Revised refractive index and absorption of In1-xGaxAsyP1-y lattice-matched to InP in transparent and absorption IR-region

Sten Seifert; Patrick Runge

doi:10.1364/OME.6.000629

1. Introduction

For the fabrication of optoelectronic devices such as detectors, lasers and modulators, the In_1-xGa_xAs_yP_1-y material system lattice-matched to InP is often used. In order to design these devices the knowledge of the refractive index and absorption is important due to its significant influence on the optical properties of an optoelectronic device. There are several models for the refractive index and absorption in the transparent regime of In_1-xGa_xAs_yP_1-y for the respective material composition [1]- [6]. In [7] an investigation and an interpolation scheme of the refractive indices based on the binary compounds of the In_1-xGa_xAs_yP_1-y material system lattice matched to InP is presented. In addition to the interpolation scheme a formular is given valid for a small region close below the band gap energy (−0.2 eV ≤ ΔE ≤ 0 eV). For larger wavelength regions around the band gap there are only two models widely accepted [8, 9]. The Afromowitz model [8] is less suitable to determine the refractive index and absorption coefficient near the band gap, since large variations in comparison with the literature values occur [7, 10]. The excitonic absorption spectrum of a semiconductor contains discrete and continuous contributions. With the help of Lorentzian broadening of the energy states the Tanguy model [9] takes the excitonic contribution into account. Thus, the Tanguy model has no major deviations from literature values near the band gap in comparison to the Afromowitz model. The Tanguy model contains several parameters, which are used to calculate the refractive index and absorption. Concerning the In_1-xGa_xAs_yP_1-y material system lattice-matched to InP Tanguy parameters exist only for In_0.88Ga_0.12As_0.25P_0.75 and for InP [11]. So far, there are no general forms of the Tanguy model for In_1-xGa_xAs_yP_1-y lattice-matched to InP.

In this publication the optical properties in the infrared transparent and absorption region are investigated with regard to the compositional change of the In_1-xGa_xAs_yP_1-y lattice-matched to InP. To determine the refractive indices and absorption coefficients of the In_1-xGa_xAs_yP_1-y material, spectroscopic ellipsometry was applied. The Tanguy parameters were determined from the evaluation of the measured ellipsometric spectra and were cross checked by high-resolution X-ray diffraction (HRXRD) and photoluminescence (PL). The relations between individual parameters of the Tanguy approximation are presented. Contrary to the results presented in [11], for the first time a general form of the Tanguy model could be derived for In_1-xGa_xAs_yP_1-y lattice-matched to InP. Thus the refractive index and the absorption coefficient for each In_1-xGa_xAs_yP_1-y lattice-matched to InP can be calculated over a wider wavelength range. In addition, the group refractive index and the chromatic material dispersion of this material system are approximated.

2. Experimental setup

To collect systematic experimental data of the refractive index and absorption for the In_1-xGa_xAs_yP_1-y-InP system, seven samples of different In_1-xGa_xAs_yP_1-y compositions lattice-matched to InP were measured by spectroscopic ellipsometry. One sample is an undoped InP substrate. Five samples composed of various undoped In_1-xGa_xAs_yP_1-y compositions lattice-matched to InP and the last one is an undoped InGaAs sample lattice-matched to InP. The different samples were grown by metal organic vapor phase epitaxy on 2 inch (001) InP substrates in a horizontal reactor. The compositions of the grown layers are chosen such that the wavelength range of In_1-xGa_xAs_yP_1-y lattice-matched on InP (920 nm to 1700 nm) is covered stepwise. Vegard’s law [12] cannot be used for determine the band gap energies because the linear approximation is not accurate enough for the complex In_1-xGa_xAs_yP_1-y material system. Instead Eq. (1) and Eq. (3) was used. The thicknesses of the grown layers are chosen such that the quality of the layers is high. Due to the production process the “undoped” layers have a silicon background doping of approximate 5E14 cm⁻³. However, the low background doping is supposed to have a negligible influence on the optical properties of the samples. For this reason, for the further investigation the background doping is not considered. After sample growth, the growth parameters have been checked by means of PL and HRXRD. Thus the band gap energy Eg as well as the lattice mismatch and the thickness were determined. E_g of In_1-xGa_xAs_yP_1-y is given in (1) [13]:

E_{g} = (\begin{array}{l} 1.35 - 0.668 x - 1.068 y + 0.758 x^{2} - 0.069 y^{2} \\ - 0.069 x y - 0.332 x^{2} y + 0.03 x y^{2} \end{array}) e V

The lattice mismatch f can be obtained as

f = \frac{a_{l} - a_{s}}{a_{s}}

where a_l denotes the lattice constant of the grown layer and a_s denotes the lattice constant of the substrate. The lattice constant a of In_1-xGa_xAs_yP_1-y layer is given in (3) [14]:

a = (5.8688 - 0.417 x + 0.1896 y + 0.0125 x y) Å .

Seven samples with associated band gap energy, wavelength, layer thickness and lattice mismatch are listed in Table 1. The wavelength (band gap near transition energy) was determined by PL and the lattice mismatch was determined by HRXRD. The values measured by PL and by HRXRD and the resulting x and y composition are listed in Table 1. The evaluating of the HRXRD measurements showed a good layer quality of each sample. The good quality was reflected in the Full Width at Half Maximum (FWHM) of the layer peaks which was less than 17 arcsec for samples A, B, C, F and G. Only sample D and E had a slightly greater FWHM since the lattice mismatch of these two samples is slightly larger. This confirms the selected layer thickness with regard to the quality.

Table 1. Values of band gap near transition energies, wavelengths, material and lattice mismatch of the grown samples

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Spectroscopic ellipsometry measurements were carried out for this publication. The measured spectra of all samples are shown in the appendix. For spectroscopic ellipsometry a Sentech SE 850 was used. The measurements were carried out in the wavelength range of 320 nm to 2500 nm and the angle of incidence was 70°. For the choosen polarizer angle the intensity in the plane of incidence and perpendicular to this plane after reflection is similar. Thus both polarization values were measured accurately. Through the selected wavelength range all samples were analyzed in near and mid-infrared transparent and near infrared absorption region. The measured data was analysed with the “SpectraRay 3” software by Sentech. For analysing the measured data of the samples, a layer model has to be created for each sample. The Tanguy-layer model was used for the In_1-xGa_xAs_yP_1-y or InGaAs layer. Between the substrates and the grown layers an interface has to be inserted due to interfaces between two compositions being formed by epitaxial growth of different compositions. The interface is to be regarded as a layer rather than a surface. The thickness of the interface extends up to 3 nm. Furthermore the interface serves as an index gradient between the successive layers. For the analysis, it is important to use a native oxide layer [7, 15] as the uppermost layer. Through contact with the atmosphere, native oxides form on the surface of a semiconductor [11]. At the time of the measurement this oxide layer had a thickness of 1 to 2 nm and a refractive index of 1.5 [15]. Figure 1 shows the layer model used for analysis.

Fig. 1 Schematic cross-sectional view of a layer model in SpectraRay 3.

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3. Measurement results

The amplitude ratio and phase shift is given by the measured ellipsometric Ψ and Δ values. By means of an appropriate model with approximate layer thickness and Tanguy model parameters of every material Ψ and Δ can be calculated. The differences between the measured (Ψexp and Δexp) and modeled (Ψmod and Δmod) spectra are quantified using the mean square error (MSE) (4) [16] where M is the number of fit parameters, N is the number of measured Ψ, Δ pairs, and σ are the standard deviations of Ψ and Δ. To extract a very precise complex refractive index and absorption coefficient, the MSE value has been minimized by tuning Tanguy model parameters. The Tanguy model (5) includes the Tanguy model parameters. The Tanguy parameters are the band gap Eg, the exciton binding energy R, the energy level broadening Γ, proportional of the Kane momentum matrix element A, a corresponds to the height of the absorption peak, b corresponds to the position of the absorption peak, the linear factor for absorption c and the second order factor for the absorption above the band gap d [8].

The Tanguy parameters and the MSE were determined by analysis of the modeled spectra by SpectraRay 3 (Table 2). To verify the correctness of the band gap energies extracted from ellipsometric data, these data were compared with PL measured band gap energies (Table 3). The evaluated band gaps from the SpectraRay software in Table 3 match fairly well with the measured PL values (Table 3). The differences between by PL measured band gap energies and the extracted band gap energies from the ellipsometric data are low. The band gap energies measured by PL is slightly higher than the band gap energies extracted from ellipsometric data. On the one hand the peak in the PL comes from states above the bandgap. The extracted band gap energies from the SpectraRay software were used for following calculations (6)- (12).

Table 2. Band gap energies of the In_1-xGa_xAs_yP_1-y compositions and Tanguy model parameters, which were determined by modelling the spectra

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Table 3. Band gap energies of the In_1-xGa_xAs_yP_1-y compositions measured by PL and band gap energies extracted from ellipsometric data

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n^{2} - 1 \approx \frac{a}{b - {(E + i Γ)}^{2}} + \frac{A \sqrt{R}}{{(E + i Γ)}^{2}} {\ln \frac{E_{g}^{2}}{E_{g}^{2} - {(E + i Γ)}^{2}} + π [2 \cot (π \sqrt{\frac{R}{E_{g}}}) - \cot (π \sqrt{\frac{R}{E_{g} - (E + i Γ)}}) - \cot (π \sqrt{\frac{R}{E_{g} + (E + i Γ)}})]}

M S E^{2} = \frac{1}{2 N - M} \sum_{i = 1}^{N} [{(\frac{Ψ_{i}^{\mod} - Ψ_{i}^{\exp}}{σ_{Ψ, i}^{\exp}})}^{2} + {(\frac{Δ_{i}^{\mod} - Δ_{i}^{\exp}}{σ_{Δ, i}^{\exp}})}^{2}]

In Fig. 2 the parameters of the Tanguy model are presented, which were obtained by the modeled spectra. There are linear relations between the Tanguy model parameters and the band gaps of In_1-xGa_xAs_yP_1-y composition lattice-matched to InP (Table 2).

Fig. 2 Parameters (R (a), Γ (b), A (c), a (d), b (e), c(f) and d(g)) of the Tanguy model in dependence of the band gap energies of the different material compositions; squares – modeled parameters from the measurements, solid line – linear approximation of the modeled parameters.

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The black squares in Fig. 2 represent the values obtained by the SpectraRay 3 software. With the values of the parameters of the Tanguy model linear fits were made. In Fig. 2 the linear fits (6), (7), (8), (9), (10), (11) and (12) of the modeled parameters of the Tanguy model are represented by the solid lines. The parameters of the Tanguy model for each In_1-xGa_xAs_yP_1-y compositions lattice-matched to InP can be determined with the help of Eqs. (1), (6)-(12).

R = - 0.00115 + 0.00191 \cdot E_{g}

Γ = - 0.000691 + 0.00433 \cdot E_{g}

A = - 0.0453 + 2.1103 \cdot E_{g}

a = 72.32 + 12.78 \cdot E_{g}

b = 4.84 + 4.66 \cdot E_{g}

c = - 0.015 + 0.02 \cdot E_{g}

d = - 0.178 + 1.042 \cdot E_{g}

Due to slight noise in the spectroscopic ellipsometry measurements not all values are consistent with the linear fits (Fig. 2) of the modeled parameters.

The refractive indices (Fig. 3) for different material compositions of In_1-xGa_xAs_yP_1-y were calculated with Eq. (5). The solid lines in Fig. 3 are the results of refractive indices obtained with the Tanguy parameters from the spectroscopic ellipsometry measurements. The dashed lines in Fig. 3 are the general form obtained from the linear fits Eqs. (1), (6)-(12). The differences in the refractive indices between the fitted parameters and the general form are negligible and can e.g. be ascribed to the lattice mismatch of the grown material compositions causing stress in the material which in turn influence the refractive index. The above mentioned ellipsometric measurements exhibit a slight noise which also leads to deviations of the linear fit. At this point no statement can be made whether the lattice mismatch or the noise of the measurement have the greater influence on the deviations. The deviations are dependent on a variety of material properties, i.e. lattice mismatch, etc.. When using more samples from the same composition a better statistic would result in less deviation. Nevertheless, a mathematical relationship in the Tanguy parameters of the In_1-xGa_xAs_yP_1-y material system lattice-matched to InP is expected that compensates the lack of statistics. In the presentation of the refractive indices (Fig. 3) the wavelength range was selected such that the resulting error (MSE) are small. The calculated refractive indices correspond to the existing values [1], [2], [4], [6] and [7]. In Fig. 3 only few literature values are shown as example to keep the Fig. clear. In [1], [2], [4], [6] and [7] more literature values are presented also join very well with our general form. A linear relationship between the refractive indices near the band gap and the band gap energies can be observed. This linear relationship is the logical consequence from the linear relationships of the parameters of the Tanguy model. The linear behavior of these parameters can be explained by the virtually linear change of the composition with respect to the band gaps [12]. The composition of the In_1-xGa_xAs_yP_1-y material significantly affects the optoelectronic properties of the layer. Thus, the change of the state of polarization upon reflection of light is directly affected. This change of the state of polarization is evaluated by the induced change of the amplitude ratio and phase shift, which were the measured ellipsometric Ψ and Δ values. Because of the linear relationship of the parameters of the Tanguy model the linear relationship of the refractive indices is caused.

Fig. 3 Refractive indices in dependence of the wavelength for different material compositions; solid lines – refractive index calculated with the Tanguy model from the ellipsometric measuremets, dashed lines – general form obtained from linear fits of the ellipsometric measured Tanguy parameters Eqs. (1), (6)-(12). ◼ [1], ● [2], ▲ [4], ✕ [6] and + [7] are literature values. The color with the specified material compositions match.

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The real part of the complex dielectric function ε(E) as function of the energy E describes the refractive index. By calculating the imaginary part of the complex dielectric function the extinction coefficient k(E) results [11]

k (E) = Im (\sqrt{ε (E) + \frac{a}{b - E^{2}}}) + c (E - E_{g}) + d {(E - E_{g})}^{2}

where a, b, c and d are the already mentioned Tanguy parameters with c and d in addition, where c, d = 0 for E ˂ E_g. The extinction coefficient k(λ) describes the strength of the interaction of radiation with the medium. It is dependent on the wavelength of the incident radiation. The extinction coefficient k(λ) as function of wavelength can be convert in the absorption coefficient α(λ) (Eq. (14). Figure 4 shows the absorption coefficients of the seven samples.

Fig. 4 Absorption coefficient in dependence of the wavelength for different material compositions.

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α (λ) = \frac{4 π \cdot k (λ)}{λ}

By means of the derived general form of the refractive index further calculations can be made in the absorption regime and near the band gap E_g. The group refractive index n_G can be calculated from the refractive index

n_{G} = n - λ \frac{d n}{d λ}

where λ is the wavelength. In Fig. 5 the group refractive indices have been evaluated from the general form of the refractive index.

Fig. 5 Group refractive indices with respect to wavelength calculated from general form of the Tanguy model.

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Furthermore, the chromatic material dispersion D can be calculated from the second derivation of the refractive index [17]

D = - \frac{λ}{c} \frac{d^{2} n}{d λ^{2}}

where c is the speed of light. The chromatic material dispersion is shown in Fig. 6 and has been calculated from the refractive index of linear fits of the parameters of the Tanguy model. The results are in good agreement with the presented chromatic dispersion in [18].

Fig. 6 Chromatic material dispersion with respect to wavelength calculated from the general form of the Tanguy model.

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4. Conclusion

Seven samples of In_1-xGa_xAs_yP_1-y lattice-matched to InP were grown and studied by spectroscopic ellipsometry. From these measurements a data set of parameters for the Tanguy model could be obtained for the refractive index of In_1-xGa_xAs_yP_1-y. The parameters showed a linear relation. From this model the refractive indices and the absorption coefficients in the transparent and in absorption regime can be calculated for this material system. Furthermore, the group refractive index and the chromatic material dispersion are estimated for the In_1-xGa_xAs_yP_1-y lattice-matched to InP.

6 Appendix

In the wavelength range above 2000 nm the measurements of Sample A, B, C, D and E show noise. These samples were measured up to the limit of the used detector (Fig. 7).

Fig. 7 Measured ellipsometric Ψ and Δ values with respect to wavelength; a) and b) Sample A; c) and d) Sample B; e) and f) Sample C; g) and h) Sample D; i) and j) Sample E; k) and l) Sample F; m) and n) Sample G; blue and green lines – measured Ψ and Δ values; red lines – fitted values

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Acknowledgment

The author would like to thank S. Peters from Sentech Instruments GmbH for fruitful discussion about analysing the ellipsometric measurements.

This work was funded by the Bundesminsiterium für Bildung und Forschung (BmBF) as a part of the HiLight project.

References and links

1. B. Broberg and S. Lindgren, “Refractive index of In1-xGaxAsyP1-y layers and InP in the transparent wavelength region,” J. Appl. Phys. 55(9), 3376–3381 (1984). [CrossRef]

2. J.-P. Weber, “Optimization of the Carrier-Induced Effective Index Change in InGaAsP Waveguide-Applications to Tunable Bragg Filters,” J. Quantum Electron 30(8), 1801–1816 (1994). [CrossRef]

3. J. Buus and M. J. Adams, “Phase and group indices for double heterostructure lasers,” Solid-State and Electron Devices 3(6), 189–195 (1979). [CrossRef]

4. C. H. Henry, L. F. Johnson, R. A. Logan, and D. P. Clarke, “Determination of the Refractive Index of InGaAsP Epitaxial Layers by Mode Line Luminescence Spectroscopy,” J. Quantum Electron QE-21(12), 1887–1892 (1985). [CrossRef]

5. Y. Suematsu, K. Kishino, S. Arai, and F. Koyama, “Chapter 4 Dynamic Single-Mode Semiconductor Laser with a Distributed Reflector,” Semiconductors and Semimetals 22, Part B, pp. 205–255 (1985).

6. F. Fiedler and A. Schlachetzki, “Optical Parameters of InP-Based Waveguides,” Solid-State Electron. 30(1), 73–83 (1987). [CrossRef]

7. H. Burkhard, H. W. Dinges, and E. Kuphal, “Optical properties of In1-xGaxP1-yAsy, InP, GaAs and GaP determined by ellipsometry,” J. Appl. Phys. 53(1), 655–662 (1982). [CrossRef]

8. M. A. Afromowitz, “Refractive Index Of Ga1-xAlxAs,” Solid State Commun. 15(1), 59–63 (1974). [CrossRef]

9. C. Tanguy, “Refractive Index of Direct Bandgap Semiconductors Near the Absorption Threshold: Influence of Excitonic Effects,” J.Quantum Electron. 32(10), 1746–1751 (1996). [CrossRef]

10. A. B. Djurisic, Y. Chan, and E. Herbert Li, “Progress in the room-temperature optical functions of semiconductors,” Mater. Sci. Eng. Rep. 38(6), 237–293 (2002). [CrossRef]

11. H. G. Bukkems, Y. S. Oei, U. Richter, and B. Gruska, “Analysis of III-V layer stacks on INR substrates using spectroscopic ellipsometry in NIR spectral range,” Thin Solid Films 364(1-2), 165–170 (2000). [CrossRef]

12. R. E. Nahory, M. A. Pollack, W. D. Johnston, and R. L. Barns, “Bandgap versus composition and demonstration of Vegard’s law for InGaAsP lattice-matched to InP,” Appl. Phys. Lett. 33(7), 659–661 (1978). [CrossRef]

13. Yu. A. Goldberg and N. M. Schmidt, “Handbook Series on Semiconductor Parameters,” 2, World Scientific, London, pp. 153–179 (1999)

14. S. Adachi, “Material Parameters of InGaAsP and Related Binaries,” J. Appl. Phys. 53(12), 8775–8792 (1982). [CrossRef]

15. S. Zollner, “Model dielectric function for native oxides on compound semiconductors,” Appl. Phys. Lett. 63(18), 2523–2524 (1993). [CrossRef]

16. C. M. Herzinger, P. Snyder, F. Celii, Y. Kao, D. Chow, B. Johs, and J. Woollam, “Studies of thin strained InAs, AlAs and AlSb layers by spectroscopic ellipsometry,” J. Appl. Phys. 79(5), 2663–2674 (1996). [CrossRef]

17. C. Pollock and M. Lipson, “Integrated Photonics”, pp.132–136, Kluwer Academic Publisher, Boston, 2003.

18. P. Runge, R. Elschner, and K. Petermann, “Chromatic Dispersion in InGaAsP Semiconductore Optical Amplifiers,” J. Quantum Electron. 46(5), 644–649 (2010). [CrossRef]

Sample	Material	PL energy [eV]	PL Wavelength [nm]	Thickness [nm]	Lattice mismatch [ppm]	x composition	y composition
A	InGaAs	0.737	1682	510	−120	0.46	1
B	InGaAsP	0.799	1550	425	−100	0.42	0.90
C	InGaAsP	0.857	1446	490	−250	0.38	0.79
D	InGaAsP	0.910	1363	495	+ 1800	0.32	0.69
E	InGaAsP	1.062	1168	510	−740	0.19	0.42
F	InGaAsP	1.170	1059	505	−190	0.11	0.24
G	InP	1.346	921	505	0	0	0

Sample	E_g [eV]	R [eV]	Γ [eV]	A [eV^1.5]	a [eV²]	b [eV²]	c	d	MSE
A	0.734	0.000294	0.00259	1.508	82.207	8.270	0.0118	1.233	0.36
B	0.796	0.000390	0.00265	1.631	82.400	8.500	0.0085	1.012	0.35
C	0.852	0.000487	0.00299	1.739	83.289	8.785	0.0058	0.914	0.35
D	0.903	0.000515	0.00304	1.864	83.514	9.116	0.0029	0.760	0.38
E	1.052	0.000798	0.00400	2.136	85.116	9.675	0.0013	0.730	0.37
F	1.167	0.001013	0.00452	2.468	87.205	10.341	0.0006	0.678	0.29
G	1.344	0.001500	0.00500	2.777	89.932	11.085	0.0002	0.544	0.31

Sample	E_g measured by PL [eV]	E_g extracted from ellipsometric data [eV]
A	0.737	0.734
B	0.799	0.796
C	0.857	0.852
D	0.910	0.903
E	1.062	1.052
F	1.170	1.167
G	1.346	1.344

Sample	Material	PL energy [eV]	PL Wavelength [nm]	Thickness [nm]	Lattice mismatch [ppm]	x composition	y composition
A	InGaAs	0.737	1682	510	−120	0.46	1
B	InGaAsP	0.799	1550	425	−100	0.42	0.90
C	InGaAsP	0.857	1446	490	−250	0.38	0.79
D	InGaAsP	0.910	1363	495	+ 1800	0.32	0.69
E	InGaAsP	1.062	1168	510	−740	0.19	0.42
F	InGaAsP	1.170	1059	505	−190	0.11	0.24
G	InP	1.346	921	505	0	0	0

Sample	E_g [eV]	R [eV]	Γ [eV]	A [eV^1.5]	a [eV²]	b [eV²]	c	d	MSE
A	0.734	0.000294	0.00259	1.508	82.207	8.270	0.0118	1.233	0.36
B	0.796	0.000390	0.00265	1.631	82.400	8.500	0.0085	1.012	0.35
C	0.852	0.000487	0.00299	1.739	83.289	8.785	0.0058	0.914	0.35
D	0.903	0.000515	0.00304	1.864	83.514	9.116	0.0029	0.760	0.38
E	1.052	0.000798	0.00400	2.136	85.116	9.675	0.0013	0.730	0.37
F	1.167	0.001013	0.00452	2.468	87.205	10.341	0.0006	0.678	0.29
G	1.344	0.001500	0.00500	2.777	89.932	11.085	0.0002	0.544	0.31

Revised refractive index and absorption of In_1-xGa_xAs_yP_1-y lattice-matched to InP in transparent and absorption IR-region

Abstract

1. Introduction

2. Experimental setup

3. Measurement results

4. Conclusion

6 Appendix

Acknowledgment

References and links

Cited By

Figures (7)

Tables (3)

Equations (16)

Optical Materials Express