The optical gain in a semiconductor laser is an essen-tial parameter in characterizing fabricated lasers andsimulating their behavior. The most commonly usedmethod to measure the gain in laser cavities is theso-called Hakki–Paoli method.1 In this method the op-tical gain is derived from the contrast ratio of themodulations in the spectrum of the amplified stimu-lated emission (ASE) caused by the resonances of thelaser cavity operating below threshold. The main ad-vantage of this method is that no external source,wideband antireflection coating of the laser facets,or accurate values of the coupling efficiency are re-quired. This intensively used method is, however,sensitive to noise, and its results are significantlyinfluenced by the response function of the spectrom-eter. Cassidy2 was first to improve the method byintroducing a ratio between the integral of the inten-sity of a mode peak and the minimum intensity. Al-though this made the method less sensitive to theresolution of the optical spectrum analyzer (OSA),both methods are still sensitive to noise. Wang andCassidy3 have recently proposed and demonstrated amethod using a nonlinear least-squares fitting of theFabry–Perot equation that is less sensitive to noise.
By taking into account all the data points of thespectrum, it improves the accuracy as compared toprevious methods. However, they had to introduce acorrection to take into account the resolution of theOSA. In this paper we present measurements takenusing an ultrahigh-resolution OSA (20 MHz, �0.16pm). Thus the effect of the response function of theOSA does not need to be compensated for. Such a highresolution provides very good accuracy in the gainmeasurement right up to the threshold of the laser.Also the cavity of the lasers under investigation canbe longer than 1 mm, which is the typical limit forstandard grating-based OSAs. This can be an issuewith devices with low gain, such as quantum dotlasers, or lasers in larger integrated circuits. We dem-onstrate this high accuracy method and combine itwith measurements of the optical transparency pointby determining the optical gain spectrum and thedifferential gain of an InP�InGaAsP Fabry–Perot la-ser structure as a function of injected carrier density.The results for a range of heat sink temperaturesat approximately room temperature are presented.These parameters will be used in our simulation mod-els for lasers fabricated with identical layer stacks.
2. Gain Measurement Method
The steady-state optical output spectrum of a Fabry–Perot laser below threshold is described by the fol-lowing Airy function equation4:
I��� � B����1 � RG��1 � R�
�1 � RG�2 � 4RG sin2�2�NgL� �, (1)
The authors are with the Institute of Communications Technol-ogy: Basic Research and Applications, Eindhoven University ofTechnology, Den Dolech 2, P.O. Box 513, 5600 MB, Eindhoven, TheNetherlands. Y. Barbarin’s e-mail address is [email protected].
Received 22 May 2006; accepted 31 July 2006; posted 15 August2006 (Doc. ID 71270).
where B��� is the total amount of spontaneous emis-sion represented by an equivalent input flux, G��� isthe single-pass modal gain, R is the laser facet reflec-tivity, L is the cavity length, and Ng is the group indexof the waveguide. The equation is rewritten and abackground level is added to obtain an equation thatis to be fitted to each mode peak in the recordedsubthreshold ASE spectrum in Eq. (2). No convolu-tion with the OSA response is necessary:
Ifit��� �C
�1 � RG�2 � 4RG sin2�2�NgL�1�
�1
�peak��
� BKG, (2)
where RG��� � RG��peak� � ��� � �peak�, C��� � B���1 � RG����1 � R� � C��peak� � ��� � �peak�, �peak isthe peak wavelength of the individual mode, and BKGis the background level. We assume that the sponta-neous emission B��� and the single-pass gain G���vary linearly with wavelength over one mode. Thusthe small asymmetry of the Fabry–Perot modes dueto the change of the gain and the ASE intensity withwavelength over the fitted range of approximatelyone free spectral range of the laser cavity is takeninto account. The fitting is done in a Matlab pro-gram using the weighted nonlinear least-squaresfitting function Isqnonlin from the optimizationtoolbox. First, a scan over the measured spectra isperformed to extract starting values for �peak andBKG. Since the spectrometer also provides accuratefrequency–wavelength differences between the modepeak positions, the group index Ng��� is calculated onthe different �peak by using this formula: Ng��peak
i�l
� �peaki��2 � ��peak
i�l � �peaki�2�8��peak
i�l � �peaki�L.
The group index as a function of wavelength is thenfitted over the total spectral range using the Cauchyformalism Ng��� � N0 � N1��2. Therefore the fittingparameters in the Airy function are C, PRG, �, �, BKG,and �peak. The weights used for the data points aredefined by
weight��� �1
��I����2 � BKG2, (3)
where I��� is the measured signal intensity, and BKG,the standard deviation of the background signal, isdetermined from areas in the spectrum with a verylow signal intensity. The parameter is chosen tominimize the residue of the fit. Tests have been per-formed for different spectrum intensities and InP-based lasers. The results show that a clear reductionof the residue of the fit is obtained for a low intensitysignal of � 0.15 � 0.02. The effect of is minor fora signal closer to the threshold.
3. Gain Curve Measurements
The measurements were performed on a Fabry–PerotInP�InGaAsP ridge waveguide laser on an InGaAsP
chip that was fixed onto a temperature-controlledcopper mount. The laser output was coupled into alensed fiber and led through an optical isolator to thehigh-resolution OSA (APEX AP2041A). The OSA isbased on a heterodyne receiver principle using asingle-mode tunable laser as a local oscillator thatenables it to achieve a resolution of 0.16 pm �20 MHz�and a wavelength accuracy of �3 pm. The spectrawere recorded in sections of 5 nm (20,000 points each)in order to have the full resolution available over thefull wavelength range of interest. The measurementof one section takes less than 5 s. All the measure-ments were performed below the lasing threshold. Nopolarizer was required since the polarization wasmeasured to be TE (30 dB polarizer extinction).
The laser layer stack consisted of a 120 nm thick� � 1.5 �m bulk InGaAsP layer between two 190 nmthick � � 1.25 �m InGaAsP layers. The structurewas clad by a 1500 nm thick p-InP layer with gradualdoping levels and a 50 nm p-InGaAs contact layer.The cavity was 1985 �m long ��1 �m�, and thewaveguide was 2 �m wide. The threshold was ap-proximately 92 mA at 16 °C and approximately113 mA at 28 °C. As can be seen in Fig. 1, the cavitymodes were very well resolved in this typical ASEspectrum. The result of a typical fit over three modesis presented in Fig. 2. The spectrometer fully resolvesthe modes that are fitted with an appropriate weight-ing of the data.
Once all the modes in the spectra are fitted, RGproducts can be plotted versus the wavelength for eachinjected current value. The results for T � 16 °C andT � 28 °C are plotted in Figs. 3(a) and 3(b). Thetypical gain shape is observed.5 Very few observedmodes did not lead to a good fit ��1%�. This happenedwhen an excess of noise was detected during the mea-surement. Those points were removed from furthercalculations on the gain curves. The gain peak shiftedover 8.0 nm from 16 °C to 28 °C. The gain peak wave-length shifted to the smaller values with an increasein carrier density as expected. The temperature ap-pears to have had an effect on the shape of the gain
Fig. 1. Typical example of a recorded subthreshold spectrum of aFabry–Perot laser on a full span range: T � 16 °C, I � 89 mA.
spectrum on the short wavelength side. To bring outthe differences in bandwidth observed at the two tem-peratures, the gain spectra recorded at 16 °C and28 °C are combined in Fig. 4. The gain spectra wereaveraged over 1 nm ��6 modes� to reduce the noise.The gain maxima of the data at the two temperatures
were overlaid to illustrate that the bandwidth in-creases slightly with temperature by approximately3 nm at the FWHM.
4. Discussion of the Method
Measurements of the gain curves by using the high-resolution spectrometer gave smooth curves for RGproduct values higher than 0.4; below this value RGproducts became inaccurate. The high resolutioninexorably limits the sensitivity of the equipment.The sensitivity of the instrument is specified for�75 dBm. However, a level down to �85 dBm hasbeen measured. On the other hand, high-quality mea-surements are obtained at current values very closeto the lasing spectrum. Wang and Cassidy reported inRef. 3 that the quality of their fitting of the modesbecame poor in the valleys in the gain regime whereRG came close to 1. To improve the fit, they tried tochange the weighting. Performing the fit to the log-arithm of the convolution of the Airy function andthereby significantly increasing the weight of thelower values’ data points, as well as including theirinstrument response function, did improve their re-sults. However, they could not successfully fit to val-ues of the RG product close to and over 0.95. Usingthe high-resolution spectrometer, we have been ableto fit on the experimental data RG products up to0.975 using weighting as given in Eq. (3), while main-taining an excellent agreement of the measured andfitted laser modes. Figure 5 shows on a logarithmicscale the results of the fit of three modes comparedwith the experimental data. The RG product is ap-proximately 0.972 for the three modes. All the mea-sured points of the mode peaks are distributed wellon both sides of the fitted curve. We observed thatabove the laser threshold the measured optical modebecomes wider and can no longer be fitted using theAiry function as is to be expected. The extracted RGproducts are clamped to 0.99, and the shape of themode starts to deviate from the Airy function.6 This is
Fig. 2. Measured and fitted spectra zoomed on three modes:T � 16 °C, I � 89 mA.
Fig. 3. RG product versus the wavelength and laser current for(a) T � 16 °C and (b) T � 28 °C.
Fig. 4. RG product for two temperatures, 16 °C and 28 °C, andthree similar RG values. The spectra for T � 28 °C have beenshifted by 8 nm to better see the changes in shape.
illustrated in Fig. 6 where a measured and a fittedmode are shown at 1 mA above threshold.
5. Differential Gain Measurement
From the previously measured gain curves, the differ-ential gain could be extracted. To improve the accuracyof the extracted differential gain per carrier, the gaincurves were smoothed over 6 modes ��1.0 nm�. Thenet modal gain �gnet� per meter for each wavelengthwas then calculated using Eq. (4). This net gain isequal to the material gain gm times the confinementfactor � minus the total optical losses of the cavity loss, which comprise the free carrier absorption withinthe active region and the losses due to scattering:
gnet �1L ln�RG
R �� �gm � loss. (4)
To determine the differential gain a relation be-tween the injected current and the carrier density isrequired. Below the laser threshold, N can be ex-tracted from the simplified rate by
N�
� BN2 � CN3 �I
qSActiveLayerL, (5)
where � is the carrier lifetime, B is the bimolecularrecombination coefficient, C is the Auger recombina-tion coefficient, I is the injected current, q is thecharge of the electron, SActiveLayer is the cross-sectionalsurface of the active layer of the semiconductor opti-cal amplifier, and L is the length of the laser cavity.All the values of the parameters used are listed inTable 1.
A contour plot of the net optical modal gain as afunction of carrier density and wavelength is pre-sented in Fig. 7. From this figure we see that the netmodal gain at one wavelength varies nonlinearlywith the carrier density, even over this small range ofcarrier densities. This is most pronounced at theshortest wavelength in the plot. A common descrip-tion for the relation between the material gain andthe number of carriers is given by7
gm � N � N0 ln� NN0
�. (6)
Here N is the differential gain factor, N is thecarrier density, and N0 is the transparency carrierdensity. The carrier density at transparency needs tobe known for each wavelength and temperature. This
Fig. 5. Measured and fitted spectra zoomed on three modes:T � 16 °C, I � 90 mA. The three modes are very well fitted.
Fig. 6. Measured and fitted spectra zoomed on a single mode:T � 28 °C, I � 114 mA. This is 1 mA above threshold; the Airyfunction is no longer valid.
Table 1. Values of the Parameters Used for the CarrierDensity Calculation
SActiveLayer Surface of the active region 0.12 �m � 2 �mL Length of the cavity 1.85 mm
Fig. 7. Contour plot of the measured net optical modal gain in thelaser as a function of carrier density and wavelength atT � 16 °C. At a fixed wavelength one can see that the gain does notincrease linearly with carrier density, especially at the shorterwavelengths.
was done by injecting light from a tunable laser (Agi-lent 81600B) that was modulated on–off at 1 kHz.The average power injected into the laser was�13 dBm. The current of the laser under test wasthen scanned, and the amplitude and phase of themodulation of the voltage over the laser as a result ofthe modulated input light was recorded using alock-in amplifier. At the transparency current theinteraction of the input light with the gain materialshould be minimal, and the amplitude of the modu-lation is at a minimum. A clear jump in the phase ofthe modulation was also observed at the transpar-ency point as the interaction of the laser light withthe gain material changes from absorption to ampli-fication. An example of an amplitude and phase sig-nal from the lockin amplifier is presented in Fig. 8. Inthis way we have measured the transparency currentfor 12 °C, 20 °C and 28 °C and for wavelengths be-tween 1510 and 1590 nm. The carrier densities werecalculated from the measured current by using Eq.(5). The results are presented in Fig. 9 for three
different temperatures. The transparency carrierdensity decreases almost linearly with the wave-length and increases significantly with temperature(�8.4 � 1021 m�3 per degree). To obtain the values ofthe transparency carrier densities required in thedifferential gain calculation for wavelength values inthe studied range and temperature, the presentedgraphs have been interpolated.
The discrete differential of the material gain�dgm�dN� is calculated in order to extract N��� with-out knowing the optical losses [Eq. (7)]. Once thedifferential gain factor is known one can extract theinternal losses from Eq. (4) using
�gm
�N ��
�N� 1L�
ln�RG�N, ��R ��� N
N0
N . (7)
Figure 10 shows the results of the differential gainfrom two temperatures (16 °C and 28 °C). We ob-served that at a fixed temperature, the differentialgain as well as the transparency carrier density de-crease linearly with the wavelength over the observedwavelength range. Figure 10 shows the wavelengths atwhich the device starts lasing for both temperatures,which indicates the maximum modal gain. This graphhas to be interpreted together with the transparencycarrier density graph for different wavelengths andtemperatures (Fig. 9). With an increase of tempera-ture, the maximum modal gain shifts to longer wave-lengths. The transparency carrier density increaseswith temperature, and this rise is larger for the short-est wavelengths. Meanwhile, the differential gain in-creases with temperature as well. We observed thatthe slope of the linear fits decreases slightly with tem-perature and that the differential gain is lower at themaximum gain. The values reported here are higherthan the typical values reported in the literature for
Fig. 10. Differential gain parameter N (�10�20 m2) as a func-tion of the wavelength and for the temperatures T � 16 °C (blackdiamonds) and T � 28 °C (gray squares). The wavelengths atwhich the device starts lasing for the two temperatures (the max-imum of the gain) are indicated.
Fig. 8. Measured amplitude and phase of the voltage modulationat the laser versus the current injection when a modulated lightfrom a tunable laser is injected. A clear transition in the phaseindicates the transparency current.
Fig. 9. Measured transparency carrier, recalculated in carrierdensities, as a function of the wavelength (at temperatures of12 °C, 20 °C, and 28 °C).
bulk InP�InGaAsP material.8–10 This difference stemsfrom the definition that we have used for this param-eter. The values listed here are the differential gains atthe transparency density. The carrier density at thelaser threshold is typically significantly higher to over-come the mirror losses. If a linear gain model is used todescribe the gain in a laser, the differential gain pa-rameter is usually determined near the laser thresh-old. Looking at Eq. (6) we see that a lower value of thedifferential gain parameter in the linear gain model isto be expected.
Once the differential gain is known, the lossescould be calculated. Values for 16 °C are 32 dB�cm,and the values for 28 °C are �33 dB�cm. We at-tribute this increase in losses with temperature to thehigher carrier concentration needed in the semicon-ductor at higher temperature.
6. Conclusion
We have demonstrated the use of an ultrahighresolution �20 MHz� spectrometer to accurately re-cord subthreshold ASE spectra from a Fabry–PerotInP�InGaAsP laser and determine the optical gain.The method is based on a nonlinear least-squaresfitting of the observed modes. The spectrometer fullyresolves the modes that could be fitted accurately,and the effect of the response function of the OSAdoes not need to be compensated for. A 1 mm longdevice is a typical limit for a standard spectrometer inthis wavelength range. Using the high-resolutionspectrometer allows us to measure devices longerthan this limit. Measurements have been performedon a 2 mm long laser cavity, and RG products up to0.975 have been measured without any discernabledifference between the measured and the fitted lasermodes. It has been observed that nearer to and abovethe lasing threshold the shape of the mode deviatesfrom the Airy function and cannot be fitted.6 Theoptical gain spectrum of the laser was measured suc-cessfully for different temperatures and subthresholdcurrent values. The net gain curves obtained confirmthe necessity of using the logarithmic relation be-
tween the gain and the carriers. The differential gainparameter in the gain relation was determined byusing the measured transparency carrier density val-ues for each wavelength and temperature. The pa-rameters determined in this paper will be used in ourlaser simulation models.
This research was supported by the NRC Photonicsprogram and the Towards Freeband CommunicationImpulse program of the Dutch Ministry of EconomicAffairs.
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