MEASUREMENT of gain from amplified spontaneous

Transcription

1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 2, FEBRUARY Fourier Series Expansion Method for Gain Measurement From Amplified Spontaneous Emission Spectra of Fabry Pérot Semiconductor Lasers Wei-Hua Guo, Qiao-Yin Lu, Yong-Zhen Huang, Senior Member, IEEE, and Li-Juan Yu Abstract A gain measurement technique, based on Fourier series expansion of periodically extended single fringe of the amplified spontaneous emission spectrum, is proposed for Fabry Pérot semiconductor lasers. The underestimation of gain due to the limited resolution of the measurement system is corrected by a factor related to the system response function. The standard deviations of the gain reflectivity product under low noise conditions are analyzed for the Fourier series expansion method and compared with those of the Hakki Paoli method and Cassidy s method. The results show that the Fourier series expansion method is the least sensitive to noise among the three methods. The experiment results obtained by the three methods are also presented and compared. Index Terms Amplified spontaneous emission (ASE), Fourier series, gain measurement, Hakki Paoli, semiconductor lasers. I. INTRODUCTION MEASUREMENT of gain from amplified spontaneous emission (ASE) spectra of Fabry Pérot semiconductor lasers is of interest for a long history [1] [7]. The Hakki Paoli (HP) method [1], which uses the intensity ratio of adjacent peaks and valleys to calculate gain, is influenced by the resolution of the measurement system [2], [3]. The measured ASE spectra are the convolution of the system response function with the intrinsic laser ASE spectra. This unavoidable convolution process yields an average effect that causes the peak intensities to decrease and the valley intensities to increase. Thus, the HP method that directly uses the peak valley ratio to derive gain would yield underestimation. To overcome this limitation, Cassidy proposed a modification scheme that uses the ratio of the average mode intensity instead of the peak intensity to the valley intensity to calculate gain [2]. Because the average mode intensity is not influenced and the valley intensity is slightly influenced by the resolution of the measurement system, Cassidy s method provides great improvement to the accuracy of measured gain and makes gain measurement above threshold possible. However, Cassidy s method is sensitive to noise provided that no averaging is applied to the valley intensities [4]. The average process would injure the accuracy of Cassidy s method unless an iterative deconvolution procedure is used [2]. Recently the Fourier transform method that makes use of Manuscript received July 17, 2003; revised October 13, This work was supported by the National Nature Science Foundation of China under Grant , Major State Basic Research Program under Grant G , and the Project of 863 plan. The authors are with the Institute of Semiconductors, Chinese Academy of Sciences, Beijing , China ( gwh@red.semi.ac.cn). Digital Object Identifier /JQE the periodic property of ASE spectra was brought forward to measure gain spectra [5]. A deconvolution process combined with the Fourier transform method can eliminate the effect of convolution with the system response function [6]. However, the derived gain spectra have fluctuations at both sides of the measured wavelength range because of the window effect [5], [6]; furthermore, it cannot be applied to several longitudinal modes. Recently, we proposed a new method that uses Fourier series expansion of periodically extended single fringe of the ASE spectrum to calculate gain [7]. In this paper, we give a detailed description of the Fourier series expansion (FSE) method, and we analyze the noise influence on the gain reflectivity product calculated by this method, the HP method, and Cassidy s method. The results manifest that the FSE method is less sensitive to noise compared with the other two methods. II. THEORY The ASE spectra of Fabry Pérot semiconductor lasers can be described by [2], [8] is the wavenumber, is the single-pass ASE, is the reflectivity, and is the gain-reflectivity product is the mode gain, is the cavity length, and is the phase defined as is the effective index. Considering one longitudinal mode as shown in Fig. 1, the phase can be expanded around as is the wavenumber corresponding to the peak of the mode. Taking the linear approximation over one longitudinal mode range, the phase can be approximated by is the group index at. is a multiple of and is assumed to be zero in the following. We periodically (1) (2) (3) (4) (5) /04$ IEEE

2 124 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 2, FEBRUARY 2004, (10) can be modified as and (11) written as. With phase as the argument, (11) can be (12) Fig. 1. Illustration of the longitudinal mode of an ASE spectrum. (13) extend the longitudinal mode and calculate the Fourier series coefficients as is the th coefficient. The single-pass ASE and the gain reflectivity product are slowly varying functions of, so that they can be approximately regarded as constants in the longitudinal mode range, and (6) can be calculated as It is seen that the zeroth coefficient is the average mode intensity used by Cassidy [2]. The gain reflectivity product can be calculated by The mode gain can be calculated from (2) and (8) provided that the reflectivity and cavity length are known. Taking wavenumber as the argument, (6) can be written as is the wavenumber interval over the longitudinal mode as shown in Fig. 1. The integration (9) is usually used to calculate the Fourier series coefficients from the measured ASE spectrum. III. INFLUENCE OF RESOLUTION In practice, the measured ASE spectra are convolutions of the response function of the measurement system with the intrinsic ASE spectra of semiconductor lasers expressed in (1). Such a convolution process can be described by (6) (7) (8) (9) (10) is half of the bandwidth (such as the 20-dB bandwidth, out of which the response function can be seen as zero) of the response function and assumed to be less than half of the mode spacing;,. Taking wavenumber as the argument and the approximations of and. We can calculate the Fourier series coefficients from the convoluted ASE spectra as Substituting (12) into (14), we have (14) (15) Based on the convolution theorem [9], (15) can be simplified as (16) (17) The gain reflectivity product can be calculated from the convoluted ASE spectra by is a correction factor defined as (18) (19) According to the following noise analysis, we find that is preferable to be used in the calculation of the gain reflectivity product. Assuming to be the resolution bandwidth of the measurement system that is generally defined as the full-width at half-maximum (FWHM) of the response function, we can calculate the correction factor for the triangle response function case as follows: and for the rectangle response function case (20) (21) is the number of resolution bandwidth of the measurement system contained in the longitu-

3 GUO et al.: FOURIER SERIES EXPANSION METHOD FOR GAIN MEASUREMENT FROM ASE SPECTRA OF FABRY PÉROT SEMICONDUCTOR LASERS 125 Assuming points sampled in one longitudinal mode, from (24) and(27) we can calculate the Fourier series coefficients with noise as (28) (29) (30) Fig. 2. Mode gain corrections versus the ratio of the longitudinal mode spacing to the FWHM of the response function for a laser with a cavity length of 250 m. The mode gain corrections calculated from practical response functions with the resolution bandwidth of 0.1, 0.2, and 0.5 nm are also plotted as symbols. (31) The gain reflectivity product with noise can be calculated by the Fourier series expansion method as dinal mode spacing and mode spacing. The mode gain correction is is the (22) In Fig. 2, we plot the mode gain correction varying with for a cavity length of 250 m for both triangle and rectangle response functions. It is shown that without the gain correction the mode gain error would exceed 1 cm if was less than 11 or 8 for the triangle or rectangle response function, respectively. (32) (8) and (28) are used in the derivation. Because the gain reflectivity product is less than 1, it is seen that decreases as increasing from (8). Hence, it is favorable to use in (32) to calculate the gain reflectivity product. Considering low noise cases, the standard deviation of in (32) can be obtained by using the error propagation equation (33) IV. INFLUENCE OF NOISE In the following, we assume that the resolution of the measurement system is high enough that we do not need to consider the resolution influence. The ASE spectra with noise can be expressed as (23) is the noise at phase, which has the mean of zero and standard deviation of and no correlation with other phases, i.e., (24) (25) (26) Considering the even symmetry of relative to, (6) can be modified as real integrations (27) For the HP method, the gain reflectivity product with noise can be calculated by (34) the noises at the peak and valley of the longitudinal mode and are assumed to be averaged from and sample points, respectively, and satisfy (29) and (31) with replaced by and. Using the error propagation equation, we obtain the standard deviation of in (34) as (35) For Cassidy s method, the gain reflectivity product with noise can be calculated by (36)

4 126 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 2, FEBRUARY 2004 integration is performed by the trapezoid integration scheme. For simplicity, the integration limits in (9) that should be the valleys determined by the fitting process are replaced by the nearest sample points. Thus, the Fourier series coefficients can be calculated as (38) Fig. 3. Proportional coefficients of the standard deviation of the calculated gain reflectivity product versus the gain reflectivity product for the HP method, Cassidy s method, and the FSE method. representing the noise of the average mode intensity in the longitudinal mode satisfies (29), and the standard deviation of in (36) is (37) for has been used in the derivation. The standard deviations of the gain reflectivity product calculated from the ASE spectra with noise are proportional to for the three methods; however, the proportional coefficients are different and depend on the number of averaging sample points and on the value of the gain reflectivity product. In Fig. 3, we plot the proportional coefficients versus the gain reflectivity product as dashed lines. Taking,, and, which are typical values used to measure gain from practical ASE spectra, it is seen that the Fourier series expansion method has the standard deviations lower than the HP method and Cassidy s method. The HP method has lower standard deviations than Cassidy s method for the gain reflectivity product values approaching 1, however Cassidy s method is better than the HP method as the gain reflectivity product values approaching zero. If no average applied to the peak and valley intensities, i.e., and, the noise influence would be much larger for the HP method and Cassidy s method, especially for Cassidy s method, as shown in Fig. 3. V. NUMERICAL SIMULATION For a Fabry Pérot semiconductor laser with a reflectivity of 0.32, cavity length of 250 m, and effective index of 3.2, we synthesize its ASE spectrum for one longitudinal mode near the wavelength of 1550 nm from (1). First, we assume that the mode gain takes the value of 30 cm and the single-pass ASE is invariant with wavelength. The ASE spectrum is sampled with a wavelength step of 0.01 nm. Three points around the peak and five points around the valleys are selected to determine the peak and valley positions by using the parabolic fit. Then the Fourier series coefficients are calculated according to (9), in which the and are the nearest sample points to the fit-determined valley positions of and, and at the valleys. Neglecting the second item on the right-hand side of (38), the error of the gain reflectivity product calculated from (8) is 0.073%, and the error reduces to 4.5 if considering the contribution of this term. Then we consider the influence of the fringe unsymmetry to its center induced by the mode gain and the single-pass ASE quickly varying with wavelength. The mode gain is assumed to vary linearly with wavelength as cm. The single-pass ASE is assumed to have a Lorentzian form with the FWHM of 40 nm and center at 1570 nm. The ASE fringe at 1550 nm is unsymmetric to its center with the intensity difference of 0.25 db between the left valley and the right one, which is similar to that in the measured ASE spectra in the short-wavelength range away from the gain peak. The error of gain reflectivity product obtained by the Fourier series expansion method increases to 0.77% under this situation. To improve accuracy the single fringe can be separated into two parts: one is from the left valley to the peak, and the other is from the peak to the right valley. For each of them the Fourier series coefficients can be calculated by (27) under the symmetry assumption. The integration accuracy can be improved by the procedure similar to (38) even if the peak and valley positions are not the sample points. The gain reflectivity product calculated from these two parts are and , and the error of the mean value reduces to 0.098%. For the HP method and Cassidy s method, the errors are % and 0.31%, respectively, if the mean value of the valley intensities is taken. Finally, we add the synthesized ASE spectra with Gaussian distributed noises. The mode gain and the single-pass ASE are assumed invariant with wavelength again. One hundred and one points are sampled over one longitudinal mode and in (33) is set to be The number of points used for averaging the peak and valley intensities are taken to be and, and and, respectively, just as in the above theoretical analysis. The obtained standard deviations of the gain reflectivity product are plotted in Fig. 3 as solid lines for the FSE method, the HP method, and Cassidy s method. The results of the standard deviation are obtained from 500 tests. It is seen that the numerical results are in good agreement with the theoretical curves. VI. MEASUREMENTS A compressively strained multiquantum-well (MQW) semiconductor laser lasing at 1.55 m with a threshold current of

5 GUO et al.: FOURIER SERIES EXPANSION METHOD FOR GAIN MEASUREMENT FROM ASE SPECTRA OF FABRY PÉROT SEMICONDUCTOR LASERS 127 limit replaced by their nearest sample points and then derive the mode gain from (18) and (2). The calculated mode gain spectra are shown in Fig. 5(a), and their differences (a) (a) (b) Fig. 4. (a) Response functions of the OSA at resolution bandwidths of 0.06, 0.1, 0.2, and 0.5 nm. (b) ASE spectra of three longitudinal modes measured with the OSA resolution bandwidths of 0.1, 0.2, and 0.5 nm. (b) 12 ma is used to measure the gain spectrum. The cavity length is measured to be 250 m and the reflectivity of the cleaved facet is calculated to be The ASE spectrum from one cleaved facet of the laser is received by a lensed single-mode fiber and passed through an Agilent 8169A Polarization Controller to pick up the TE mode emission and then recorded by an Agilent B optical spectrum analyzer (OSA). We use the OSA with different resolution bandwidths to measure the spectra of an Agilent A tunable laser that is emitting at 1550 nm with an output power of 1 mw. These spectra are considered to be the response functions of the OSA because the linewidth of the tunable laser is much narrower than the resolution bandwidths of the OSA, of which the smallest is 0.06 nm. The obtained response functions for the resolution bandwidths are shown in Fig. 4(a). The ASE spectra of the laser at the injection current of 10 ma are measured by the OSA with a resolution bandwidth of 0.06, 0.1, 0.2, and 0.5 nm and a wavelength step of 0.01 nm. Three fringes of the measured ASE spectra under the resolution bandwidth of 0.1, 0.2, and 0.5 nm are plotted in Fig. 4(b). After determining the peak and valley positions of the ASE spectra by parabolic fit of 5 and 21 points selected from around the peaks and valleys, we calculate the Fourier series coefficients from (9) with the integration Fig. 5. (a) Calculated mode gain spectra by the FSE method with the gain correction for the resolution bandwidths of 0.06, 0.1, 0.2, and 0.5 nm. (b) Mode gain differences with reference to the gain spectrum obtained at the resolution bandwidth of 0.06 nm. with reference to the mode gain spectrum obtained under the resolution bandwidth of 0.06 nm are shown in Fig. 5(b). The maximal mode gain difference for the whole spectrum is only 1.3 cm even with a 0.5-nm resolution bandwidth, which is about 1/3 of the longitudinal mode spacing. At the center of the mode gain spectrum the difference about 0.5 cm is obtained for the 0.5-nm resolution bandwidth and 0.2 cm for the 0.2-nm resolution bandwidth. The mode gain corrections calculated by (17), (19), and (22) from the response functions shown in Fig. 4(a) with the mode spacing of 1.3 nm are plotted as symbols in Fig. 2. The obtained mode gain corrections are in agreement with the line of the rectangle function at the 0.2- and 0.5-nm resolution bandwidth and the line of the triangle function at the 0.1-nm resolution bandwidth, respectively. As shown in Fig. 4(a), the response functions with the resolution bandwidth of 0.2 and 0.5 nm is similar to rectangle functions and the 0.1-nm response function is close to the triangle one. With the same reference as the above, we plot the mode gain difference for the 0.1-, 0.2-, and 0.5-nm resolution bandwidths without the

6 128 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 2, FEBRUARY 2004 Fig. 8. Gain reflectivity-product spectra obtained by the FSE method with the wavelength step of 0.01 and 0.1 nm. Fig. 6. Mode gain differences for the FSE method without the gain correction and the HP method with reference to the gain spectrum at the 0.06-nm resolution bandwidth. increases, thus the noise influence is increased as indicated by (33) and the obtained gain reflectivity-product spectra are less smooth, which can be seen from Fig. 8. However, the integration accuracy is still high enough even for the wavelength step of 0.1 nm by the trapezoid integration scheme. VII. CONCLUSION Fig. 7. Gain reflectivity-product spectra calculated by the FSE method, the HP method, and Cassidy s method at the resolution bandwidth of 0.1 nm. gain correction in Fig. 6. Without the gain correction, the measured mode gain will be underestimated by cm at the resolution bandwidth of 0.5 nm, and the underestimation is cm and within 1 cm at the resolution bandwidth of 0.2 and 0.1 nm, respectively. The mode gain differences obtained by the HP method are also plotted in Fig. 6, which has a magnitude larger than that obtained by the FSE method under the same resolution bandwidth. In Fig. 7, we plot the gain reflectivity-product spectra obtained by the HP method, Cassidy s method, and the FSE method without the gain correction at the resolution bandwidth of 0.1 nm. We use the parabolic fit to determine the peak and valley intensities [8], which has the average effect but keeps the peak and valley intensities largely unaffected. The numbers of points used for fit are 5 for peaks and 21 for valleys. It can be seen that the HP method and the FSE method yield smoother spectra than Cassidy s method does, which is in agreement with the theoretical results in Fig. 3. Finally, we consider the effect of wavelength step on the obtained gain reflectivity-product spectra by the FSE method. The wavelength step takes the values of 0.01 and 0.1 nm, the resolution bandwidth is 0.1 nm, and the trapezoid integration scheme is used in (9) to calculate the Fourier series coefficients. The number of points sampled in one longitudinal mode decreases as the wavelength step We propose a new method based on the ratios of Fourier series coefficients to calculate gain from the measured ASE spectra of Fabry Pérot semiconductor lasers. With a correction factor derived from the response function of the measurement system, the gain can be obtained with high accuracy even if a very loose requirement of the resolution of the measurement system is applied. Without correction, this method results in better accuracy than the HP method does. The noise effect on the gain reflectivity product is also discussed. It is found that the Fourier series expansion method is less sensitive to noise than the HP method and Cassidy s method. REFERENCES [1] B. W. Hakki and T. L. Paoli, Gain spectra in GaAs double-heterostructure injection lasers, J. Appl. Phys., vol. 46, no. 3, pp , [2] D. T. Cassidy, Technique for measurement of the gain spectra of semiconductor diode lasers, J. Appl. Phys., vol. 56, no. 11, pp , [3] L. A. L. S. Cho, P. M. Smowton, and B. Thomas, Spectral gain measurements for semiconductor laser diodes, Proc. Inst. Elect. Eng., pt. J, vol. 137, no. 1, pp , [4] V. Jordan, Gain measurement of semiconductor laser diodes: Requirements for the wavelength resolution and sensitivity to noise, Proc. Inst. Elect. Eng., pt. J, vol. 141, no. 1, pp , [5] D. Hofstetter and J. Faist, Measurement of semiconductor laser gain and dispersion curves utilizing Fourier transforms of the emission spectra, IEEE Photon. Technol. Lett., vol. 11, pp , Nov [6] W. H. Guo, Y. Z. Huang, C. L. Han, and L. J. Yu, Measurement of gain for Fabry-Pérot semiconductor lasers by the fourier transform method with a deconvolution process, IEEE J. Quantum Electron, vol. 39, pp , June [7] W.-H. Guo, Q.-Y. Lu, Y.-Z. Huang, and L.-J. Yu, Measurement of gain spectrum for semiconductor lasers utilizing integrations of product of emission spectrum and a phase function over one mode interval, IEEE Photon. Technol. Lett., vol. 15, pp , Nov [8] C. S. Chang, S. L. Chuang, J. R. Minch, W. W. Fang, Y. K. Chen, and T. Tanbun-Ek, Amplified spontaneous emission spectroscopy in strained quantum-well lasers, IEEE J. Select. Topics Quantum Electron., vol. 1, no. 4, pp , [9] R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. New York: McGraw-Hill, 1978, p. 108.

7 GUO et al.: FOURIER SERIES EXPANSION METHOD FOR GAIN MEASUREMENT FROM ASE SPECTRA OF FABRY PÉROT SEMICONDUCTOR LASERS 129 Wei-Hua Guo was born in Hubei Province, China, in He received the B.Sc. degree in physics from Nanjing University, Nanjing, China, in He is currently working toward the Ph.D. degree at the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China. He is currently studying the FDTD simulation and fabrication of photonic microcavities, photonic crystals, and semiconductor optical amplifiers. Yong-Zhen Huang (M 95 SM 01) was born in Fujian Province, China, in He received the B.Sc., M.Sc., and Ph.D. degrees in physics from Peking University, Beijing, China, in 1983, 1986, and 1989, respectively. In 1989, he joined the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, he worked on the tunneling time for quantum barriers, asymmetric Fabry Perot cavity light modulators, and vertical-cavity light-emitting lasers (VCSEL). In 1994, he was a Visiting Scholar at BT Laboratories, Ipswich, U.K., he was involved in the fabrication of the 1550-nm InGaAsP VCSEL. Since 1997, he has been a Professor with the Institute of Semiconductors, Chinese Academy of Sciences, and is the Vice Director of the Optoelectronic R & D center. His current research interests are microcavity lasers, semiconductor optical amplifiers, and photonic crystals. Qiao-Yin Lu was born in Jiangsu Province, China, in She received the B.Sc. and M.Sc. degrees in optics from Changchun Institute of Optics and Fine Mechanics, Changchun, China, in 1998 and 2001, respectively. She is currently working toward the Ph.D. degree at the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China. She is currently studying the design and the fabrication of microcavity lasers. Li-Juan Yu was born in Heilongjiang Province, China, in She received the B.S. and M.Sc. degrees in solid physics from Jilin University, Changchun, China, in 1986 and 1989, respectively, and the Ph.D. degree in microelectronics from Xi an Jiaotong University, Xi an, China, in She joined the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, in 2000, working on material growth by metal organic chemical vapor deposition and the fabrication process of semiconductor optical amplifiers. She is currently an Associate Research Professor at the Institute of Semiconductors, Chinese Academy of Sciences, her research interest is in the material growth and the process techniques of optoelectronic devices.