Comparison of different methods for optical gain spectroscopy

The internal losses of green and blue laser diodes are challenging to determine because of the narrow longitudinal mode spacing. Furthermore, the internal losses of state-of-the-art blue and green laser diodes are in the range of only a few inverse centimeter. Therefore the dynamical range given by the maxima and minima of the longitudinal mode spectrum is very large, even for moderate optical gain. Under these conditions, the usually employed, so-called Hakki–Paoli method to determine the optical gain becomes inaccurate. Now, we compare this with two other methods, the Cassidy method and an evaluation based on a Fourier transformation for a green laser diode. An error estimation as well as a correction of the systematic error caused by the spectral resolution of the setup were established. The overall highest gain was measured with the Cassidy method in the range of the lasing wavelength, as this method is least affected by the spectral resolution. In comparison of all methods, the highest gain for the wavelengths above the lasing wavelength is observed for one variation of the Fourier method, because background noise has the least influence on this method. For wavelengths below lasing wavelength we see similiar optical gain for all methods.


Introduction
Laser diodes made from GaN are important for many applications, for example for laser projectors, augmented and virtual reality and automotive lighting. 1,2) For the optimization and simulation of these laser diodes it is necessary to characterize the optical gain spectra as function of the driving current. 1,[3][4][5][6][7][8][9][10][11] To get information about the optical gain, the common method is to measure the modulation depth of the longitudinal mode spectrum of the laser diode, as shown in Fig. 1. The formation of the longitudinal mode spectrum can be described as interference in a Fabry-Pérot-etalon. 12) From the finesse of the peaks, it is possible to calculate the gain with the method of Refs. 3, 4. The optical gain is an important parameter for the quality of a laser diode, because it is possible to determine the internal losses, differential gain, and gain dispersion. 6,7) For state-ofthe-art blue and green laser diodes, there occur some challenges. The mode spacing is only about 0.06-0.08 nm, so a spectrometer with extremely high resolution is required to measure modulation depth of the peaks. 1) Additionally, the laser diodes have a broad spectrum below threshold, which lead to an other challenge, the intensities for the minima and the regions outside the peak wavelength are comparably small and therefore the signal to noise ratio becomes as well. 13) For known reflectivities and resonator length, the modal gain g can be calculated from the maxima and minima of the measured longitudinal mode spectrum.
and L is the resonator length, R 1 and R 2 are the mirror reflectivities and I max and I min is the intensity in the maxima and minima of the longitudinal mode spectrum, respectively.
The determination of the minima is a significant challenge and strongly dependent on the noise level. The wavelength resolution influences the spectrum in a way that the maxima are decreased while the peaks are widened. To mitigate the influence of the resolution, Cassidy suggests to take into account integrated intensity of a peak instead of the maxima to minima ratio. 14) This method is intended to be less dependent from the resolution of the spectrometer, as the integrated intensity does weakly depend on the resolution In this equation the numerator stands for the area below the peak and the denominator for the area between the minimum and the x-axis, respectively. A third way to determine the optical gain is to use a Fourier transformation of parts of the spectrum. This Fourier spectrum consists of a series of equidistant peaks. For high gain, the longitudinal gain spectrum is similar to a series of delta peaks, and the amplitude of the peaks in Fourier space are slowly decreasing with increasing order. For low gain the longitudinal mode spectrum is nearly constant with a small sinusoidal modulation, resulting in a Fourier spectrum where the peak amplitudes decrease fast with the order of the peak. References 15,16 showed that the slope of the decrease of these Fourier coefficients on logarithmic scale is proportional to the modal gain. However, due to the measurement errors, the amplitude of the peaks deviates from this expected exponential behavior. Therefor the calculated gain depends on the Fourier coefficients which are being used to calculate the gain by Fourier method. In particular, the zeroth Fourier peak is influenced by a constant background and by noise. The other peaks depend on the shape of the longitudinal mode peaks, and are therefore influenced by the resolution of the spectrometer (Fig. 5).
Different methods to evaluate the longitudinal mode spectrum and derive the optical gain have been compared for laser diodes in the telecommunication wavelength range. 17,18) Here, we compare these methods for a laser diode in the green spectral range.

Setup
Our setup for the measurement consists of the laser diode, lenses to collimate and focus the beam, a polarization filter, a chopper wheel and a pinhole. The setup is shown in Fig. 2. The last lens (L2) focuses the beam on the entry slit of the monochromator, which is a SPEX 1404 double spectrometer of 0.85 m focal length and a very high spectral resolution. The layout of the monochromator is drawn in Fig. 3. We measure the spectrum at the output slit with a photodiode (OPT301). The signal of the photodiode is amplified with a lock-in amplifier, which is locked to the frequency of the chopper wheel. The lock-in amplifier also reduces the noise. The noise level depends on the sensitivity, which we need to adjust to use the full dynamic range. To avoid temperature drift during our measurements, we regulate the temperature of our laser diode with two Peltier elements.
For an accurate measurement it is necessary that light from only one transverse mode of the laser cavity is collected. Any emission into other modes, out of plane, or of wrong polarization will contribute to background, reduce the observed modulation of the spectrum and thus decrease the measured optical gain. Therefore, a linear polarization filter is inserted to suppress TM modes. Furthermore, a m 200 m pinhole is inserted as spatial filter to select the light from the waveguide and suppress the influence of spontaneous emission into substrate. L1 and L3 form a telescope with a 1.25 × magnification. The pinhole therefore selects a area which is small compared to the size of the substrate, and most of the light scattered from rough sidewalls and bond interface towards the spectrometer can thus be suppressed. The double spectrometer has an additional slits between the two stages to suppress scattered light intensity and therefore improve the dynamic range.

Measurements
The measurements were done with a generic commercial green laser diode with a resonator length of 600 μm and the reflectivity of the mirror is given with 0.85. The threshold current of this diode is in the range of 39 mA and the slope efficiency is about -0.45 W A 1 . First, we measure the dark spectrum which depends on the sensitivity of the lock-in amplifier as described above. It is important to subtract the dark spectrum from the measured spectra to correct the light of the environment and the dark current of the photodiode without light. In the final step, we measure the longitudinal mode spectra below threshold.
Small wavelength sections of the longitudinal mode spectrum shown in Fig. 1 for wavelengths below, at, and above peak gain are presented in Fig. 4. The methods of Hakki-Paoli and Cassidy are illustrated at the lasing wavelength (middle picture) of Fig. 4. Hakki-Paoli calculates the gain from the maximum to minimum ratio, while Cassidy is based on the ratio of the area below the peak (light blue) to the area below the minima (dark blue).
To illustrate the used Fourier methods and the behavior at different sections of the longitudinal spectrum, the Fourier transformed spectra of these sections are plotted in Fourier space (Fig. 5). For wavelengths below lasing wavelength λ lase the influence of the resolution and also the noise level is small. At λ lase we see a strong dependence on the resolution  of the spectrometer: the peaks of higher order decreased in Fourier faster than linear. Above λ lase the low signal-tobackground ratio rises the zeroth order peak in Fourier space and lower dependence on the resolution. Because of these reasons, the slope of the linear fit to the peaks in Fourier space depend on the number and order of the peaks taken into account. We compared three versions of the Fourier method. In the first case, only the zeroth and first peak in Fourier space were taken into account. This minimizes the impact of the spectral resolution, however, it is most sensitive to background noise contributing to the zeroth peak. The second method takes the first and second peak into account. The error due to spectral resolution is still small, and the effect of background noise is mostly eliminated. For the third method, the Fourier peaks 1-4 were taken into account in the linear fit.

Error due to finite resolution
The resolution of the spectrometer can be determined, when we measure a single longitudinal mode of the laser diode spectrum above threshold. As the spectral width of the mode is small compared to the spectrometer resolution, the shape of the measured peak reflects the spectral resolution. We fit this spectral response function by a Gaussian function, where the full width half maximum (FWHM) can be calculated from the standard deviation σ with s = · · · FWHM 2 2 ln 2 . For a slit width of m 30 m, we obtain a spectral resolution of about 9 pm.
To estimate the error caused by the finite resolution of the spectrometer for the different methods, we simulate longitudinal mode spectra by convolution of an Airy function for given finesse with a Gaussian of given FWHM. Then the  error is determined by calculating the optical gain from these simulated curves for all three methods. In Fig. 7 the calculated optical gain is plotted as function of the finesse. For all methods the derived gain is lower than the ideal gain which was used as input for the simulated longitudinal mode spectra. For better visibility, this systematic error of the determined optical gain is plotted in Fig. 8, again as function of the ideal finesse. As expected, Cassidy is less sensitive to the resolution than Hakki-Paoli and Fourier. 14) For Hakki-Paoli and Cassidy the influence of the resolution differs for different finesses, while for Fourier the influence is constant for all finesses, but depends on the number of Fourier orders taken into account to calculate the optical gain.
The resolution of the spectrometer decreases the calculated gain for all methods. With the results shown in Fig. 6 we can correct this systematic error, which is caused by the resolution. The inaccuracy of the fit with the Gaussian function results in a secondary error. This is less than the systematic error and can decrease or increase the gain. It is therefore added to the standard deviation of the measured gain.

Standard deviations for dark spectra, minima and maxima
Another factor for the calculation of the gain is the accuracy of the used measuring devices. Besides resolution, the detector noise is an important source for measurement errors. This intensity noise depends on the settings of the sensitivity of the lock-in amplifier. This noise leads to an error for the mean of the dark spectra, for the determined maxima, minima and also for the integral used in the Cassidy method. Consequently we have three contributions to statistical error, where the variations affect the calculated gain for both Hakki-Paoli and Cassidy and one parameter (the dark spectrum) for Fourier including the zeroth peak. We calculate the standard deviation for the measured dark spectra and weighted it for the different parameters of influence.
where N is the number of points, for which we need the standard deviation. To keep the error of the mean of the dark spectra small, we take the mean for 2000 points. In deriving the error bars for the different methods, we take into account that the background noise level is added to only one point for the maximum in Hakki-Paoli method, but for all points contributing to the integral in the Cassidy method. Thus we multiply this error with the number of points between two neighboring peaks. The error of the minima is determined in a different way, as we fit points of the minima applying a parabolic fit, as shown in Fig. 4. The standard deviation of this fit is also divided by the square root of the number of points fitted.
We did some longer dark measurements over several hours with up to 40 000 points to prove that we observe a statistic error with Gaussian normal distribution (see Fig. 9). We divided these spectra in pieces of equal length and compared the resolved distributions for each part with a Gaussian distribution and see a very good agreement. The differences between the mean values, standard deviations and weighted deviations for the parts are relatively small. We also measure the dark spectra for the same settings before and after the measurement and do not observe significant differences. If we compare dark measurements from different times of the working process for the same settings of the sensitivity, we also get similar results.

Error calculation
We made an error calculation for all three methods with errors as determined in Sect. 3 and thus derived error bars for the gain.
We can divide the gain spectrum in three parts, one part with the wavelengths below the lasing wavelength λ lase , one part in the range of the lasing wavelength, and the last part for wavelengths above the lasing wavelength (Fig. 10). We see some differences for the used methods at the different parts.   For short wavelengths the influence of the resolution of the spectrometer increases for increasing wavelength towards λ lase . This leads to lower gain for the Fourier methods and the highest gain for Cassidy. At the lasing wavelength Hakki-Paoli is about 0.3 cm −1 less than Cassidy and Fourier with the zeroth and first peak is about 0.5 cm −1 less than Cassidy. Fourier with the first and second peak is 1.0 cm −1 less and Fourier first to fourth peak is about 1.2 cm −1 less than Cassidy. We suppose that Cassidy reflects the gain best, because of the smallest influence of the resolution (Fig. 8). At a certain wavelength above λ lase there is a crossover of the gain spectra as calculated by the different methods. Above this crossover wavelength the two Fourier methods without the zeroth peak produce a higher gain than the other three methods. In this area the influence of noise and offset increases due to the lower intensity (Fig. 4). Because of this, the gain calculated with Cassidy, Hakki-Paoli and Fourier with the zeroth peak decreases. For the Fourier transformed spectrum the influence of noise and underground is only in the zeroth peak. Thus Fourier first to second and first to fourth peak are not influenced and produce a higher gain.

Conclusion
For wavelengths below the lasing wavelength all methods result in similar gains. For increasing wavelength the differences increase until the lasing wavelength. The Fourier methods without the zeroth peak result in the lowest gain.
In the range of the lasing wavelength the correction of the resolution error leads to a convergence of all Fourier methods and Hakki-Paoli to Cassidy method without reaching the level of Cassidy. Therefore, we assume that the Cassidy method is the best method for this part of the spectrum.
The Fourier methods without the zeroth peak result in the highest gain for wavelengths above the crossover point. Hakki-Paoli, Cassidy and Fourier with the zeroth peak decrease much more due to the increasing noise and underground level. In a Fourier transformed spectrum only the zeroth peak is influenced by the noise and underground, thus the methods without this peak are not decreased.
We conclude that Fourier with the zeroth peak results in the smoothest curve below lasing wavelength. At lasing wavelength Cassidy obtains the best result because of the lowest influence of the resolution of the spectrometer and for wavelengths above the lasing wavelength Fourier method without respect to the zeroth peak evaluate the best gain curve because this is not influenced by the noise and underground.