Intracavity diamond heatspreaders in lasers: the effects of birefringence

doi:10.1364/OE.14.009250

Intracavity diamond heatspreaders in lasers: the effects of birefringence

Francesco van Loon, Alan J. Kemp, Alexander J. Maclean, Stephane Calvez, John-Mark Hopkins, Jennifer E. Hastie, Martin D. Dawson, and David Burns »View Author Affiliations

Optics Express, Vol. 14, Issue 20, pp. 9250-9260 (2006)
http://dx.doi.org/10.1364/OE.14.009250

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Abstract

The birefringence of a number of commercially-available diamond platelets is assessed in the context of their use for intracavity thermal management in lasers. Although diamond is normally thought of as isotropic, significant birefringence is found to be present in some samples, with considerable variation from batch to batch, and in some cases across an individual sample. Nonetheless, low-loss operation is achieved in a laser cavity containing a Brewster element, either by rotating the sample or by using a diamond platelet with low birefringence.

1. Introduction

Efficient heat removal is a prerequisite for power scaling in most lasers; nowhere more so than in the high-specification finesse lasers required for advanced sensing, environmental monitoring and spectroscopy. Higher powers can enable higher scan rates and larger signal to noise ratios, but this should not come at the price of reduced functionality. For this reason, thermal management remains a crucial topic in advanced laser science.

One important recent development in this field has been the use of diamond to improve heat removal [1

R. L. Fork, W. W. Walker, R. L. Laycock, J. J. A. Green, and S. T. Cole, “Integrated diamond sapphire laser,” Opt. Express 11, 2532–2548 (2003). [CrossRef] [PubMed]

–3

Y. Tzuk, A. Tal, S. Goldring, Y. Glick, E. Lebiush, G. Kaufman, and R. Lavi, “Diamond cooling of highpower diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 40, 262–269 (2004). [CrossRef]

]. Used as an intracavity heatspreader [4

W. J. Alford, T. D. Raymond, and A. A. Allerman, “High power and good beam quality at 980 nm from a vertical external-cavity surface-emitting laser,” J. Opt. Soc. Am. B 19, 663–666 (2002). [CrossRef]

], this material has had a particularly strong impact on optically pumped vertical-external-cavity-surface emitting lasers (VECSELs [5

M. Kuznetsov, F. Hakimi, R. Sprague, and A. Mooradian, “Design and characteristics of high-power (> 0.5-W CW) diode- pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM₀₀ beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999). [CrossRef]

, 6

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 39, R74–R85 (2004).

], see Fig. 1), widening the wavelength coverage of these high brightness semiconductor lasers by enabling high power demonstrations at 675nm [7

J. E. Hastie, S. Calvez, M. D. Dawson, T. Leinonen, A. Laakso, J. Lyytikainen, and M. Pessa, “High power CW red VECSEL with linearly polarised TEM₀₀ output beam,” Opt. Express 13, 77–81 (2005). [CrossRef] [PubMed]

], 850nm, [8

J. E. Hastie, “High power surface emitting semiconductor lasers,” Ph.D. Thesis, University of Strathclyde, (2004).

, 9

J. E. Hastie, J. M. Hopkins, S. Calvez, C. W. Jeon, D. Burns, R. Abram, E. Riis, A. I. Ferguson, and M. D. Dawson, “0.5-W single transverse-mode operation of an 850-nm diode- pumped surface-emitting semiconductor laser,” IEEE Photon. Technol. Lett. 15, 894–896 (2003). [CrossRef]

] 980nm [10

J. E. Hastie, J. M. Hopkins, C. W. Jeon, S. Calvez, D. Burns, M. D. Dawson, R. Abram, E. Riis, A. I. Ferguson, W. J. Alford, T. D. Raymond, and A. A. Allerman, “Microchip vertical external cavity surface emitting lasers,” Electron. Lett. 39, 1324–1326 (2003). [CrossRef]

], 1060nm [11

K. S. Kim, J. R. Yoo, S. H. Cho, S. M. Lee, S. J. Lim, J. Y. Kim, J. H. Lee, T. Kim, and Y. J. Park, “1060 nm vertical-external-cavity surface-emitting lasers with an optical-to-optical efficiency of 44% at room temperature,” Appl. Phys. Lett. 88, 091107 (2006). [CrossRef]

], 1320nm [2

J. M. Hopkins, S. A. Smith, C. W. Jeon, H. D. Sun, D. Burns, S. Calvez, M. D. Dawson, T. Jouhti, and M. Pessa, “0.6 W CW GaInNAs vertical external-cavity surface emitting laser operating at 1.32 µm,” Electron. Lett. 40, 30–31 (2004). [CrossRef]

], 1550nm [12

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Sel. Top. Quantum Electron. 11, 1126–1134 (2005). [CrossRef]

] and 2300nm [13

J.-M. Hopkins, A. J. Maclean, D. Burns, N. Schulz, M. Rattunde, C. Manz, K. Koehler, and J. Wagner, “Tunable, Single-frequency, Diode-pumped 2.3µm VECSEL,” presented at Conference on Lasers and Electro-Optics, (Long Beach, 2006).

, 14

N. Schulz, M. Rattunde, C. Manz, K. Koehler, C. Wild, J. Wagner, S.-S. Beyertt, U. Brauch, T. Kuebler, and A. Giesen, “Optically Pumped GaSb-Based VECSEL Emitting 0.6 W at 2.3 µm,” IEEE Photonics Technol. Lett. 18, 1070–1072 (2006). [CrossRef]

]. The heatspreader facilitates this spectral coverage by removing heat directly from the gain region (consisting of quantum wells separated by pump absorbing barriers), bypassing any thermal impedance in the substrate or distributed Bragg reflector (DBR) [15

A. J. Kemp, G. J. Valentine, J. M. Hopkins, J. E. Hastie, S. A. Smith, S. Calvez, M. D. Dawson, and D. Burns, “Thermal management in vertical-external-cavity surface-emitting lasers: Finite-element analysis of a heatspreader approach,” IEEE J. Quantum Electron. 41, 148–155 (2005). [CrossRef]

, 12

Fig. 1. Schematic diagram of a VECSEL with an intracavity heatspreader; the inset shows the heat removal paths. QWs - quantum wells. (Not to scale)

The exceptional thermal conductivity of diamond (~2W/(mm.K) [16

D. Nikogosyan, Handbook of Properties of Optical Materials (John Wiley and Sons Ltd, London, 1997).

]) is key to its utility for thermal management in lasers; however, for intracavity use, its optical properties are also very important and diamond has proven to be sufficiently low loss to enable highly efficient laser operation with intracavity heatspreaders (see e.g. [11

]). The extended cavity implicit in the VECSEL format offers the potential to add new functionality by adding extra components — an opportunity not afforded by most semiconductor laser geometries. A number of these functions, for example nonlinear frequency conversion, have a built-in polarisation dependence and thus the birefringence of any intracavity components becomes important. Diamond is normally considered to be an isotropic material [16

D. Nikogosyan, Handbook of Properties of Optical Materials (John Wiley and Sons Ltd, London, 1997).

]; however, there have been reports of birefringence in certain circumstances (see e.g. [17

A. R. Lang, “Causes of Birefringence in Diamond,” Nature 213, 248–251 (1967). [CrossRef]

]). This paper describes the assessment of the birefringence in commercially-sourced diamond platelets used as heatspreaders in VECSELs and, more importantly, the the effect this birefringence has on laser performance. It begins by discussing qualitative, whole sample, polarisation micrographs of the diamond (section 2); quantitative, single point polarimetric measurements are outlined in section 3; this work is compared to in-laser assessments in section 4; and in section 5, Jones matrix analysis is used to assess the implications of any birefringence for laser operation.

2. Qualitative polarisation microscopy on diamond heatspreader samples

To provide a qualitative assessment of the birefringence of the samples, simple polarisation micrographs were taken using a conventional polarising microscope (SP-200-XM from Brunel Microscopes Ltd., UK) [18

M. Abramowitz, Reflected Light Microscopy: An Overview (Olympus America Inc., 1990).

]. The white light illumination source (a 20W halogen bulb) was polarised with a thin film polariser; the sample was then viewed in reflection through a x4 objective and a crossed polariser. The resulting micrographs for the samples discussed in this report are shown in Fig. 2 and Fig. 3. The diamond was purchased at different times in batches of around four or five pieces with the same nominal specification. Whilst the samples in batches 1, 3 and 4 are type IIa natural diamond (Fig. 2), those in batch 2 are single crystal synthetic diamond (Fig. 3) (the optical properties of similar synthetic diamond has also recently been investigated by Turri and co-workers [19

G. Turri, Y. Chen, M. Bass, D. Orchard, J. E. Butler, S. Magana, T. Feygelson, D. Thiel, K. Fourspring, S. Hawkins, M. Baronowski, R. V. Dewees, M. D. Seltzer, A. Guenthner, and D. C. Harris, “Optical Properties of Single-Crystal Chemical-Vapour-Deposited Diamond,” presented at 11th DoD Electromagnetic Windows Symposium, (San Diego, 2006).

].) The samples were ordered as 4mm diameter optical windows with thicknesses of either 250 or 500µm. The large faces had lasergrade polishes. No explicit specification was placed on the birefringence of the samples. Detailed specifications are provided in the appendix. The pieces making up a single batch were found to have similar properties and thus the testing of one representative sample per batch is described in this paper (two samples in the case of batch 2).

Fig. 2. Polarisation micrographs of type IIa natural diamond samples from batches 1, 3 and 4. (The sample from batch 1 is bonded to a piece of VECSEL wafer and mounted for laser operation. The visible aperture is 3mm. The others are free-standing and 4mm in diameter.)

Fig. 3. Polarisation micrographs of the synthetic diamond from batch 2. Both samples are free-standing and 4mm in diameter.

Changes in colour in a polarisation micrograph result from changes in the magnitude and orientation of any birefringence in the sample [18

M. Abramowitz, Reflected Light Microscopy: An Overview (Olympus America Inc., 1990).

]. (In our case, isotropic samples have a greenish hue because the polarisers are not perfect and their residual transmission when crossed peaks in the green.) Thus, whilst the samples from batches 3 and 4 appear relatively isotropic; qualitatively at least, considerable birefringence is observed in batches 1 and 2. (Note that the sample from batch 1 is pictured bonded to a VECSEL sample and mounted for laser operation; the others are pictured free-standing) The video in Fig. 4 shows the rotation of sample (a) from batch 2 as viewed through the polarising microscope. Variation of the colour pattern is observed indicating that there is both significant birefringence and significant variation in its magnitude and orientation across the sample. On rotating the sample from batch 1, the strong stripes visible in Fig. 2 disappeared, suggesting associated birefringence. By contrast, no significant variation of the colour pattern was observed on rotation of the samples from batches 3 and 4, indicating that any birefringence is small rather than merely uniform.

Fig. 4. (1.79Mb) Video of rotation of sample (a) from batch 2

While the synthetic diamond gives the most visible signature under the polarising microscope, it is interesting to note that there is considerable variation between the natural diamond samples (batches 1, 3 and 4 in Fig. 2). Micrographs of other type IIa natural diamond samples indicated significant variation in birefringence between, and to much a lesser extent within, batches.

Given the qualitative evidence of birefringence in diamond heatspreader samples, and the batch to batch variation, it is vital to get a quantitative measure of the birefringence and, more importantly, to characterize its effect on laser performance. Such experiments are described in sections 3 and 4.

3. Quantitative polarimetry on diamond heatspreader samples

3.1 Estimation of birefringence using polarimetry

To quantify the birefringence, a simple set of polarimetric measurements were made. These measurements were taken both in transmission (bare diamond) and in reflection (diamond bonded to the VECSEL sample; bare VECSEL sample). The reflection set up is shown in Fig. 5. The input polarisation was linear and horizontal (in the plane of the diagram); it was then rotated using a half-waveplate. At each half-waveplate angle, the output polarisation was analyzed using a cube polariser: the maximum and minimum power transmitted through the polariser, and the polariser angles at which these occurred were recorded. Such measurements of the sample from batch one, liquid-capillary bonded [4

, 20

Z. L. Liau, “Semiconductor wafer bonding via liquid capillarity,” Appl. Phys. Lett. 77, 651–653 (2000). [CrossRef]

] to a 1060nm VECSEL sample (for full details see [11

]) and mounted for laser operation, are shown in Fig. 6. These measurements were taken in reflection.

The birefringence manifests itself as the dips and peaks in Fig. 6(a) and in the s-shapes in Fig. 6(b). An isotropic sample would give horizontal lines at one and zero in former case — the polarisation remains linear so normalised maxima and minima of 1 and 0 are expected — and straight lines of gradient -2 in the latter (following the sign convention indicated in Fig. (5). (Note that the angle of the polarisation after the half waveplate is twice the angle to which the half wavplate is set).

Fig. 5. Polarimetry set-up

The magnitude and orientation of the birefringence in the sample is quantified using a Jones matrix model (see e.g. Ref. [21

A. Yariv, “Jones Calculus and its Applications to Propagation in Optical Systems with Birefringent Crystals,” in Optical Electronics, (Saunders College Publishing, 1991), pp. 16–29.

]). The phase retardation between components polarised along the axes of a birefringent element is given by:

δ = \frac{2 \cdot π \cdot t \cdot Δ n}{λ}

(1)

where t is the plate thickness, Δn is the difference between the reflective indices along the two axes (the birefringence) and λ is the free space wavelength. The Jones matrix representing the birefringent element is then:

M (δ) = [\begin{matrix} e^{\frac{i \cdot δ}{2}} & 0 \\ 0 & e^{\frac{- i \cdot δ}{2}} \end{matrix}]

(2)

(the overall phase factor is omitted since it is not relevant in these calculations). If this matrix is combined with those for the half-waveplate and polariser, then a fit to the experimental data can be got by varying δ. (The matrix M appears once in transmission measurements; in reflection it appears twice. Also, in a reflection-based experiment, the fraction of the incident light reflected off the front surface of the sample must be factored in.) Adding 2.n.π (where n is a natural number) to the value of δ does not materially change M and hence the predicted polarisation at the output. Thus, this measurement does not return Δn, nor δ, but δ modulo 2π. (For the purposes of this paper, this quantity will be referred to as the ‘waveplate retardation’ and expressed as a fraction of 2.π.) That is to say this methodology cannot determine the order of the waveplate that the sample represents. Thus, an isotropic sample would have a waveplate retardation of zero; whilst that of a full waveplate would be 1. This analysis implicitly assumes that the overall transmission of the sample is not dependent on input polarisation. This is not necessarily the case where the birefringence is strong since the sample is effectively an etalon and any change in the sampled refractive index (due to a change in the input polarisation) will change its transmission. In most cases this was not found to be a significant effect; but, where necessary, the analysis was adjusted to accommodate for this.

There is, however, a further ambiguity: for an arbitrary waveplate measured in transmission, two values of the waveplate retardation will fit the experimental data equally well; in reflection this rises to four values. For consistency, the lowest value that fits the data is quoted throughout this paper. This is not, however, a problem for the laser-based calculations (see section 5), since each of these values will return the same answer in such calculations and will affect the laser operation in the same way (for a narrow laser bandwidth at least).

By fitting to the data in Fig. 6, a waveplate retardation of about 0.19 is calculated for the diamond sample from batch 1 bonded to the VECSEL sample. In Fig. 6(a), the zeros are at 0° and 45° (half-waveplate angle, incident polarization angles of 0° and 90°), hence the axes of the waveplate that the sample represents, and so the axes its birefringence, are parallel and perpendicular to the horizontal input polarisation. (Given that the composite had been rotated in a laser arrangement to minimize loss at a Brewster surface, this is to be expected). The angle between the axes of the waveplate that the diamond represents and the input polarisation will be referred to as the waveplate orientation; in this case that angle is 0°.

Fig. 6. Polarimetry measurements taken in reflection on the diamond sample from batch 1 bonded to a VECSEL sample: (a) variation in the maximum and minimum power transmitted through the polariser, (b) variation in the angle at which maximum and minimum transmission are observed; both with half-waveplate angle. (N.B. the angle of the polarisation incident on the sample is twice the angle of the half-waveplate.)

As the experimental results in section 4 and the theoretical predictions in section 5, will illustrate, this is a significant waveplate retardation in the context of a laser cavity containing polarisation selective elements. Thus, it will be important to confirm its origin — is it intrinsic to the diamond, or a result of the bonding and mounting process? — and to quantify the variability indicated by the polarisation micrographs in section 2.

3.2 Origin of the birefringence

The polarisation microscopy discussed in section two suggests that the birefringence observed in the bonded composite is intrinsic to the diamond. To confirm this, a series of polarimetric measurement were undertaken. Measurements at four points on the bare VECSEL sample put an upper limit on its waveplate retardation of 0.02; confirming that was not the origin of the birefringence. Two diamond samples were then measured in transmission (the sample from batch 3 and sample (a) from batch 2). A series of point measurements were undertaken along a line across the width of the diamond sample. These samples were then bonded to identical VECSEL samples and mounted for laser operation in a brass mount with indium foil at the interfaces. The characteristics of the measured birefringence were the same before and after bonding and mounting: a birefringence of <0.02 was measured before and after bonding and mounting for the diamond from batch 3, while significant and variable birefringence was measured for sample (a) from batch 2 in both cases (the measurements of the free-standing diamond are shown in Fig. 7). Thus, one can conclude that the observed birefringence is a property of the diamond sample and does not arise, for example, from stresses induced by the bonding and mounting process.

3.3 Batch to batch variability

The polarisation microscopy indicates that the birefringence varies considerably between batches of diamond samples. Polarimetric measurements were used to quantify this variation. The diamond samples were measured in transmission (with the exception of the sample from batch one which was bonded and mounted and hence measured in reflection). The samples were translated in a direction perpendicular to the probe beam to take a series of measurements. The probe laser spot radius on the sample was about 40µm, similar to the pump and laser mode sizes in the laser experiments.

The measurements in Fig. 7 indicate that for batch 2 (the synthetic diamond samples) there is both significant birefringence and significant variation in the magnitude and orientation of that birefringence. Indeed, when a measurement was made of a ‘hot-spot’ identified on the polarisation micrograph in Fig. 3, an even higher waveplate retardation of 0.16 was recorded. Some of the type IIa natural diamond samples show considerably smaller birefringence and no measureable variation in its orientation; for the sample from batch 3 in particular, the birefringence is equal to or less than the minimum sensitivity level of the apparatus (a waveplate retardation of <0.02: black dotted line in Fig. 7(a). This was assessed by running the experiment with no sample present). The sample from batch 4 shows some evidence of slightly larger birefringence, but this is not conclusive. The sample from batch 1 has a variable birefringence, presumably related to the stripes observed in the polarisation microscopy. (A larger value of 0.19 was measure for a different position on this sample, see section 3.1.)

The orientation of the birefringence is difficult to measure for waveplate retardations <0.03; thus, although it is clear that there is significant variation in the orientation of the waveplating in the synthetic diamond samples (batch 2), there is no such evidence for the type IIa natural diamond samples (batches 1, 3 and 4).

The birefringence varies considerably from batch to batch. This would be of concern, for example, in volume production of a laser with an intracavity polarization sensitive component, say a birefringent filter for tuning or a nonlinear crystal for frequency conversion. However, it is clear that lower and more uniform birefringence can be achieved (batches 3 and 4 for example). This material will be more appropriate for laser operation (see section 4).

Fig. 7. Variation in the waveplate retardation (a) and its orientation (b) on a line across the width of various diamond samples. The black dotted line part a represents an estimate of the minimum measureable waveplate retardation.

4. In-laser assessment of diamond heatspreader samples

Having measured the birefringence in the various samples, it is necessary to relate it to the laser performance. To do this, a series of measurements were taken using a cavity configuration similar to that in Fig. 1. First, a bare VECSEL sample was used; the polarisation was found to be linear and horizontal for all positions on the sample. Next, the sample was bonded in turn to various diamond samples: batch 2 sample a; the sample from batch 3 (this wedged sample was AR coated on the outer surface before the laser tests); and the sample from batch 4. A series of measurements were undertaken with a glass Brewster plate inserted into the cavity; the output power and Brewster loss (the fraction of the intracavity power reflected out of the cavity by the surfaces of the Brewster plate) were assessed as a function of position on the sample (Fig. 8). A 9% output coupling was found to be optimal for the plane parallel samples from batches 2 and 4; 5% was optimal for the wedged and AR coated sample from batch 3. This is to be expected given the lower effective gain in the latter case due to the suppression of the Fabry-Perot resonance between the DBR and the front of the diamond — where there is a significant resonance, the effective gain is higher because the laser field is enhanced in the vicinity of the gain region. No attempt was made to optimize the rotational orientation of the VECSEL diamond composite. The variation in the output power without the Brewster plate is plotted in Fig. 9 for comparison.

Fig. 8. The output power (a) and Brewster loss per surface (b) in a laser containing a glass Brewster plate as a function of position for various diamond heatspreaders

Fig. 9. The output power as a function of position with no Brewster plate present

For the synthetic diamond sample (a) from batch 2, the strong and variable birefringence observed in the polarisation microscopy and polarimetry measurements is reflected in the large and variable Brewster loss. In some cases, the birefringence is such that the laser ceases to operate; in others, the Brewster loss is significantly increased and the output power consequently reduced. By contrast, where the diamond has been seen to have low birefringence in the previous measurements — the sample from batch 3 — very low Brewster losses, <0.01% per surface, are observed. The sample from batch 4 gives a performance level between the two extremes, in keeping with the earlier assessments of its birefringence — the variability in output power being as much due to the poorer surface roughness of this sample. (N.B. The absolute position of a measurement on the sample surface is not consistent between measurements sets; hence, the position (x) axes in Fig. 8 and Fig. 9 do not read directly across to the position axis in Fig. 7. The reduction in output power at around 2mm observed in Fig. 8 is thought to result from a small crack in the surface of the VECSEL sample.)

More important than the magnitude of the birefringence in a laser context is its orientation. If a diamond sample has strong birefringence, it should nevertheless be possible to achieve low loss operation in a laser containing a Brewster plate provided the axes of the birefringence and those of the Brewster plate are aligned. Practically, this will be difficult if the orientation of the birefringence is strongly dependent on position (for example in the synthetic samples from batch 2). However, the sample from batch 1 has significant birefringence which the polarisation micrographs and polarimetry measurements suggest is orientated in the same direction across the sample, although the magnitude varies somewhat (see Figs. 2 and 7). Thus, an experiment was undertaken whereby the VECSEL-diamond composite was rotated around the cavity axis of a laser containing a Brewster plate.

As Fig. 10 illustrates, it is possible to reduce the Brewster loss to less than 0.06% per surface provided the angle is set to with ±2°. Outside this range, the Brewster loss quickly increases, with a 2% round-trip loss (0.5% per surface) resulting from an angular misorientation of around 10°. The minimum Brewster loss achieved was 0.04% per surface; this is comparable to that achieved with the low birefringence diamond sample from batch 3 (<0.01% per surface without rotational correction). Also shown on Fig. 10 is the theoretically predicted loss based on the experimental measurement of the waveplate retardation of the VECSEL-diamond composite summarized in Fig. 6 (the loss was calculated from the eigenvalues of the round trip Jones matrix for the cavity [22

T. Y. Fan, “Single-Axial Mode, Intracavity Doubled Nd:YAG Laser,” IEEE J. Quantum Electron. 27, 2091–2093 (1991). [CrossRef]

]; a small offset has been added to the theoretical curve to account for the non-zero measured loss at zero angular misorientation.). The good correspondence between experiment and the theoretical predictions confirms the potential of birefringence in diamond to cause significant intracavity loss and also the potential for that loss to be eliminated with sufficiently accurate alignment. (Experiment and theory are in poorer agreement on the right of Fig. 10 because the centre wavelength of the laser changed to reduce the loss.) However, where the orientation of the birefringence varies considerably with position — the synthetic samples from batch 2 for example — such an optimization would be considerably more problematic: unless the axis of rotation is very precisely aligned with the cavity axis, the position of the cavity mode on the diamond, and hence the orientation of the birefringence experienced, will vary as the composite is rotated.

Fig. 10. Percentage Brewster loss per surface (four surfaces per round trip) on rotation of the VECSEL-diamond composite around the cavity axis (diamond sample from batch one)

5. Modelling and discussion of the effects of birefringence on laser performance

To assist in assessing the significance of the observed birefringence, some simple modelling was undertaken. Via the eigenvalues of the round-trip Jones matrix for the laser cavity, the round-trip loss was estimated as a function of both the angular misorientation of the axes of the birefringence in the VECSEL-diamond composite from the plane of incidence of the Brewster plate and the magnitude of its waveplate retardation. The round trip Jones Matrix is:

M_{RT} = R (- φ) \cdot M (δ) \cdot R (φ) \cdot B \cdot B \cdot R (- φ) \cdot M (δ) \cdot R (φ)

(4)

where and R(φ) is the matrix representing a rotation through an angle φ (the misorientation of the axes of the birefringence):

R (φ) = [\begin{matrix} \cos (φ) & - \sin (φ) \\ \sin (φ) & \cos (φ) \end{matrix}]

(5)

B is the matrix representing a Brewster plate with a transmission T in for the high loss polarisation:

B = [\begin{matrix} 1 & 0 \\ 0 & T \end{matrix}]

(6)

Figure 11(a) indicates that significant round trip loss can result from a relatively small angular misorientation of the axes of the birefringence for a waveplate retardation equal to that measured experimentally in Fig. 6 (diamond from batch 1). Hence, for many diamond samples, it will be necessary to align the axes of the birefringence to any intracavity polarisation selective element to within a few degrees.

Fig. 11. Percentage round trip loss resulting from VECSEL-diamond composite birefringence as a function of: (a) angular misalignment of the gain element from the Brewster plate for a fractional waveplate retardation of 0.19 (that for the sample from batch 1); (b) the fractional waveplate retardation for four angular misorientations (0° solid red line; 5° dotted blue line; 10° dashed green line; 15° dot-dash brown line).

Figure 11(b) illustrates the sort of waveplate retardations that will cause significant problems in a laser context. For waveplate retardations between about 0.1 and 0.4 (these are single pass retardations — a double pass occurs on a round trip and hence the loss returns to zero at 0.5) — the loss is essentially constant and mainly a function of angle. The waveplate retardation needs to be below about 0.01 to avoid significant additional loss (>0.5% per round trip) at reasonable angular misorientations (15° or less). For a 0.5mm thick diamond sample and operation at 1064nm, this is a birefringence of ~2×10^-6. For the worst case scenario of a 45° misorientation, the waveplate retardation would need to be <0.005 to keep the round trip loss below 0.5%; a birefringence of <1×10^-6. The results presented in section 4 demonstrate that that this is achievable for some natural diamond samples at least.

6. Conclusion

The work presented in this paper indicates that birefringence has the potential to be a significant issue for lasers using diamond for intracavity thermal management, especially if polarization selective elements are present. The issue is not so much the existence or magnitude of this birefringence, but the variation of its orientation across a single sample and the sample to sample variability. However, diamond need not have significant birefringence — it is in principle isotropic — hence, the most important implication of this work is the need to specify the maximum birefringence of the diamond as part of the procurement process. In addition, where birefringence is present, but has uniform orientation, it is possible to eliminate the Brewster loss by rotating the sample. For volume manufacture of lasers, the more predictable birefringence properties and lower cost of silicon carbide, its availability in large diameter wafer format and its high thermal conductivity, make it an attractive alternative to diamond.

Whilst birefringence in diamond is potentially problematic for intracavity heatspreader applications, the work in this paper demonstrates that with appropriate procurement and testing, diamond heatspreaders need not have a detrimental effect on the laser performance; indeed, the unique thermal management properties of diamond are likely to see it grow in importance for power-scaling high brightness doped-dielectric and semiconductor lasers.

Appendices

Appendix: sample specifications

Table 1 provides a summary of the manufacturer’s specifications for the diamond samples bought for this work. (No birefringence specification was provided when they were purchased.)

Table 1: Manufacture’s specifications of the diamond used in this work

	Batch 1	Batch 2	Batch 3	Batch 4
Diamond type	IIa	Synthetic	IIa	IIa
Thickness (mm)	0.25	0.5	0.5	0.5
Diameter (mm)	4	4	4	4
Flatness (fringes at 632.8nm)	<0.25	<0.25	<0.25	3-6
Roughness (nm Ra measured on 1mm²)	<5	<5	<5	<5
Parallelism (µm/mm)	<0.33	<0.33	2° wedge	<2

All diamond was purchased from a single commercial supplier who also undertook the polishing to the specifications listed in table 1. The samples were purchased as ‘optical windows’. The various batches were purchased at different times, with each batch typically consisting of around five pieces. The relationship between the pieces within a batch is unknown, but their similar birefringence properties mean it is possible they were cut from the same ‘parent’ diamond.

Type IIa natural diamond contains very low levels of nitrogen impurities and is thus well suited for optical applications (for further information on natural diamond classification see e.g. Ref. [23

Element 6, “Diamond Types,” (Element 6 Ltd., 2006), http://www.e6.com/e6/page.jsp?pageid=400602030

]). The synthetic samples are single crystal diamond grown by chemical vapour deposition (designated type IIIa by the supplier).

Acknowledgments

The authors would like to thank Jun-Youn Kim, Kisung Kim and Taek Kim (Samsung Advanced Institute of Technology, Korea) for providing the VECSEL material used in this work. AJK and JEH gratefully acknowledge support from the Royal Society of Edinburgh and the Royal Academy of Engineering respectively in the form of personal research fellowships.

References and links

1.	R. L. Fork, W. W. Walker, R. L. Laycock, J. J. A. Green, and S. T. Cole, “Integrated diamond sapphire laser,” Opt. Express 11, 2532–2548 (2003). [CrossRef] [PubMed]
2.	J. M. Hopkins, S. A. Smith, C. W. Jeon, H. D. Sun, D. Burns, S. Calvez, M. D. Dawson, T. Jouhti, and M. Pessa, “0.6 W CW GaInNAs vertical external-cavity surface emitting laser operating at 1.32 µm,” Electron. Lett. 40, 30–31 (2004). [CrossRef]
3.	Y. Tzuk, A. Tal, S. Goldring, Y. Glick, E. Lebiush, G. Kaufman, and R. Lavi, “Diamond cooling of highpower diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 40, 262–269 (2004). [CrossRef]
4.	W. J. Alford, T. D. Raymond, and A. A. Allerman, “High power and good beam quality at 980 nm from a vertical external-cavity surface-emitting laser,” J. Opt. Soc. Am. B 19, 663–666 (2002). [CrossRef]
5.	M. Kuznetsov, F. Hakimi, R. Sprague, and A. Mooradian, “Design and characteristics of high-power (> 0.5-W CW) diode- pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM₀₀ beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999). [CrossRef]
6.	A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D 39, R74–R85 (2004).
7.	J. E. Hastie, S. Calvez, M. D. Dawson, T. Leinonen, A. Laakso, J. Lyytikainen, and M. Pessa, “High power CW red VECSEL with linearly polarised TEM₀₀ output beam,” Opt. Express 13, 77–81 (2005). [CrossRef] [PubMed]
8.	J. E. Hastie, “High power surface emitting semiconductor lasers,” Ph.D. Thesis, University of Strathclyde, (2004).
9.	J. E. Hastie, J. M. Hopkins, S. Calvez, C. W. Jeon, D. Burns, R. Abram, E. Riis, A. I. Ferguson, and M. D. Dawson, “0.5-W single transverse-mode operation of an 850-nm diode- pumped surface-emitting semiconductor laser,” IEEE Photon. Technol. Lett. 15, 894–896 (2003). [CrossRef]
10.	J. E. Hastie, J. M. Hopkins, C. W. Jeon, S. Calvez, D. Burns, M. D. Dawson, R. Abram, E. Riis, A. I. Ferguson, W. J. Alford, T. D. Raymond, and A. A. Allerman, “Microchip vertical external cavity surface emitting lasers,” Electron. Lett. 39, 1324–1326 (2003). [CrossRef]
11.	K. S. Kim, J. R. Yoo, S. H. Cho, S. M. Lee, S. J. Lim, J. Y. Kim, J. H. Lee, T. Kim, and Y. J. Park, “1060 nm vertical-external-cavity surface-emitting lasers with an optical-to-optical efficiency of 44% at room temperature,” Appl. Phys. Lett. 88, 091107 (2006). [CrossRef]
12.	H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, “Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers,” IEEE J. Sel. Top. Quantum Electron. 11, 1126–1134 (2005). [CrossRef]
13.	J.-M. Hopkins, A. J. Maclean, D. Burns, N. Schulz, M. Rattunde, C. Manz, K. Koehler, and J. Wagner, “Tunable, Single-frequency, Diode-pumped 2.3µm VECSEL,” presented at Conference on Lasers and Electro-Optics, (Long Beach, 2006).
14.	N. Schulz, M. Rattunde, C. Manz, K. Koehler, C. Wild, J. Wagner, S.-S. Beyertt, U. Brauch, T. Kuebler, and A. Giesen, “Optically Pumped GaSb-Based VECSEL Emitting 0.6 W at 2.3 µm,” IEEE Photonics Technol. Lett. 18, 1070–1072 (2006). [CrossRef]
15.	A. J. Kemp, G. J. Valentine, J. M. Hopkins, J. E. Hastie, S. A. Smith, S. Calvez, M. D. Dawson, and D. Burns, “Thermal management in vertical-external-cavity surface-emitting lasers: Finite-element analysis of a heatspreader approach,” IEEE J. Quantum Electron. 41, 148–155 (2005). [CrossRef]
16.	D. Nikogosyan, Handbook of Properties of Optical Materials (John Wiley and Sons Ltd, London, 1997).
17.	A. R. Lang, “Causes of Birefringence in Diamond,” Nature 213, 248–251 (1967). [CrossRef]
18.	M. Abramowitz, Reflected Light Microscopy: An Overview (Olympus America Inc., 1990).
19.	G. Turri, Y. Chen, M. Bass, D. Orchard, J. E. Butler, S. Magana, T. Feygelson, D. Thiel, K. Fourspring, S. Hawkins, M. Baronowski, R. V. Dewees, M. D. Seltzer, A. Guenthner, and D. C. Harris, “Optical Properties of Single-Crystal Chemical-Vapour-Deposited Diamond,” presented at 11th DoD Electromagnetic Windows Symposium, (San Diego, 2006).
20.	Z. L. Liau, “Semiconductor wafer bonding via liquid capillarity,” Appl. Phys. Lett. 77, 651–653 (2000). [CrossRef]
21.	A. Yariv, “Jones Calculus and its Applications to Propagation in Optical Systems with Birefringent Crystals,” in Optical Electronics, (Saunders College Publishing, 1991), pp. 16–29.
22.	T. Y. Fan, “Single-Axial Mode, Intracavity Doubled Nd:YAG Laser,” IEEE J. Quantum Electron. 27, 2091–2093 (1991). [CrossRef]
23.	Element 6, “Diamond Types,” (Element 6 Ltd., 2006), http://www.e6.com/e6/page.jsp?pageid=400602030

OCIS Codes
(140.6810) Lasers and laser optics : Thermal effects
(160.6840) Materials : Thermo-optical materials

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: July 27, 2006
Revised Manuscript: September 1, 2006
Manuscript Accepted: September 21, 2006
Published: October 2, 2006

Citation
Francesco van Loon, Alan J. Kemp, Alexander J. Maclean, Stephane Calvez, John-Mark Hopkins, Jennifer E. Hastie, Martin D. Dawson, and David Burns, "Intracavity diamond heatspreaders in lasers: the effects of birefringence," Opt. Express 14, 9250-9260 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9250

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References

R. L. Fork, W. W. Walker, R. L. Laycock, J. J. A. Green, and S. T. Cole, "Integrated diamond sapphire laser," Opt. Express 11, 2532-2548 (2003). [CrossRef] [PubMed]
J. M. Hopkins, S. A. Smith, C. W. Jeon, H. D. Sun, D. Burns, S. Calvez, M. D. Dawson, T. Jouhti, and M. Pessa, "0.6 W CW GaInNAs vertical external-cavity surface emitting laser operating at 1.32 μm," Electron. Lett. 40, 30-31 (2004). [CrossRef]
Y. Tzuk, A. Tal, S. Goldring, Y. Glick, E. Lebiush, G. Kaufman, and R. Lavi, "Diamond cooling of high-power diode-pumped solid-state lasers," IEEE J. Quantum Electron. 40, 262-269 (2004). [CrossRef]
W. J. Alford, T. D. Raymond, and A. A. Allerman, "High power and good beam quality at 980 nm from a vertical external-cavity surface-emitting laser," J. Opt. Soc. Am. B 19, 663-666 (2002). [CrossRef]
M. Kuznetsov, F. Hakimi, R. Sprague, and A. Mooradian, "Design and characteristics of high-power (> 0.5-W CW) diode- pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams," IEEE J. Sel. Top. Quantum Electron. 5, 561-573 (1999). [CrossRef]
A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, "Vertical-external-cavity semiconductor lasers," J. Phys. D 39,R74-R85 (2004).
J. E. Hastie, S. Calvez, M. D. Dawson, T. Leinonen, A. Laakso, J. Lyytikainen, and M. Pessa, "High power CW red VECSEL with linearly polarised TEM00 output beam," Opt. Express 13, 77-81 (2005). [CrossRef] [PubMed]
J. E. Hastie, "High power surface emitting semiconductor lasers," Ph.D. Thesis, University of Strathclyde, (2004).
J. E. Hastie, J. M. Hopkins, S. Calvez, C. W. Jeon, D. Burns, R. Abram, E. Riis, A. I. Ferguson, and M. D. Dawson, "0.5-W single transverse-mode operation of an 850-nm diode- pumped surface-emitting semiconductor laser," IEEE Photon. Technol. Lett. 15, 894-896 (2003). [CrossRef]
J. E. Hastie, J. M. Hopkins, C. W. Jeon, S. Calvez, D. Burns, M. D. Dawson, R. Abram, E. Riis, A. I. Ferguson, W. J. Alford, T. D. Raymond, and A. A. Allerman, "Microchip vertical external cavity surface emitting lasers," Electron. Lett. 39, 1324-1326 (2003). [CrossRef]
K. S. Kim, J. R. Yoo, S. H. Cho, S. M. Lee, S. J. Lim, J. Y. Kim, J. H. Lee, T. Kim, and Y. J. Park, "1060 nm vertical-external-cavity surface-emitting lasers with an optical-to-optical efficiency of 44% at room temperature," Appl. Phys. Lett. 88, 091107 (2006). [CrossRef]
H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, "Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers," IEEE J. Sel. Top. Quantum Electron. 11, 1126-1134 (2005). [CrossRef]
J.-M. Hopkins, A. J. Maclean, D. Burns, N. Schulz, M. Rattunde, C. Manz, K. Koehler, and J. Wagner, "Tunable, Single-frequency, Diode-pumped 2.3μm VECSEL," presented at Conference on Lasers and Electro-Optics, (Long Beach, 2006).
N. Schulz, M. Rattunde, C. Manz, K. Koehler, C. Wild, J. Wagner, S.-S. Beyertt, U. Brauch, T. Kuebler, and A. Giesen, "Optically Pumped GaSb-Based VECSEL Emitting 0.6 W at 2.3 μm," IEEE Photonics Technol. Lett. 18, 1070-1072 (2006). [CrossRef]
A. J. Kemp, G. J. Valentine, J. M. Hopkins, J. E. Hastie, S. A. Smith, S. Calvez, M. D. Dawson, and D. Burns, "Thermal management in vertical-external-cavity surface-emitting lasers: Finite-element analysis of a heatspreader approach," IEEE J. Quantum Electron. 41, 148-155 (2005). [CrossRef]
D. Nikogosyan, Handbook of Properties of Optical Materials (John Wiley and Sons Ltd, London, 1997).
A. R. Lang, "Causes of Birefringence in Diamond," Nature 213, 248-251 (1967). [CrossRef]
M. Abramowitz, Reflected Light Microscopy: An Overview (Olympus America Inc., 1990).
G. Turri, Y. Chen, M. Bass, D. Orchard, J. E. Butler, S. Magana, T. Feygelson, D. Thiel, K. Fourspring, S. Hawkins, M. Baronowski, R. V. Dewees, M. D. Seltzer, A. Guenthner, and D. C. Harris, "Optical Properties of Single-Crystal Chemical-Vapour-Deposited Diamond," presented at 11th DoD Electromagnetic Windows Symposium, (San Diego, 2006).
Z. L. Liau, "Semiconductor wafer bonding via liquid capillarity," Appl. Phys. Lett. 77, 651-653 (2000). [CrossRef]
A. Yariv, "Jones Calculus and its Applications to Propagation in Optical Systems with Birefringent Crystals," in Optical Electronics, (Saunders College Publishing, 1991), pp. 16-29.
T. Y. Fan, "Single-Axial Mode, Intracavity Doubled Nd:YAG Laser," IEEE J. Quantum Electron. 27, 2091-2093 (1991). [CrossRef]
Element 6, "Diamond Types," (Element 6 Ltd., 2006), http://www.e6.com/e6/page.jsp?pageid=400602030.

Abram, R.
Alford, W. J.
Allerman, A. A.
Bengtsson, J.
Beyertt, S.-S.
Brauch, U.
Burns, D.
Calvez, S.
Cho, S. H.
Cole, S. T.
Dawson, M. D.
Fan, T. Y.
Ferguson, A. I.
Foreman, H. D.
Fork, R. L.
Garnache, A.
Gerster, E.
Giesen, A.
Glick, Y.
Goldring, S.
Green, J. J. A.
Hakimi, F.
Hastie, J. E.
Hoogland, S. H.
Hopkins, J. M.
Jeon, C. W.
Jouhti, T.
Kaufman, G.
Kemp, A. J.
Kim, J. Y.
Kim, K. S.
Kim, T.
Koehler, K.
Kuebler, T.
Kuznetsov, M.
Laakso, A.
Lang, A. R.
Larsson, A.
Lavi, R.
Laycock, R. L.
Lebiush, E.
Lee, J. H.
Lee, S. M.
Leinonen, T.
Liau, Z. L.
Lim, S. J.
Lindberg, H.
Lyytikainen, J.
Manz, C.
Mooradian, A.
Park, Y. J.
Pessa, M.
Rattunde, M.
Raymond, T. D.
Riis, E.
Schulz, N.
Smith, S. A.
Sprague, R.
Strassner, M.
Sun, H. D.
Tal, A.
Tropper, A. C.
Tzuk, Y.
Valentine, G. J.
Wagner, J.
Walker, W. W.
Wilcox, K. G.
Wild, C.
Yoo, J. R.

Appl. Phys. Lett.

K. S. Kim, J. R. Yoo, S. H. Cho, S. M. Lee, S. J. Lim, J. Y. Kim, J. H. Lee, T. Kim, and Y. J. Park, "1060 nm vertical-external-cavity surface-emitting lasers with an optical-to-optical efficiency of 44% at room temperature," Appl. Phys. Lett. 88, 091107 (2006). [CrossRef]
Z. L. Liau, "Semiconductor wafer bonding via liquid capillarity," Appl. Phys. Lett. 77, 651-653 (2000). [CrossRef]

Electron. Lett.

J. E. Hastie, J. M. Hopkins, C. W. Jeon, S. Calvez, D. Burns, M. D. Dawson, R. Abram, E. Riis, A. I. Ferguson, W. J. Alford, T. D. Raymond, and A. A. Allerman, "Microchip vertical external cavity surface emitting lasers," Electron. Lett. 39, 1324-1326 (2003). [CrossRef]
J. M. Hopkins, S. A. Smith, C. W. Jeon, H. D. Sun, D. Burns, S. Calvez, M. D. Dawson, T. Jouhti, and M. Pessa, "0.6 W CW GaInNAs vertical external-cavity surface emitting laser operating at 1.32 μm," Electron. Lett. 40, 30-31 (2004). [CrossRef]

IEEE J. Quantum Electron.

Y. Tzuk, A. Tal, S. Goldring, Y. Glick, E. Lebiush, G. Kaufman, and R. Lavi, "Diamond cooling of high-power diode-pumped solid-state lasers," IEEE J. Quantum Electron. 40, 262-269 (2004). [CrossRef]
A. J. Kemp, G. J. Valentine, J. M. Hopkins, J. E. Hastie, S. A. Smith, S. Calvez, M. D. Dawson, and D. Burns, "Thermal management in vertical-external-cavity surface-emitting lasers: Finite-element analysis of a heatspreader approach," IEEE J. Quantum Electron. 41, 148-155 (2005). [CrossRef]
T. Y. Fan, "Single-Axial Mode, Intracavity Doubled Nd:YAG Laser," IEEE J. Quantum Electron. 27, 2091-2093 (1991). [CrossRef]

IEEE J. Sel. Top. Quantum Electron.

H. Lindberg, M. Strassner, E. Gerster, J. Bengtsson, and A. Larsson, "Thermal management of optically pumped long-wavelength InP-based semiconductor disk lasers," IEEE J. Sel. Top. Quantum Electron. 11, 1126-1134 (2005). [CrossRef]
M. Kuznetsov, F. Hakimi, R. Sprague, and A. Mooradian, "Design and characteristics of high-power (> 0.5-W CW) diode- pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams," IEEE J. Sel. Top. Quantum Electron. 5, 561-573 (1999). [CrossRef]

IEEE Photon. Technol. Lett.

J. E. Hastie, J. M. Hopkins, S. Calvez, C. W. Jeon, D. Burns, R. Abram, E. Riis, A. I. Ferguson, and M. D. Dawson, "0.5-W single transverse-mode operation of an 850-nm diode- pumped surface-emitting semiconductor laser," IEEE Photon. Technol. Lett. 15, 894-896 (2003). [CrossRef]

IEEE Photonics Technol. Lett.

N. Schulz, M. Rattunde, C. Manz, K. Koehler, C. Wild, J. Wagner, S.-S. Beyertt, U. Brauch, T. Kuebler, and A. Giesen, "Optically Pumped GaSb-Based VECSEL Emitting 0.6 W at 2.3 μm," IEEE Photonics Technol. Lett. 18, 1070-1072 (2006). [CrossRef]

J. Opt. Soc. Am. B

W. J. Alford, T. D. Raymond, and A. A. Allerman, "High power and good beam quality at 980 nm from a vertical external-cavity surface-emitting laser," J. Opt. Soc. Am. B 19, 663-666 (2002). [CrossRef]

J. Phys. D

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, "Vertical-external-cavity semiconductor lasers," J. Phys. D 39,R74-R85 (2004).

Nature

A. R. Lang, "Causes of Birefringence in Diamond," Nature 213, 248-251 (1967). [CrossRef]

Opt. Express

Other

J. E. Hastie, "High power surface emitting semiconductor lasers," Ph.D. Thesis, University of Strathclyde, (2004).
M. Abramowitz, Reflected Light Microscopy: An Overview (Olympus America Inc., 1990).
G. Turri, Y. Chen, M. Bass, D. Orchard, J. E. Butler, S. Magana, T. Feygelson, D. Thiel, K. Fourspring, S. Hawkins, M. Baronowski, R. V. Dewees, M. D. Seltzer, A. Guenthner, and D. C. Harris, "Optical Properties of Single-Crystal Chemical-Vapour-Deposited Diamond," presented at 11th DoD Electromagnetic Windows Symposium, (San Diego, 2006).
D. Nikogosyan, Handbook of Properties of Optical Materials (John Wiley and Sons Ltd, London, 1997).
J.-M. Hopkins, A. J. Maclean, D. Burns, N. Schulz, M. Rattunde, C. Manz, K. Koehler, and J. Wagner, "Tunable, Single-frequency, Diode-pumped 2.3μm VECSEL," presented at Conference on Lasers and Electro-Optics, (Long Beach, 2006).
Element 6, "Diamond Types," (Element 6 Ltd., 2006), http://www.e6.com/e6/page.jsp?pageid=400602030.
A. Yariv, "Jones Calculus and its Applications to Propagation in Optical Systems with Birefringent Crystals," in Optical Electronics, (Saunders College Publishing, 1991), pp. 16-29.

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