 together with the strong field component I y exp x; y and their simulated counterparts. These two-dimensional intensity distributions on a linear projection basis only have been calculated from the experimentally obtained spatially resolved Stokes parameters S 0 exp x; y and S 1 exp x; y using the following mathematical expression: 1 I x x; y S 0 x; y S 1 x; y 2
Fig. 3. Spatially resolved Stokes parameters: experimental (left) and numerically simulated (right) results.
and
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Fig. 4. Spatially resolved I x x; y and I y x; y calculated from the experimentally obtained Stokes parameters (left) in comparison with their numerically simulated counterparts (right).
In conclusion, we demonstrated what we believe is a new technique to measure spatially resolved Stokes parameters of VCSELs. Furthermore, we successfully compared our experimentally obtained results of spatially resolved Stokes parameters of a small-area surface-grating VCSEL with their numerically simulated counterparts. This also confirms the rigorous functionality of the VELM code with respect to the spatial distribution of polarization in terms of the Stokes formalism and demonstrates that the measured Sx; y patterns are consequently caused by the modal features of the small-sized VCSEL. This comparison and further investigations on spatiospectrally resolved Stokes parameter measurements also of multitransverse mode VCSELs will offer insight into the complex polarization behavior of VCSELs. This knowledge combined with the powerful VELM simulation tool can then be used to design VCSELs with particular polarization properties for special applications.
(5b)
We thank Dr. Johannes Michael Ostermann and Dr. Rainer Michalzik for providing excellent VCSEL structures.
The calculations were performed on the one hand to visualize the four-lobe intensity distribution in the x direction as it has been reported already in [12] and on the other hand to determine the ratio between the intensity of the dominant and the weak component to be 15 10−4 , to be comparable with the theoretical findings. The maximal and minimal values of the simulated results (maximal value of S 0 sim x; y 1; minimal value of S 1 sim x; y −1) are in excellent agreement with the experimentally obtained results, confirming the dominant linear vertical polarization character of the emission. The patterns of S 2 exp x; y and S 3 exp x; y are also fully reproduced by the simulations. Also, the maximal and minimal values of S 2 sim x; y and S 3 sim x; y compare very well with their experimental counterparts, being in the range between −0.02 and 0.02 for both S 2 x; y and S 3 x; y. As a consequence, also the weak mode component I x exp x; y and I x sim x; y compare very well. By comparing the experimentally obtained results and the numerically simulated results generated by VELM code, we were able prove that, in spite the fact that we have a polarization controlling surface grating, a nonzero amount of (i) linear polarization S 2 x; y and (ii) circular polarization component S 3 x; y show up. The amount of S 2 x; y and S 3 x; y and their complex four-lobe and eight-lobe pattern in terms of change of sign, respectively, can be explained by the superposition of the two complex orthogonal linearly polarized electromagnetic field components, originating from the solution of Maxwell’s equations in our VCSEL structure [12]. Indeed, the VELM code applied to a small-area, surfacegrating VCSEL with a slightly oblong oxide aperture (whose major axis is perpendicular to the field brackets), excellently reproduces the bracket structure, the fourand eight-lobe pattern as well as their normalized values.
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1 I y x; y S 0 x; y − S 1 x; y: 2
