- Email: [email protected]
Nonlinear responses and optical limiting behavior of fast green FCF dye under a low power CW He–Ne laser irradiation
Optics Communications 271 (2007) 551–554 www.elsevier.com/locate/optcom
Nonlinear responses and optical limiting behavior of fast green FCF dye under a low power CW He–Ne laser irradiation Kazem Jamshidi-Ghaleh a
a,*
, Somaieh Salmani a, Mohammad Hossain Majles Ara
b
Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran Department of Physics, Teacher Training University of Tehran, Tehran, Iran
b
Received 13 August 2006; received in revised form 17 October 2006; accepted 18 October 2006
Abstract The nonlinear responses and optical limiting performance of a dye type acid, called fast green FCF, are investigated under irradiation of 35 mW continuous wave He–Ne Laser. The second order refractive index and nonlinear absorption coefficient are measured by use of z-Scan technique. The optical limiting behavior is investigated by transmission measurement through the sample. Linear absorption coefficient (a), nonlinear absorption coefficient (b) and second order refractive index (n2) of fast green FCF are measured at different concentrations. 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear optics; Nonlinear refraction; Nonlinear absorption; Optical limiting; Fast green FCF
1. Introduction Since the development of laser technology, the research done on materials exhibiting strong nonlinear optical properties has been studied intensively because of their potential applications in photonics devises such as optical limiter [1– 3]. Optical limiting refers to a decrease of the optical transmittance of a material with increased incident light intensity. Wide ranges of materials with various nonlinear optical mechanisms contributing for the optical limiting and nonlinear absorption have been investigated [4–10]. Many publications on optical limiting materials were focused on nonlinear optical responses of organo-metallic compounds [4,5], semiconductor materials [6] and most recently, nanoparticles of metals and semiconductors [7– 9] because of their large nonlinearity and ultra-fast response time [7–10]. The optical limiting behavior can be achieved by one or more of the nonlinear optical mechanisms such as excited*
Corresponding author. E-mail address: [email protected] (K. Jamshidi-Ghaleh).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.037
state absorption reverse saturable absorption, free-carrier absorption, multi-photon, absorption, thermal defocusing/scattering, photo refraction, nonlinear refraction and induced scattering [11]. Optical limiting performance will be enhanced by coupling two or more of the nonlinear optical mechanisms. ESA and RSA are the most common mechanisms for the nonlinear optical behavior of organic materials. In this paper, the nonlinear optical properties of a dye type acid, called fast green FCF (acid blue 3), have been studied under low power continuous wave (CW) He–Ne laser irradiation. The FCF dye belongs to the triphenylmethane groups [12]. All the dyes of this series are derived from the hydrocarbon, triphenylmethane, and the tertiary alcohol, triphenylcarbinol, both of which are colorless. The chemical structure of fast green FCF dye is depicted in Fig. 1. The UV–vis absorption spectrum of fast green FCF dye is obtained and it exhibits the peak absorption at 630 nm as in Fig. 2. The goal of this work is to measure the nonlinear absorption coefficient, nonlinear refractive index and optical limiting of the fast green FCF dye by irradiation of a focused 35 mW CW He–Ne at 632.8 nm
552
K. Jamshidi-Ghaleh et al. / Optics Communications 271 (2007) 551–554
Fig. 1. The molecular structure of fast green FCF dye.
at the sample was m 30. With this optical geometry, the maximum applied irradiance on the sample target was 2800 W cm2. Two large area power meters (P1 and P2) were used to measure the incident and transmitted power of the laser beam. In transmittance measurement, the output energy monitor is located close to the glass sample without aperture. This optical geometry provides a transmittance measurement that is sensitive to nonlinear absorption. In closed aperture z-scan measurement, a 25% transmittance aperture was placed at far-field of the lens. Fast green FCF dye samples at three concentrations of 105 M, 104 M and 2 · 104 M were prepared as a water solution in a glass cell with 1 mm thickness. In the case of z-scan measurements, the sample was moved by a micrometer translating stage along the z-axis. The closed aperture allowed measurement of both sign and value of nonlinear refraction in the sample. Open aperture verifies again nonlinear absorption and can be used to calculate the nonlinear absorption coefficient. 3. Experimental results
Fig. 2. The UV–vis absorption spectra of fast green FCF dye.
laser irradiation. Nonlinear transmission measurement [13] and z-scan method [14] have been applied to measure the nonlinear absorption and the nonlinear refractive index at the applied wavelength. 2. Experimental The optical geometry used for transmittance measurement is shown in Fig. 3. A CW He–Ne laser was used for the light source. The maximum measurable output power of the laser was 35 mW. An attenuator was used to control the incident power on the sample. A small part of the input beam was split using a glass plate to monitor the input power. The major part of the laser beam was focused with 10 cm focal length quartz lens, to increase the beam irradiance at the sample. The laser beam size was measured by edge scan method. The beam waist diameter (FW1/e2M)
We have measured the beam spot size behind the sample at different concentrations. Fig. 4 shows images of the beam spot size when the sample is placed on the focal length of the lens. As it is seen clearly, the beam spot size increases with increasing FCF concentrations, which indicates divergence on the beam passing through the sample. In other words, the sample acts as a negative lens when laser beam passes through it. The same behavior is observed for different incident intensities at a given concentration. In addition, as it is shown clearly, diffraction and interference type fringes are formed when the nonlinear effects are important. It may be due to the nonlinear phase shift due to the intensity dependence of the refractive index. Further investigations about these observations are in progress. For optical limiting measurements, the sample is put near the focal plane of the lens and the input power is varied. Fig. 5 shows as an example the typical results of the transmission measurements in FCF sample at 2 · 104 M concentration as a function of incident power, varying from 0.5 mW up to 30 mW. As it is shown clearly for incident beam power of above 7 mW, the transmission
Fig. 3. The optical geometry used to investigate the nonlinear optical responses in the fast green FCF dye.
K. Jamshidi-Ghaleh et al. / Optics Communications 271 (2007) 551–554
553
Fig. 4. Beam spot size behind the sample with different concentrations. In the first two images, the white circle is due to the saturation effect on the camera.
Fig. 5. The nonlinear transmission behavior in the fast green FCF dye sample at 2 · 104 M concentration exposed to 35 mW CW He–Ne laser at 632.8 nm wavelength. The line curve shows the linear transmission of fast green FCF dye.
becomes nonlinear. At low incident powers, up to 7 mW, the output power varies linearly with a ratio of 10, which from low Beer gives the linear absorption coefficient of 4 mm1. Considerable decrease (about 90%) was observed in transmitted power at incident powers above 14 mW. The FCF transmission measurement result verifies that it may be a good candidate for optical limiting at 633 nm continuous wave lasers. The same procedure is followed for other concentrations and the results are summarized in Table 1. The measurements of nonlinear absorption coefficient (b) and nonlinear refractive index (n2) were conducted using open and closed aperture z-scan techniques, respectively [5]. A closed aperture scheme allowed us to determine both the values of the nonlinear refractive index of the sample and an open aperture scheme is applied in order to determine the value of the nonlinear absorption coefficient. The technique was based on the variation of trans-
mitted radiation intensity by alteration of the geometrical parameters of the interaction region. The experimental data were recorded by gradually moving a sample along the axis of propagation (z) of a focused Gaussian beam through its focal plane and measuring the transmission of the sample for each z-position. As the sample experiences different intensities at different positions, the recording of the transmission as a function of the z-coordinate provides accurate information about the nonlinear effects present. Fig. 6 displays the open aperture z-scan results of Fast Green FCF dye at three different concentrations. As it is seen, the transmission at the focus decreases with increasing sample concentration. One obtains that higher concentration of the sample gives better nonlinear optical properties. The same behavior was also observed for different input powers. The normalized transmittance for open aperture z-scan is [14] 1 X ðq0 Þm T ðzÞ ¼ ð1Þ 3=2 m¼0 ðm þ 1Þ where q0 ðzÞ ¼ bI 0 Leff =ð1 þ z2 =z20 Þ; jq0 j < 1; z0 ¼ px20 =k is the diffraction length of the beam, Leff = [1 exp(aL)]/ a, and I0 is the intensity of the laser beam at the focus (z = 0). The calculated values of a and b are summarized in Table 1. The full curves depicted in Fig. 6 are obtained from fitting by using Eq. (1). Fig. 7 shows the result of the closed aperture z-scan experimental measurement. The results have been obtained at an input laser power density of about 2800 W cm2 for
Table 1 Calculated values of linear absorption coefficient, nonlinear absorption coefficient and nonlinear refractive index at three concentrations Fast green FCF, 105 M Fast green FCF, 104 M Fast green FCF, 2 · 104 M Organic dye nile bluea a
See [15].
a (mm1)
b (cm W1)
n2 (cm2 W1)
0.41 2.56 4.18 1.24
0.65 · 105 2.80 · 105 6.50 · 105 1.35 · 105
0.1 · 108 0.8 · 108 3.2 · 108 0.42 · 108
Fig. 6. Normalized open aperture z-scan transmittance of fast green FCF dye at 632.8 nm. The full curves show the calculated results from Eq. (1).
554
K. Jamshidi-Ghaleh et al. / Optics Communications 271 (2007) 551–554
ear refraction in the fast green FCF dye at 105 M, 104 M and 2 · 104 M concentrations exposed to a low power CW He–Ne laser irradiation at 632.8 nm. Input–output measurement is applied to optical limiting behavior investigation. Power limiting threshold at 2 · 104 concentration was measured as 7 mW. Conventional z-scan open and closed aperture curves are used to investigate and measure nonlinear absorption and refraction coefficient in fast green FCF dye. The values of linear absorption coefficient (a), nonlinear absorption coefficient (b) and nonlinear refractive index (n2) are measured at three different concentrations. Fig. 7. Close aperture z-scan results at 632.8 nm, CW He–Ne laser with applied intensity level of 2800 W cm2; aperture linear transmittance was S = 0.25.
closed aperture parameter of S = 0.25. We find that fast green FCF dye sample exhibits strong self-defocusing behavior as revealed in the peak–valley shaped curves [5]. The peak–valley configuration of the transmittance curve depicts the negative sign of n2 for fast green FCF dye. The peak–valley, DPpv, is related to nonlinear refractive index n2 by DT pv ¼ ð1 SÞ
0:25
kn2 I 0 Leff
ð2Þ
where k = 2p/k, I0 is the input power density and Leff is given by Leff = [1 exp(La)]/a. We have measured the peak-to-valley difference for each concentration and calculated the values of n2 as summarized in Table 1. For comparison, the experimental results obtained for organic dye Nile Blue by CW He–Ne laser are also included [15]. 4. Conclusions We have presented the measurement results of the optical limiting performance, nonlinear absorption and nonlin-
References [1] T.J. Bunning, I.V. Natarajan, M.G. Schmitt, B.L. Epling, R.L. Crane, Appl. Opt. 30 (30) (1991) 4341. [2] L.W. Tutt, A. Kost, Nature 356 (1992) 225. [3] R. Chari, S.R. Mishra, H.R. Rawat, S.M. Oak, Appl. Phys. B 62 (1996) 293. [4] B.L. Jusuts, A.L. Huston, Z.H. Kafafi, Opt. Lett. 18 (19) (1993) 1603. [5] L.W. Tutt, S.W. Mccahon, Opt. Lett. 15 (1990) 700. [6] S.K. Han, Z.G. Huo, R. Srivastava, R.V. Ramaswanry, J. Opt. Soc. Am. B 4 (1989) 663. [7] G.X. Chen, M.H. Hong, T.C. Chong, J. Appl. Phy. 95 (3) (2004) 1455. [8] R.A. Ganeev, A.I. Ryasnyansky, Sh.R. Kamalov, M.K. Kodirov, T.U. Smanov, J. Phys. D. Appl. Phys. 34 (2001) 1602. [9] L. Quanshui, L. Chunling, L. Zhengang, G. Qihuang, Opt. Express 13 (6) (2005) 1833. [10] H. Xu, J. Wang, P. Zhu, Y. Song, Y. Li, Y. Sun, F. Cui, S. Zhang, Z. Xia, Y. Zou, C. Yu, SPIE 2321(1994) 10–3. [11] L.W. Tutt, T.F. Boggess, Prog. Quant. Electron. 17 (1993) 299. [12]. [13] K. Jamshidi-Ghaleh, N. Mansour, D. Ashkenasi, H.J. Hoffmann, Opt. Commun. 246 (2005) 213. [14] M. Sheik-Bahae, A.A. Said, T.H. Wei, D. Hagan, E.W. Styland, IEEE J. Quantum Electron. 26 (4) (1990) 760. [15] A. Qusay Mohammed, P.K. Palanisamy, Mod. Phys. Lett. B 20 (11) (2006) 623.
Nonlinear responses and optical limiting behavior of fast green FCF dye under a low power CW He–Ne laser irradiation Kazem Jamshidi-Ghaleh a
a,*
, Somaieh Salmani a, Mohammad Hossain Majles Ara
b
Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran Department of Physics, Teacher Training University of Tehran, Tehran, Iran
b
Received 13 August 2006; received in revised form 17 October 2006; accepted 18 October 2006
Abstract The nonlinear responses and optical limiting performance of a dye type acid, called fast green FCF, are investigated under irradiation of 35 mW continuous wave He–Ne Laser. The second order refractive index and nonlinear absorption coefficient are measured by use of z-Scan technique. The optical limiting behavior is investigated by transmission measurement through the sample. Linear absorption coefficient (a), nonlinear absorption coefficient (b) and second order refractive index (n2) of fast green FCF are measured at different concentrations. 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear optics; Nonlinear refraction; Nonlinear absorption; Optical limiting; Fast green FCF
1. Introduction Since the development of laser technology, the research done on materials exhibiting strong nonlinear optical properties has been studied intensively because of their potential applications in photonics devises such as optical limiter [1– 3]. Optical limiting refers to a decrease of the optical transmittance of a material with increased incident light intensity. Wide ranges of materials with various nonlinear optical mechanisms contributing for the optical limiting and nonlinear absorption have been investigated [4–10]. Many publications on optical limiting materials were focused on nonlinear optical responses of organo-metallic compounds [4,5], semiconductor materials [6] and most recently, nanoparticles of metals and semiconductors [7– 9] because of their large nonlinearity and ultra-fast response time [7–10]. The optical limiting behavior can be achieved by one or more of the nonlinear optical mechanisms such as excited*
Corresponding author. E-mail address: [email protected] (K. Jamshidi-Ghaleh).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.037
state absorption reverse saturable absorption, free-carrier absorption, multi-photon, absorption, thermal defocusing/scattering, photo refraction, nonlinear refraction and induced scattering [11]. Optical limiting performance will be enhanced by coupling two or more of the nonlinear optical mechanisms. ESA and RSA are the most common mechanisms for the nonlinear optical behavior of organic materials. In this paper, the nonlinear optical properties of a dye type acid, called fast green FCF (acid blue 3), have been studied under low power continuous wave (CW) He–Ne laser irradiation. The FCF dye belongs to the triphenylmethane groups [12]. All the dyes of this series are derived from the hydrocarbon, triphenylmethane, and the tertiary alcohol, triphenylcarbinol, both of which are colorless. The chemical structure of fast green FCF dye is depicted in Fig. 1. The UV–vis absorption spectrum of fast green FCF dye is obtained and it exhibits the peak absorption at 630 nm as in Fig. 2. The goal of this work is to measure the nonlinear absorption coefficient, nonlinear refractive index and optical limiting of the fast green FCF dye by irradiation of a focused 35 mW CW He–Ne at 632.8 nm
552
K. Jamshidi-Ghaleh et al. / Optics Communications 271 (2007) 551–554
Fig. 1. The molecular structure of fast green FCF dye.
at the sample was m 30. With this optical geometry, the maximum applied irradiance on the sample target was 2800 W cm2. Two large area power meters (P1 and P2) were used to measure the incident and transmitted power of the laser beam. In transmittance measurement, the output energy monitor is located close to the glass sample without aperture. This optical geometry provides a transmittance measurement that is sensitive to nonlinear absorption. In closed aperture z-scan measurement, a 25% transmittance aperture was placed at far-field of the lens. Fast green FCF dye samples at three concentrations of 105 M, 104 M and 2 · 104 M were prepared as a water solution in a glass cell with 1 mm thickness. In the case of z-scan measurements, the sample was moved by a micrometer translating stage along the z-axis. The closed aperture allowed measurement of both sign and value of nonlinear refraction in the sample. Open aperture verifies again nonlinear absorption and can be used to calculate the nonlinear absorption coefficient. 3. Experimental results
Fig. 2. The UV–vis absorption spectra of fast green FCF dye.
laser irradiation. Nonlinear transmission measurement [13] and z-scan method [14] have been applied to measure the nonlinear absorption and the nonlinear refractive index at the applied wavelength. 2. Experimental The optical geometry used for transmittance measurement is shown in Fig. 3. A CW He–Ne laser was used for the light source. The maximum measurable output power of the laser was 35 mW. An attenuator was used to control the incident power on the sample. A small part of the input beam was split using a glass plate to monitor the input power. The major part of the laser beam was focused with 10 cm focal length quartz lens, to increase the beam irradiance at the sample. The laser beam size was measured by edge scan method. The beam waist diameter (FW1/e2M)
We have measured the beam spot size behind the sample at different concentrations. Fig. 4 shows images of the beam spot size when the sample is placed on the focal length of the lens. As it is seen clearly, the beam spot size increases with increasing FCF concentrations, which indicates divergence on the beam passing through the sample. In other words, the sample acts as a negative lens when laser beam passes through it. The same behavior is observed for different incident intensities at a given concentration. In addition, as it is shown clearly, diffraction and interference type fringes are formed when the nonlinear effects are important. It may be due to the nonlinear phase shift due to the intensity dependence of the refractive index. Further investigations about these observations are in progress. For optical limiting measurements, the sample is put near the focal plane of the lens and the input power is varied. Fig. 5 shows as an example the typical results of the transmission measurements in FCF sample at 2 · 104 M concentration as a function of incident power, varying from 0.5 mW up to 30 mW. As it is shown clearly for incident beam power of above 7 mW, the transmission
Fig. 3. The optical geometry used to investigate the nonlinear optical responses in the fast green FCF dye.
K. Jamshidi-Ghaleh et al. / Optics Communications 271 (2007) 551–554
553
Fig. 4. Beam spot size behind the sample with different concentrations. In the first two images, the white circle is due to the saturation effect on the camera.
Fig. 5. The nonlinear transmission behavior in the fast green FCF dye sample at 2 · 104 M concentration exposed to 35 mW CW He–Ne laser at 632.8 nm wavelength. The line curve shows the linear transmission of fast green FCF dye.
becomes nonlinear. At low incident powers, up to 7 mW, the output power varies linearly with a ratio of 10, which from low Beer gives the linear absorption coefficient of 4 mm1. Considerable decrease (about 90%) was observed in transmitted power at incident powers above 14 mW. The FCF transmission measurement result verifies that it may be a good candidate for optical limiting at 633 nm continuous wave lasers. The same procedure is followed for other concentrations and the results are summarized in Table 1. The measurements of nonlinear absorption coefficient (b) and nonlinear refractive index (n2) were conducted using open and closed aperture z-scan techniques, respectively [5]. A closed aperture scheme allowed us to determine both the values of the nonlinear refractive index of the sample and an open aperture scheme is applied in order to determine the value of the nonlinear absorption coefficient. The technique was based on the variation of trans-
mitted radiation intensity by alteration of the geometrical parameters of the interaction region. The experimental data were recorded by gradually moving a sample along the axis of propagation (z) of a focused Gaussian beam through its focal plane and measuring the transmission of the sample for each z-position. As the sample experiences different intensities at different positions, the recording of the transmission as a function of the z-coordinate provides accurate information about the nonlinear effects present. Fig. 6 displays the open aperture z-scan results of Fast Green FCF dye at three different concentrations. As it is seen, the transmission at the focus decreases with increasing sample concentration. One obtains that higher concentration of the sample gives better nonlinear optical properties. The same behavior was also observed for different input powers. The normalized transmittance for open aperture z-scan is [14] 1 X ðq0 Þm T ðzÞ ¼ ð1Þ 3=2 m¼0 ðm þ 1Þ where q0 ðzÞ ¼ bI 0 Leff =ð1 þ z2 =z20 Þ; jq0 j < 1; z0 ¼ px20 =k is the diffraction length of the beam, Leff = [1 exp(aL)]/ a, and I0 is the intensity of the laser beam at the focus (z = 0). The calculated values of a and b are summarized in Table 1. The full curves depicted in Fig. 6 are obtained from fitting by using Eq. (1). Fig. 7 shows the result of the closed aperture z-scan experimental measurement. The results have been obtained at an input laser power density of about 2800 W cm2 for
Table 1 Calculated values of linear absorption coefficient, nonlinear absorption coefficient and nonlinear refractive index at three concentrations Fast green FCF, 105 M Fast green FCF, 104 M Fast green FCF, 2 · 104 M Organic dye nile bluea a
See [15].
a (mm1)
b (cm W1)
n2 (cm2 W1)
0.41 2.56 4.18 1.24
0.65 · 105 2.80 · 105 6.50 · 105 1.35 · 105
0.1 · 108 0.8 · 108 3.2 · 108 0.42 · 108
Fig. 6. Normalized open aperture z-scan transmittance of fast green FCF dye at 632.8 nm. The full curves show the calculated results from Eq. (1).
554
K. Jamshidi-Ghaleh et al. / Optics Communications 271 (2007) 551–554
ear refraction in the fast green FCF dye at 105 M, 104 M and 2 · 104 M concentrations exposed to a low power CW He–Ne laser irradiation at 632.8 nm. Input–output measurement is applied to optical limiting behavior investigation. Power limiting threshold at 2 · 104 concentration was measured as 7 mW. Conventional z-scan open and closed aperture curves are used to investigate and measure nonlinear absorption and refraction coefficient in fast green FCF dye. The values of linear absorption coefficient (a), nonlinear absorption coefficient (b) and nonlinear refractive index (n2) are measured at three different concentrations. Fig. 7. Close aperture z-scan results at 632.8 nm, CW He–Ne laser with applied intensity level of 2800 W cm2; aperture linear transmittance was S = 0.25.
closed aperture parameter of S = 0.25. We find that fast green FCF dye sample exhibits strong self-defocusing behavior as revealed in the peak–valley shaped curves [5]. The peak–valley configuration of the transmittance curve depicts the negative sign of n2 for fast green FCF dye. The peak–valley, DPpv, is related to nonlinear refractive index n2 by DT pv ¼ ð1 SÞ
0:25
kn2 I 0 Leff
ð2Þ
where k = 2p/k, I0 is the input power density and Leff is given by Leff = [1 exp(La)]/a. We have measured the peak-to-valley difference for each concentration and calculated the values of n2 as summarized in Table 1. For comparison, the experimental results obtained for organic dye Nile Blue by CW He–Ne laser are also included [15]. 4. Conclusions We have presented the measurement results of the optical limiting performance, nonlinear absorption and nonlin-
References [1] T.J. Bunning, I.V. Natarajan, M.G. Schmitt, B.L. Epling, R.L. Crane, Appl. Opt. 30 (30) (1991) 4341. [2] L.W. Tutt, A. Kost, Nature 356 (1992) 225. [3] R. Chari, S.R. Mishra, H.R. Rawat, S.M. Oak, Appl. Phys. B 62 (1996) 293. [4] B.L. Jusuts, A.L. Huston, Z.H. Kafafi, Opt. Lett. 18 (19) (1993) 1603. [5] L.W. Tutt, S.W. Mccahon, Opt. Lett. 15 (1990) 700. [6] S.K. Han, Z.G. Huo, R. Srivastava, R.V. Ramaswanry, J. Opt. Soc. Am. B 4 (1989) 663. [7] G.X. Chen, M.H. Hong, T.C. Chong, J. Appl. Phy. 95 (3) (2004) 1455. [8] R.A. Ganeev, A.I. Ryasnyansky, Sh.R. Kamalov, M.K. Kodirov, T.U. Smanov, J. Phys. D. Appl. Phys. 34 (2001) 1602. [9] L. Quanshui, L. Chunling, L. Zhengang, G. Qihuang, Opt. Express 13 (6) (2005) 1833. [10] H. Xu, J. Wang, P. Zhu, Y. Song, Y. Li, Y. Sun, F. Cui, S. Zhang, Z. Xia, Y. Zou, C. Yu, SPIE 2321(1994) 10–3. [11] L.W. Tutt, T.F. Boggess, Prog. Quant. Electron. 17 (1993) 299. [12]