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Simple method for estimating shape functions of optical spectra
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Optics Communications 281 (2008) 368–373 www.elsevier.com/locate/optcom
Simple method for estimating shape functions of optical spectra K. Wada *, J. Fujita, J. Yamada, T. Matsuyama, H. Horinaka Department of Physics and Electronics, Graduate School of Engineering, Osaka Prefecture University, 1-1, Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan Received 7 December 2006; received in revised form 21 September 2007; accepted 21 September 2007
Abstract A simple method for estimating shape functions of optical spectra is proposed, based on the numerical correlation with an assumed optical spectrum composed of a central main rectangular component and two right-angled triangular wings on either side of the main component. The degrees of correlation between observed and assumed spectra were examined using spectral and coherence widths of those spectra and pulsewidths of Fourier-transform-limited pulses calculated from those spectra. By using this method, shape functions of output spectra from superluminescent diodes and a self-pulsating laser diode were evaluated in detail over a relatively wide injection current range beyond their rated currents. Ó 2007 Elsevier B.V. All rights reserved. PACS: 47.50.Ef; 42.55.Px Keywords: Broadband spectrum; Spectral shape function; Gaussian spectrum; Superluminescent diode; Self-pulsation; Gain-narrowing
1. Introduction In recent years, broadband light sources, such as femtosecond lasers and superluminescent diodes, have been used in a wide range of fields, including metrology [1], biophotonics [2], and laser processing [3]. Although the broadband spectra of such devices are often evaluated simply by using only the spectral width (usually the full width at half maximum, FWHM), their shape functions also include important information about the spectra. In OCT (optical coherence tomography) measurements, for example, the coherence length of a light source, corresponding to the longitudinal spatial resolution, is determined not only by the spectral width but also by the spectral shape function [4]. When the spectral width is constant, spectral shape functions that widen toward the bottom, such as a Lorentzian spectrum, make the coherence length shorter. It is recognized, however, that a Gaussian spectrum is preferable *
Corresponding author. Tel.: +81 72 254 9264; fax: +81 72 254 9908. E-mail address: [email protected] (K. Wada).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.09.042
for use in OCT measurements [5] because the autocorrelation functions obtained by the inverse Fourier-transform of such Lorentz-like spectra have fatter tails, resulting in the masking of weak signals in high-contrast images. Therefore, the shape functions of optical spectra should be carefully estimated. In spectroscopy, a valuable numerical technique for fitting emission or absorption spectra to determine the influence of Doppler broadening on intrinsic emission or absorption is to use Voigt-profiles, which are the convolution of Lorentzian and Gaussian profiles; this allows the temperature of the emitting or absorbing layers [6] or the total pressure of a gas atmosphere [7] to be estimated. When estimating the spectral shape functions of optical spectra, however, in many cases the observed spectra are numerically fitted using typical functions (Lorentzian, Gaussian, hyperbolic secant squared, etc.) without such a theoretical grounding. With such an approach, however, it seems to be difficult to obtain a quantitative evaluation of the difference between shape functions of observed and assumed spectra.
K. Wada et al. / Optics Communications 281 (2008) 368–373
In this Letter, we propose a simple method for estimating shape functions of observed optical spectra using three FWHM values, namely, spectral widths and coherence widths of observed spectra and pulsewidths of Fouriertransform-limited pulses calculated from those spectra. Using this method, the features of the shape function of an observed spectrum are extracted and compared with those extracted from the assumed functions. 2. Numerical model
pffiffiffiffiffiffiffiffiffiffi P ðf Þ expði/Þ (/: spectrum phase). Because phase problems are of no concern in the estimation of spectral shape, for simplicity, a Fourier-transform-limited relation between time and frequency domains is assumed to hold in this numerical study. Under this assumption, the value of / becomes constant for all spectrum data, and Fourier-transform-limited pulses are found in the time domain as a result of the inverse complex Fourier-transform of the complex spectra. Then, pulsewidths Dt (FWHM) of the obtained pulses are measured to get values of TBP (=Dt Æ Df0). Thus, the method of computation is just the same for both the TBP and CBP cases, except for the input spectrum data type. Although both parameters are used in discussions of pulse- or spectrum-shape functions, they have different uses. To make this clear, we show the following numerical results. Fig. 2 shows variations of the TBP (a) and the CBP (b) as a function of the wing width when the wing height is set as a running parameter. It is found from Fig. 2 that the CBP is sensitive to the changes of the wing height, whereas the TBP is not sensitive. This difference enables us to estimate spectral shape functions. In Fig. 3, an example can be seen for the loci of combination values (wing height, wing width) satisfying the conditions of TBP = 0.44 and CBP = 0.88. The two loci have only one intersecting point of (0.30, 0.87), based on the above different sensitivities to the changes of the wing height. This means that there exists a one-to-one correspondence between the combination values of (wing height, wing width) and (CBP, TBP). By estimating the CBP and TBP values from an experimentally observed optical spectrum, we can therefore uniquely determine a particular spectral
a
TBP
Initially, we assume an optical power spectrum composed of a central main rectangular component with a bandwidth of Df0 and two right-angled triangular wings on either side of the main component, as shown in Fig. 1. The height of the main component is normalized to be 1.0 and the wing height and width are also normalized by 1.0 and Df0, respectively. Varying the wing height and width allows a variety of spectral shapes to be modeled. We expect that the shape functions of observed spectra can be estimated by examining the degrees of correlation between those assumed shapes and the actual shapes of experimentally observed spectra. To this end, two dimensionless parameters are used: one is the timebandwidth product (TBP) and the other is the product of coherence time and bandwidth, termed here the coherence time-bandwidth product (CBP). The former has ordinarily been used in pulse chirp estimation problems [8,9], and the latter has often been used in OCT measurements in terms of spatial resolution [4,10]. According to the Wiener– Khintchine theorem, coherence widths Ds are measured as the FWHM of autocorrelation functions C(s) obtained by the inverse Fourier-transform of power spectrum data R1 (CðsÞ ¼ 1 P ðf Þ expði2pftÞdf , P(f): power spectrum data, f: frequency). The values of CBP are then calculated simply by Ds Æ Df0. In the case of TBP, the power spectrum data are transformed into complex spectrum data in the form
0.8
Wing height
0.6
0.1 0.15 0.2
0.4
0.3 0.4 0.5
0.2 1.0
0 0
Spectral intensity
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0.5
1.0
1. 5
2.0
2. 5
3.0
b 1.5
Δf0
Wing height 0.1
CBP
Wings
Wing height
1.0
0.15 0.2
0.5 0.3
0
Wing width
Frequency
Fig. 1. Optical power spectrum model prepared for numerical correlation with observed spectra. Upper figure shows variation of spectral shape when wing width is varied.
0.4
0.5
0 0
0.5
1.0
1.5
2.0
2. 5
3.0
Wing width Fig. 2. Variations of (a) TBP and (b) CBP as a function of wing width when wing height is set as a running parameter, in the range of 0.1–0.5.
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Normalized spectral intensityy
Wing width
1.2
1.0
0.5
Δf0
0.3 0
TBP = 0.44
Frequency
0.87
Fig. 5. Comparison of Gaussian spectrum with numerical spectrum model having the same combination value (CBP, TBP) as Gaussian spectrum.
0.8
CBP = 0.88 0.6 0.20
0.25
0.30
0.35
0.40
0.45
0.50
Wing height Fig. 3. Loci of combination values (wing height and wing width) satisfying conditions of TBP = 0.44 (solid) and CBP = 0.88 (dashed).
shape, in the form of a rectangular component and accompanying right-angled triangular wings, which corresponds best to the observed one. Fig. 4 shows an example of the assumed spectral shapes. The three spectra at the left have the same TBP value of 0.44 and the ones at the right have the same CBP value of 0.88 when the wing heights are set to 0.15, 0.3, and 0.5, respectively, under a constant bandwidth Df0. The middle spectra for the wing height of 0.3, which are identical to each other, have a combination value (CBP, TBP) of (0.88, 0.44), corresponding to that of a Gaussian spectrum [8] and also to the above intersecting point. As shown in Fig. 5, the Gaussian spectrum with a bandwidth of Df0 and a normalized height of 1.0 is then TBP = 0.44
superimposed on the middle spectrum in Fig. 4, showing a good match in terms of shape. We then plot loci of the combination values as coordinates on the TBP–CBP plane in Fig. 6 by varying the wing width when the wing height is set as a running parameter. The wider the wing width, the smaller the TBP and the CBP. The combination values corresponding to typical shape functions (rectangular, Gaussian, hyperbolic secant squared (sech2), and Lorentzian) are listed in Table 1 and plotted in Fig. 6. It is found that the transition of the shape
CBP = 0.88
1.0
Normalized spectral intensity
0.5
Δf0
Δff0
Fig. 6. Loci of combination values depicted by varying wing width when wing height is set as a running parameter, in the range of 0.1–0.5. The combination values of four typical shape functions are plotted on TBP– CBP plane.
0
1.0
0.79Δf0
0.69Δf0
Table 1 TBP and CBP values of typical shape functions
0.5 0.3 0
1.0
/2 0.87Δf0
0.87Δf0
0.5 0.15 0
1.01Δf0
Frequency
1.69Δf0
Frequency
Fig. 4. Spectral shapes obtained under conditions of TBP = 0.44 (left) and CBP = 0.88 (right) when wing heights are set to 0.5, 0.3, and 0.15, respectively.
/2
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function from rectangular to Lorentzian, via Gaussian and sech2, on the TBP–CBP plane shows a tendency to increase the wing width and decrease the wing height, simultaneously. By plotting the combination value (CBP, TBP) of an experimentally observed optical spectrum on the TBP–CBP plane and comparing it with those of typical assumed shape functions, we can roughly estimate the shape function of the observed optical spectrum without deriving the wing height and width from the combination value (CBP, TBP). 3. Experimental
16 12 8 4 0 30
4. Results and discussion
35
40
45
50
55
60
Injection current (mA)
Fig. 7 shows variations of spectral widths Df of the SLD1 (squares) and SLD2 (triangles) outputs, respectively, when the injection current was varied. By setting the ratios of the instrumental resolutions to observed spectral widths less than 1% throughout the experiment, estimation errors on the TBP and CBP values were suppressed within ±2%. In the case of SLD1, the spectral width was approximately constant to 6.5 THz over the range of injection currents. As for SLD2, the spectral width became narrower as the injection current was increased. This tendency was particularly significant at injection currents beyond the rated current (90 mA). This is thought to be due to gain-narrowing caused by the increase in optical oscillation at high injection currents. We then calculated pulsewidths Dt and coherence widths Ds from the observed spectra and plotted combination values (CBP, TBP) on the TBP–CBP plane in Fig. 8. In the case of SLD1, the locus of the combination value started with a point (at 60 mA) near the sech2-point and ended with a point (at 120 mA) near the Gaussianpoint, following a small arc. From this locus, it was found
Spectral width (THz)
Fig. 8. Loci of combination values of output spectra for SLD1 (squares) and SLD2 (triangles) when injection current was varied. The values indicate injection current in mA.
Spectral width (THz)
We prepared two commercially available 840-nm superluminescent diodes (SLD1: Anritsu, AS8C1110Z40M, and SLD2: made by a different manufacturer) and a 660-nm self-pulsating laser diode (PLD: Matsushita, LNCQ08). The output spectra from those diodes were observed with an optical spectrum analyzer (Yokogawa, AQ6317C) at wavelength resolutions of 0.01–0.1 nm (= 4.25–42.5 GHz at 840 nm). The injection current was varied over a relatively wide range beyond their rated currents.
Fig. 9. Variation of spectral width of PLD output spectra with respect to injection current.
that the spectra observed around the rated current (75 mA) had slightly lower wing heights than the sech2-profile in terms of our spectrum model. As for SLD2, the locus of the combination value started with a point (at 50 mA) below the sech2-point and ended with a point (at 120 mA) near the Lorentzian-point, following a relatively long linear curve with a small jump. The combination value at the rated current (90 mA) was found to occupy almost the midpoint between the sech2-point and the Lorentzianpoint on the TBP–CBP plane. Thus, the gain-narrowing effect caused notable spectral shape variation. The PLD was also examined in the same way. Fig. 9 shows the variation of spectral width with respect to the
10 8 6 4
SLD1
2
SLD2
0 50
60
70
80
90
100
110
120
Injection current (mA) Fig. 7. Variation of spectral widths of output spectra for SLD1 (squares) and SLD2 (triangles) with respect to injection current.
Fig. 10. Loci of combination values of PLD output spectra when injection current was varied. The values indicate injection current in mA.
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injection current, and Fig. 10 shows the locus of the combination value on the TBP–CBP plane. It is interesting to note that the locus followed a deformed elliptical orbit whose apsides were located in the vicinity of the sech2point and the Lorentzian-point. The PLD operated as a light-emitting diode when the injection current was below the threshold of 54 mA. In this current region (specifically, 30–54 mA), gain-narrowing took place in the PLD, similarly to the SLD2, showing a marked decrease of the spectral width from 14 to 4 THz in Fig. 9. The corresponding locus is depicted as the upper part of the elliptical orbit in Fig. 10, showing the transition of the shape function from near sech2 to almost Lorentzian. When the injection current was increased beyond the threshold, the PLD oscillated in self-pulsation mode, and the spectral width became approximately constant (around 1 THz), as seen in the current range of 56–60 mA in Fig. 9. Although the variation of spectral width in the current range of 54–60 mA was small in contrast to the case in the above gain-narrowing region, the corresponding combination value changed significantly and formed the lower part of the elliptical orbit in Fig. 10. This means that the transition of the oscillation mode from
1.0
a Δf0 = 6.6 THz
0.5 0.2 0
Normalized spectral intensity
1.09Δf0
1.0
b
spontaneous emission to self-pulsation was accompanied by a strong spectral shape variation. It is clear from Fig. 10 that different mechanisms of spectral shape variation brought about different routes between positions near the sech2-point and the Lorentzian-point. By examining the values of the TBP and CBP parameters, we can thus easily obtain detailed information about the spectral shape of the optical spectra. To confirm the validity of this method, we show examples of observed spectra and numerical fitting spectra in Fig. 11. In the case of multi-longitudinal mode spectra shown in Fig. 11b and c, the TBP and CBP values were estimated from envelopes of those spectra. To get fitting spectra, the wing heights and widths were derived from the TBP and CBP values of observed spectra by the same way as shown in Fig. 3. The numerical fitting spectra corresponded well with the observed spectra and were found to be applicable even to some observed spectra with asymmetrical shapes because the combination values of those asymmetrical spectra were also uniquely determined. 5. Conclusions We proposed a simple method for estimating shape functions of observed optical spectra based on numerical correlation with an assumed power spectrum-shape composed of a central main rectangular component and two right-angled triangular wings on either side of the main component. Using this method, we examined the shape functions of output spectra from superluminescent diodes and a self-pulsating laser diode. As a result, the shape functions of the observed spectra were estimated in detail by comparing them with typical assumed shape functions, and the influence of gain-narrowing or self-pulsation, occurring in those emitting diodes, on the spectral shape variation was also evaluated.
12.6 THz
0.5
Acknowledgements 0.2 0 2.00Δf0
1.0
c 4.0 THz
0.5
The authors are indebted to Dr. M. Yuri of Matsushita Electric Co. for providing the laser diode samples and to Mr. S. Hiroshima of Koyo Electric Co. for lending the optical spectrum analyzer used in this work. This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology under Grant-in-Aid for Scientific Research (C) No. 17560035. References
0.15 0
Frequency
4.19Δf0
Fig. 11. Examples of observed spectra and corresponding numerical fitting spectra: (a) SLD1 output spectrum at 100 mA, and PLD output spectra at (b) 40 mA and (c) 52 mA, under spectral resolutions of (a), (b) 42.5 GHz, and (c) 21.25 GHz, respectively.
[1] Th. Udem, R. Holzwarth, T.W. Ha¨nsch, Nature 416 (2002) 233. [2] D. Huang, E.A. Swanson, C.P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C.A. Puliafito, J.G. Fujimoto, Science 254 (1991) 1178. [3] S. Kawata, H.-Bo. Sun, T. Tanaka, K. Takada, Nature 412 (2001) 697. [4] Y. Zhang, M. Sato, N. Tanno, Opt. Lett. 26 (2001) 205.
K. Wada et al. / Optics Communications 281 (2008) 368–373 [5] R. Tripathi, N. Nassif, J.S. Nelson, B.H. Park, J.F. de Boer, Opt. Lett. 27 (2002) 406. [6] R.J. Leiweke, B.N. Ganguly, Appl. Phys. Lett. 88 (2006) 131501. [7] Z. Bacsik, A. Gyivicsa´n, K. Horva´th, J. Mink, Anal. Chem. 78 (2006) 2382.
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[8] K.L. Sala, G.A. Kenny-Wallace, G.E. Hall, IEEE J. Quantum Electron. QE-16 (1980) 990. [9] K. Wada, Y. Cho, Opt. Lett. 19 (1994) 1633. [10] K. Wada, K. Hirata, M. Yoshida, K. Kawai, K. Yoshioka, T. Matsuyama, H. Horinaka, Y. Cho, Opt. Commun. 228 (2003) 49.
Optics Communications 281 (2008) 368–373 www.elsevier.com/locate/optcom
Simple method for estimating shape functions of optical spectra K. Wada *, J. Fujita, J. Yamada, T. Matsuyama, H. Horinaka Department of Physics and Electronics, Graduate School of Engineering, Osaka Prefecture University, 1-1, Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan Received 7 December 2006; received in revised form 21 September 2007; accepted 21 September 2007
Abstract A simple method for estimating shape functions of optical spectra is proposed, based on the numerical correlation with an assumed optical spectrum composed of a central main rectangular component and two right-angled triangular wings on either side of the main component. The degrees of correlation between observed and assumed spectra were examined using spectral and coherence widths of those spectra and pulsewidths of Fourier-transform-limited pulses calculated from those spectra. By using this method, shape functions of output spectra from superluminescent diodes and a self-pulsating laser diode were evaluated in detail over a relatively wide injection current range beyond their rated currents. Ó 2007 Elsevier B.V. All rights reserved. PACS: 47.50.Ef; 42.55.Px Keywords: Broadband spectrum; Spectral shape function; Gaussian spectrum; Superluminescent diode; Self-pulsation; Gain-narrowing
1. Introduction In recent years, broadband light sources, such as femtosecond lasers and superluminescent diodes, have been used in a wide range of fields, including metrology [1], biophotonics [2], and laser processing [3]. Although the broadband spectra of such devices are often evaluated simply by using only the spectral width (usually the full width at half maximum, FWHM), their shape functions also include important information about the spectra. In OCT (optical coherence tomography) measurements, for example, the coherence length of a light source, corresponding to the longitudinal spatial resolution, is determined not only by the spectral width but also by the spectral shape function [4]. When the spectral width is constant, spectral shape functions that widen toward the bottom, such as a Lorentzian spectrum, make the coherence length shorter. It is recognized, however, that a Gaussian spectrum is preferable *
Corresponding author. Tel.: +81 72 254 9264; fax: +81 72 254 9908. E-mail address: [email protected] (K. Wada).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.09.042
for use in OCT measurements [5] because the autocorrelation functions obtained by the inverse Fourier-transform of such Lorentz-like spectra have fatter tails, resulting in the masking of weak signals in high-contrast images. Therefore, the shape functions of optical spectra should be carefully estimated. In spectroscopy, a valuable numerical technique for fitting emission or absorption spectra to determine the influence of Doppler broadening on intrinsic emission or absorption is to use Voigt-profiles, which are the convolution of Lorentzian and Gaussian profiles; this allows the temperature of the emitting or absorbing layers [6] or the total pressure of a gas atmosphere [7] to be estimated. When estimating the spectral shape functions of optical spectra, however, in many cases the observed spectra are numerically fitted using typical functions (Lorentzian, Gaussian, hyperbolic secant squared, etc.) without such a theoretical grounding. With such an approach, however, it seems to be difficult to obtain a quantitative evaluation of the difference between shape functions of observed and assumed spectra.
K. Wada et al. / Optics Communications 281 (2008) 368–373
In this Letter, we propose a simple method for estimating shape functions of observed optical spectra using three FWHM values, namely, spectral widths and coherence widths of observed spectra and pulsewidths of Fouriertransform-limited pulses calculated from those spectra. Using this method, the features of the shape function of an observed spectrum are extracted and compared with those extracted from the assumed functions. 2. Numerical model
pffiffiffiffiffiffiffiffiffiffi P ðf Þ expði/Þ (/: spectrum phase). Because phase problems are of no concern in the estimation of spectral shape, for simplicity, a Fourier-transform-limited relation between time and frequency domains is assumed to hold in this numerical study. Under this assumption, the value of / becomes constant for all spectrum data, and Fourier-transform-limited pulses are found in the time domain as a result of the inverse complex Fourier-transform of the complex spectra. Then, pulsewidths Dt (FWHM) of the obtained pulses are measured to get values of TBP (=Dt Æ Df0). Thus, the method of computation is just the same for both the TBP and CBP cases, except for the input spectrum data type. Although both parameters are used in discussions of pulse- or spectrum-shape functions, they have different uses. To make this clear, we show the following numerical results. Fig. 2 shows variations of the TBP (a) and the CBP (b) as a function of the wing width when the wing height is set as a running parameter. It is found from Fig. 2 that the CBP is sensitive to the changes of the wing height, whereas the TBP is not sensitive. This difference enables us to estimate spectral shape functions. In Fig. 3, an example can be seen for the loci of combination values (wing height, wing width) satisfying the conditions of TBP = 0.44 and CBP = 0.88. The two loci have only one intersecting point of (0.30, 0.87), based on the above different sensitivities to the changes of the wing height. This means that there exists a one-to-one correspondence between the combination values of (wing height, wing width) and (CBP, TBP). By estimating the CBP and TBP values from an experimentally observed optical spectrum, we can therefore uniquely determine a particular spectral
a
TBP
Initially, we assume an optical power spectrum composed of a central main rectangular component with a bandwidth of Df0 and two right-angled triangular wings on either side of the main component, as shown in Fig. 1. The height of the main component is normalized to be 1.0 and the wing height and width are also normalized by 1.0 and Df0, respectively. Varying the wing height and width allows a variety of spectral shapes to be modeled. We expect that the shape functions of observed spectra can be estimated by examining the degrees of correlation between those assumed shapes and the actual shapes of experimentally observed spectra. To this end, two dimensionless parameters are used: one is the timebandwidth product (TBP) and the other is the product of coherence time and bandwidth, termed here the coherence time-bandwidth product (CBP). The former has ordinarily been used in pulse chirp estimation problems [8,9], and the latter has often been used in OCT measurements in terms of spatial resolution [4,10]. According to the Wiener– Khintchine theorem, coherence widths Ds are measured as the FWHM of autocorrelation functions C(s) obtained by the inverse Fourier-transform of power spectrum data R1 (CðsÞ ¼ 1 P ðf Þ expði2pftÞdf , P(f): power spectrum data, f: frequency). The values of CBP are then calculated simply by Ds Æ Df0. In the case of TBP, the power spectrum data are transformed into complex spectrum data in the form
0.8
Wing height
0.6
0.1 0.15 0.2
0.4
0.3 0.4 0.5
0.2 1.0
0 0
Spectral intensity
369
0.5
1.0
1. 5
2.0
2. 5
3.0
b 1.5
Δf0
Wing height 0.1
CBP
Wings
Wing height
1.0
0.15 0.2
0.5 0.3
0
Wing width
Frequency
Fig. 1. Optical power spectrum model prepared for numerical correlation with observed spectra. Upper figure shows variation of spectral shape when wing width is varied.
0.4
0.5
0 0
0.5
1.0
1.5
2.0
2. 5
3.0
Wing width Fig. 2. Variations of (a) TBP and (b) CBP as a function of wing width when wing height is set as a running parameter, in the range of 0.1–0.5.
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K. Wada et al. / Optics Communications 281 (2008) 368–373 1.0
Normalized spectral intensityy
Wing width
1.2
1.0
0.5
Δf0
0.3 0
TBP = 0.44
Frequency
0.87
Fig. 5. Comparison of Gaussian spectrum with numerical spectrum model having the same combination value (CBP, TBP) as Gaussian spectrum.
0.8
CBP = 0.88 0.6 0.20
0.25
0.30
0.35
0.40
0.45
0.50
Wing height Fig. 3. Loci of combination values (wing height and wing width) satisfying conditions of TBP = 0.44 (solid) and CBP = 0.88 (dashed).
shape, in the form of a rectangular component and accompanying right-angled triangular wings, which corresponds best to the observed one. Fig. 4 shows an example of the assumed spectral shapes. The three spectra at the left have the same TBP value of 0.44 and the ones at the right have the same CBP value of 0.88 when the wing heights are set to 0.15, 0.3, and 0.5, respectively, under a constant bandwidth Df0. The middle spectra for the wing height of 0.3, which are identical to each other, have a combination value (CBP, TBP) of (0.88, 0.44), corresponding to that of a Gaussian spectrum [8] and also to the above intersecting point. As shown in Fig. 5, the Gaussian spectrum with a bandwidth of Df0 and a normalized height of 1.0 is then TBP = 0.44
superimposed on the middle spectrum in Fig. 4, showing a good match in terms of shape. We then plot loci of the combination values as coordinates on the TBP–CBP plane in Fig. 6 by varying the wing width when the wing height is set as a running parameter. The wider the wing width, the smaller the TBP and the CBP. The combination values corresponding to typical shape functions (rectangular, Gaussian, hyperbolic secant squared (sech2), and Lorentzian) are listed in Table 1 and plotted in Fig. 6. It is found that the transition of the shape
CBP = 0.88
1.0
Normalized spectral intensity
0.5
Δf0
Δff0
Fig. 6. Loci of combination values depicted by varying wing width when wing height is set as a running parameter, in the range of 0.1–0.5. The combination values of four typical shape functions are plotted on TBP– CBP plane.
0
1.0
0.79Δf0
0.69Δf0
Table 1 TBP and CBP values of typical shape functions
0.5 0.3 0
1.0
/2 0.87Δf0
0.87Δf0
0.5 0.15 0
1.01Δf0
Frequency
1.69Δf0
Frequency
Fig. 4. Spectral shapes obtained under conditions of TBP = 0.44 (left) and CBP = 0.88 (right) when wing heights are set to 0.5, 0.3, and 0.15, respectively.
/2
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function from rectangular to Lorentzian, via Gaussian and sech2, on the TBP–CBP plane shows a tendency to increase the wing width and decrease the wing height, simultaneously. By plotting the combination value (CBP, TBP) of an experimentally observed optical spectrum on the TBP–CBP plane and comparing it with those of typical assumed shape functions, we can roughly estimate the shape function of the observed optical spectrum without deriving the wing height and width from the combination value (CBP, TBP). 3. Experimental
16 12 8 4 0 30
4. Results and discussion
35
40
45
50
55
60
Injection current (mA)
Fig. 7 shows variations of spectral widths Df of the SLD1 (squares) and SLD2 (triangles) outputs, respectively, when the injection current was varied. By setting the ratios of the instrumental resolutions to observed spectral widths less than 1% throughout the experiment, estimation errors on the TBP and CBP values were suppressed within ±2%. In the case of SLD1, the spectral width was approximately constant to 6.5 THz over the range of injection currents. As for SLD2, the spectral width became narrower as the injection current was increased. This tendency was particularly significant at injection currents beyond the rated current (90 mA). This is thought to be due to gain-narrowing caused by the increase in optical oscillation at high injection currents. We then calculated pulsewidths Dt and coherence widths Ds from the observed spectra and plotted combination values (CBP, TBP) on the TBP–CBP plane in Fig. 8. In the case of SLD1, the locus of the combination value started with a point (at 60 mA) near the sech2-point and ended with a point (at 120 mA) near the Gaussianpoint, following a small arc. From this locus, it was found
Spectral width (THz)
Fig. 8. Loci of combination values of output spectra for SLD1 (squares) and SLD2 (triangles) when injection current was varied. The values indicate injection current in mA.
Spectral width (THz)
We prepared two commercially available 840-nm superluminescent diodes (SLD1: Anritsu, AS8C1110Z40M, and SLD2: made by a different manufacturer) and a 660-nm self-pulsating laser diode (PLD: Matsushita, LNCQ08). The output spectra from those diodes were observed with an optical spectrum analyzer (Yokogawa, AQ6317C) at wavelength resolutions of 0.01–0.1 nm (= 4.25–42.5 GHz at 840 nm). The injection current was varied over a relatively wide range beyond their rated currents.
Fig. 9. Variation of spectral width of PLD output spectra with respect to injection current.
that the spectra observed around the rated current (75 mA) had slightly lower wing heights than the sech2-profile in terms of our spectrum model. As for SLD2, the locus of the combination value started with a point (at 50 mA) below the sech2-point and ended with a point (at 120 mA) near the Lorentzian-point, following a relatively long linear curve with a small jump. The combination value at the rated current (90 mA) was found to occupy almost the midpoint between the sech2-point and the Lorentzianpoint on the TBP–CBP plane. Thus, the gain-narrowing effect caused notable spectral shape variation. The PLD was also examined in the same way. Fig. 9 shows the variation of spectral width with respect to the
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SLD2
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70
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90
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Injection current (mA) Fig. 7. Variation of spectral widths of output spectra for SLD1 (squares) and SLD2 (triangles) with respect to injection current.
Fig. 10. Loci of combination values of PLD output spectra when injection current was varied. The values indicate injection current in mA.
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injection current, and Fig. 10 shows the locus of the combination value on the TBP–CBP plane. It is interesting to note that the locus followed a deformed elliptical orbit whose apsides were located in the vicinity of the sech2point and the Lorentzian-point. The PLD operated as a light-emitting diode when the injection current was below the threshold of 54 mA. In this current region (specifically, 30–54 mA), gain-narrowing took place in the PLD, similarly to the SLD2, showing a marked decrease of the spectral width from 14 to 4 THz in Fig. 9. The corresponding locus is depicted as the upper part of the elliptical orbit in Fig. 10, showing the transition of the shape function from near sech2 to almost Lorentzian. When the injection current was increased beyond the threshold, the PLD oscillated in self-pulsation mode, and the spectral width became approximately constant (around 1 THz), as seen in the current range of 56–60 mA in Fig. 9. Although the variation of spectral width in the current range of 54–60 mA was small in contrast to the case in the above gain-narrowing region, the corresponding combination value changed significantly and formed the lower part of the elliptical orbit in Fig. 10. This means that the transition of the oscillation mode from
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spontaneous emission to self-pulsation was accompanied by a strong spectral shape variation. It is clear from Fig. 10 that different mechanisms of spectral shape variation brought about different routes between positions near the sech2-point and the Lorentzian-point. By examining the values of the TBP and CBP parameters, we can thus easily obtain detailed information about the spectral shape of the optical spectra. To confirm the validity of this method, we show examples of observed spectra and numerical fitting spectra in Fig. 11. In the case of multi-longitudinal mode spectra shown in Fig. 11b and c, the TBP and CBP values were estimated from envelopes of those spectra. To get fitting spectra, the wing heights and widths were derived from the TBP and CBP values of observed spectra by the same way as shown in Fig. 3. The numerical fitting spectra corresponded well with the observed spectra and were found to be applicable even to some observed spectra with asymmetrical shapes because the combination values of those asymmetrical spectra were also uniquely determined. 5. Conclusions We proposed a simple method for estimating shape functions of observed optical spectra based on numerical correlation with an assumed power spectrum-shape composed of a central main rectangular component and two right-angled triangular wings on either side of the main component. Using this method, we examined the shape functions of output spectra from superluminescent diodes and a self-pulsating laser diode. As a result, the shape functions of the observed spectra were estimated in detail by comparing them with typical assumed shape functions, and the influence of gain-narrowing or self-pulsation, occurring in those emitting diodes, on the spectral shape variation was also evaluated.
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Acknowledgements 0.2 0 2.00Δf0
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The authors are indebted to Dr. M. Yuri of Matsushita Electric Co. for providing the laser diode samples and to Mr. S. Hiroshima of Koyo Electric Co. for lending the optical spectrum analyzer used in this work. This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology under Grant-in-Aid for Scientific Research (C) No. 17560035. References
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Fig. 11. Examples of observed spectra and corresponding numerical fitting spectra: (a) SLD1 output spectrum at 100 mA, and PLD output spectra at (b) 40 mA and (c) 52 mA, under spectral resolutions of (a), (b) 42.5 GHz, and (c) 21.25 GHz, respectively.
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