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The spatially and spectrally resolved mode imaging method (S2) and lock-in detection technique are combined to allow for low signal gain measurements in double clad, Nd3+- doped fiber in the spectral region of 900 nm. The combination of these methods gives us the opportunity to measure the low signal gain, without disruption of the result by the amplified spontaneous emission (ASE). Results of the modal gain measurements are compared to numerical calculations.
© 2014 Optical Society of America
1. Introduction
During past several years we can observe growing interest in the modal analysis of optic fibers, either passive or active. Such investigations seem to be of great importance in the context of broadband fiber amplifiers, fiber lasers and new techniques of optical communication and information coding [1–3].
Determining the modal gain in the multimode fiber amplifiers is especially important issue in the field of high power lasers or telecommunication fiber amplifiers. Directly opposite requirements stand for modal gain distribution in such cases. In order to achieve the output beam of high quality it is crucial to maintain possibly low gain of higher order modes of large mode area fiber amplifiers as it is hard to avoid their presence at all [4–8]. On the other hand fiber amplifiers designed for space division multiplexing (SDM) transmission have to exhibit possibly equal gain for all transmitted modes [9–11]. There are two interferometric methods addressed to distinguish and imagine the fiber modes: C2 [12] (Cross-Correlated imaging) and S2 (Spatially and Spectrally resolved imaging). In the present work we focus our attention on S2 method, a bit modified in comparison to that used in [1], and farther combined with the lock-in technique.
Recently there were just few attempts to use the S2 mode imaging technique [1,13] to measure modal gain in large core fiber amplifiers [14,15]. This technique appeared to be very useful in the case of high power amplifiers, where due to the high power of transmitted signal, the amplified spontaneous emission (ASE) could be neglected and did not disrupted the measurements results. To be able to use this technique in the case of small signals and small gain one have to find the way to avoid the ASE contribution to the result of measurement. The S2 method is based on the interference of the fundamental mode and the higher order modes. The modal decomposition of the transmitted signal is based on the Fourier transform of the transmission spectra, where the interference of the modes appears as a modulation of the light intensity. The larger is the time delay of the higher order modes relative to the fundamental mode, the higher is the frequency of the modulation of the transmitted spectrum. The unmodulated part of the transmission spectrum, which appears at the zero time delay in the Fourier transform of the spectrum is regarded as a fundamental mode contribution to the signal. In general, the signal at the zero time delay is the sum of the intensities of all the modes, but in the most cases the fundamental mode contribution to this part of the signal is overwhelming. Spontaneous emission poses a problem in estimation of the relative modal gain of the signals of intensities small relative to the ASE intensity by simple comparison of the S2 measurements results for the fiber under excitation and with the pump turned off. Since the ASE is not coherent with the signal, it contributes only to the unmodulated part of the transmission spectrum and could be identified as a fundamental mode intensity during the S2 data analysis. As long as the intensity of the transmitted signal is not much higher than the intensity of the ASE, this will lead to the overestimation of the gain of the fundamental mode relative to the gain of the higher order modes. The way to get rid of the ASE contribution is to use a detection technique which allows to distinguish between the ASE and the transmitted signal even if they appear at the same wavelength. Such a kind of signal discrimination is offered by the lock-in detection and the intensity modulation of the signal beam. Using the modulated signal beam, and the lock-in amplifier operating at the reference frequency of the signal beam modulation, we are able to remove any signal which does not undergo such a modulation. Above all we can easily remove the CW ASE signal, which appears under the CW pump beam excitation.
In this paper we describe our solution for the modal gain decomposition of small signal in the presence of relatively strong ASE. As an example we have chosen a weak gain at about 900 nm, in double clad Nd3+ doped fiber with active core diameter 13.2 μm and numerical aperture 0.1. The energy level diagram together with transitions specific for the Nd3+ ions, both used in forthcoming calculations and interpretations, are presented in Fig. 1.
2. Experimental setup
To determine the modal composition of the propagated beam we used setup depicted in Fig. 2(a). As a broadband light source served superluminescent diode (SLD) Superlum T-870-HP, and spectral range of the operation was from 780 nm to 1000 nm. Light emitted from the SLD was spectrally filtered with 500 mm monochromator (Andor Shamrock SR 500i), equipped with 1200 lines/mm grating. It resulted in spectral resolution of about 0.03 nm in the region of 900 nm. Wavelength was altered by the monochromator with 0.05 nm step and at every wavelength a picture of 13 times magnified output beam was registered by CMOS camera (Thorlabs DCC1645M, 5.2 μm pixel size). By switching the single mode fibers and replacing camera with the fiber end (core of the fiber plays a role of the single pixel of the camera [1], we will call it “probe fiber”), without touching any optics directly connected to the tested fiber, we can reconfigure our setup to the gain measurement option Fig. 2(b). As a source of the excitation we used an Ar ion laser working at 514.5 nm. An avalanche photodiode (APD) served as a detector coupled to the monochromator output. The advantage of using such a configuration is the possibility to perform the S2 and gain measurements of the beam propagated in the identical conditions and to eliminate the ASE contribution to the result by use of the lock-in detection technique. Once the beating frequencies of different higher order modes are assigned to the particular modes during S2 measurement, we will use just the beat frequencies (not all images) to distinguish different modes during the gain measurement.
Maintaining the same conditions (the modal composition of the probe beam) when switching from the S2 measurement to the gain measurement is not crucial in this case, but this characteristics of the setup is required during the other experiments, which we are going to perform. Using single point detector instead of camera allows us to use the fast intensity modulation of the probe beam and the lock-in detection. By using the modulated probe beam, lock-in detection [16,17], and CW excitation we are able to eliminate the ASE signal, since it is constant in time, and then the lock-in output signal is proportional only to the modulated part of the APD signal.
3. Results of S2 measurements
First we performed S2 measurements of the 1.15 m long section of the tested fiber for three spectral ranges: 820 nm - 850 nm, 850 nm - 870 nm and 890 nm - 930 nm. The first range was divided into four equal parts, the second one was divided into two parts, and the third one was divided into four parts, and in each part we analyzed mode group delays with respect to the fundamental mode (LP01) by recognizing their images related to different beat frequencies. Results of this operation (dots and squares) and results of the calculations, performed with the sequence of the analytical calculations (approximate analytic solution of the characteristic equation for the step index cylindrical core fiber) proposed in the paragraphs 2-4 Eq. (1) - Eq. (23) in [18] are depicted in Fig. 3. It is seen, that the group delay for LP11 and LP02 undergo different dispersion rules.
Fig. 3 Group delays of LP11 and LP02 modes with respect to the fundamental mode LP01 (measured - dots and squares, calculated - solid lines).
Exemplary result of mode imaging, performed in the spectral range from 820 nm to 850 nm is shown in Fig. 4.
Fig. 4 S2 measurement results in spectral range from 820 nm to 850 nm for 1.15 meter long section of the fiber:(a) - sum of the modules of Fourier transforms of spectra taken from every camera pixel, (b, c, d) - measured electric field distribution for LP01, LP11 and LP02 mode.
Analyzed fiber, beside the fundamental mode, supports LP11 and LP02 modes propagation. According to the model [7] we expected LP21 appearance, but calculations, performed with beam propagation method (BPM), and taking into account collimation optics we used, suggest very weak LP21 excitation. Looking at Fig. 3 we can expect a problem with distinguishing LP11 and LP02 modes around 905 nm because their beat frequencies, caused by their interference with fundamental mode, are equal. One can say, that LP11 and LP02 modes are degenerated at 905 nm. Actually, we can avoid this problem by adjusting the collimation optics properly to excite the LP11 or LP02 mode separately and performing the S2 imaging to verify the result. Then we can perform the gain measurement using probe beam propagated in LP01 and LP11 or LP01 and LP02 modes. This procedure requires a reasonable assumption, that the probe fiber end is not placed in the region of the output beam image, where intensityof the minimized higher order mode dominates the overall maximized other one. For example, this would be the case, if we maximized LP11 (in relation to LP02) and placed the probe fiber end exactly at the center of the beam image. Despite the fact that in practice there is little chance to make such a mistake, we would like to have a tool to exclude this case. The problem with LP11 and LP02 identification can be solved taking a closer look at Fig. 3(a), where LP11 and LP02 peaks exhibit different broadening. Larger broadening of LP02 than LP11 function is due to the higher group velocity dispersion (GVD) of LP02 than LP11, what is confirmed in Fig. 3, where the slope of LP02 group delay versus wavelength is steeper than the slope of LP11 group delay function. In general such a broadening is treated as detrimental for the imaging. Then we can take the advantage on the GVD and its different influence on the LP11 and LP02 Fourier transform peaks. In the optical coherence tomography (OCT) based on Michelson or Mach-Zehnder interferometer to diminish the impact of the object and the reference arms dispersion mismatch on the image, we can use the numerical dispersion compensation [19]. In principle our setup is the Mach-Zehnder interferometer, where the fundamental mode plays a role of the reference arm, and we can apply the same procedure. The procedure of numerical dispersion compensation relies on multiplying the complex intensity spectrum (imaginary part is generated with Hilbert transform of real part) by the function: , where: a is proportional to the GVD. Since the signs of the GVDs of the LP11 and LP02 differ (see Fig. 3), if we choose such a parameter a in the above formula which minimizes the width of the peak of one of the modes in the Fourier spectrum, the other will experience the broadening. The larger is the GVDs difference of the modes, the larger the effect. To give a clear example of using such a procedure we have used the spectrum presented in Fig. 4(a), where the LP11 and LP02 peaks are seen on one spectrum, and we have compensated their GVDs separately. The result is presented in Fig. 5. One can see completely different behavior of the peaks related to the LP11 and LP02 modes, and this difference is even bigger in 900 nm region. It is important, that when the width of the peak decreases, the height of this peak increases, and the product of these values is constant. It indicates, that the procedure of the numerical dispersion compensation does not change the total power of the signal related to the particular mode. Especially in the case of the large value of the GVD one should use the integrated value of the peak around its center in the Fourier spectrum to estimate the content of the mode in the analyzed beam during the S2 data processing. In the case of small GVD values, the widths of the peaks related to different modes are almost equal and the S2 processing might depend only on the heights of the peaks. In our case, the appropriate measure of the mode content is reflected by the integral of the peak (regardless of the used value a) or its maximum value if a is chosen to equalize the widths of the LP11 and LP02 (the closest case is presented in Fig. 5(c)). We perform the S2 measurement just to recognize the modes and the positions of their centers in the Fourier spectrum, so in our case it is not crucial to deliver the accurate modal composition of the beam during the S2 data processing. Applying this procedure, when we perform the S2 measurement, and finding only the LP11 or LP02 content (beside the fundamental mode), and switching our experimental setup to the gain measurement, we will be able to prove, that the mode beating observed in the probe fiber position is related to the desired higher order mode, even in case of degenerated mode group delays.
Fig. 5 Numerical dispersion compensation effect on the 820 nm - 850 nm spectra: (a) - uncompensated, (b) compensated to minimize width of LP11 peak, (c) compensated to minimize width of LP02 peak.
Such a qualitative procedure of minimizing the widths of the peaks in Fig. 4, by changing the value of the parameter a, can provide a crude estimation of the GVD differences between the modes. The difference between the GVDs of the modes, estimated by the manual adjustment of the parameter a is + 1.4 ps2/km for LP11 and LP01, and −3.2 ps2/km for LP02 and LP01, whereas the values calculated with the analytical model described in [18] are + 1.0 ps2/km, and −3.2ps2/km respectively.
4. Results of the excited state transmission measurement
To choose the most interesting spectral region for excited state transmission (EST) measurements we performed the preliminary mode-independent EST characteristic in the wide spectral range from 775 nm to 1025 nm, additionally we measured ground state absorption and the amplified spontaneous emission. The length of the tested fiber was 1.7 m, excitation source was 514 nm, 0.6 W argon laser beam. The measurement was performed with the setup depicted in Fig. 2(b). This time, to avoid mode interference impact on the spectra, we used as a probe fiber a 200 μm core diameter, multimode fiber instead of single mode, and placed diffuser in front of its input. Using the probe fiber of the large core diameter and diffuser, we averaged the intensity of the interfering modes over the whole cross-section of the tested fiber. For each wavelength, the signals coming from the regions of the output beam image, where the higher order modes interfering constructively or destructively with the fundamental mode add up, to give the smooth spectrum as a result. Mode interference would disrupt the GSA measurement, where we divide the transmission spectra of the fiber sections of different lengths. Frequency of the mode interference pattern in the transmission spectrum changes with the length of the fiber, so if we divide the transmission spectra modulated by the interference of the modes, appearing at the different frequencies we will get the false result, manifested by the presence of oscillating regions in the absorption spectrum. First we collected the transmission spectrum with excitation source turned off, next we repeated the measurement with 514 nm laser turned on, then we divided transmission spectrum under excitation by the transmission spectrum of unexcited fiber and took the natural logarithm of the result, Fig. 6(c). Thus, we obtained excited state optical density spectrum. This result reflects all the characteristic features of the EST spectra in the Nd3+-doped glasses [20] and does not indicate any photodarkening effects, so the same section of the fiber could be used during further measurements. Ground state absorption (GSA) spectrum was obtained by dividing the measured transmission spectrum of 1.7 m long section by the transmission spectrum of 1.15 m long section of the fiber. First, we measured the transmission spectrum of the 1.7 m long section of the fiber, then we cut off the section of the length of 0.55 m (without touching the input of the fiber), and we measured the transmission of the remaining 1.15 m long section of the fiber.
Fig. 6 GSA (a), ASE (b), and EST (c) in 1.7 m long Nd3+ doped double clad fiber, energy levels diagram and optical transitions of interest.
For further examination we choose spectral range from 890 nm to 905 nm, where transmission of the fiber grows under excitation due to the gain and ground state absorption bleaching. Since our purpose was to find differences in interaction of the modes with the excited core area rather than to find exact value of the gain, we focused our attention on changes in transmission, without distinguishing, if it is due to gain or bleaching. Both these processes act in exactly the same way on the propagated beam, and especially on the modal composition of the transmitted beam. For simplicity the overall transmission increase will be further called “gain”. Choosing the spectral range from 890 nm to 905 nm we avoided the ambiguity in LP11 and LP02 signals distinguishing, and we were able to measure their gain at the same time or separately. In this spectral region, the LP11 signal was related to the peak at 1.4 ps/m group delay on the Fourier transform, and LP02 signal appeared at 1.7 ps/m group delay.
We compared three cases: 1.15 meter long, straight section of the fiber, cladding pumped, and a probe beam consisting of LP01 and LP02 (case 1), 1.7 meter long, coiled (R = 10cm) section of the fiber, core pumped, and a probe beam consisting of LP01 and LP11 (case 2), 1.7 meter long, straight section of the fiber, cladding pumped, and a probe beam consisting of LP01, LP11, and LP02 (case 3). In the case 2, we used the coiled fiber to diminish the content of LP02 in the probe beam [21, 22], the coil radius of 10 cm appeared to be small enough to eliminate the LP02 from the probe beam.
In each case the 514 nm pump power ranged from 300 mW to 700 mW with 100 mW interval. Electric field of each mode was extracted from the Fourier transform of the transmitted spectra by solving the following set of Eqs.:
Next, the gain was calculated as a ratio of the intensity of a given mode propagated in the presence of the pump (IP) and the mode intensity (I0) measured in absence of excitation.The set of Eqs. (1) becomes unambiguous, when we assume the dominant content of the fundamental mode in the considered region of the output beam image (chosen by the position of the probe fiber). Gain measurements were preceded by the S2 measurements in each case, so we were justified to make such an assumption. Results of the gain measurements in all three cases are presented in Fig. 7.Fig. 7 Measured (squares and points) and calculated (lines) gain: (a) - case 1, (b) - case 2, (c) - case 3.
The solid lines in Fig. 7 reflect the results of calculations, based on simplified diagram of the energy levels and transitions of the Nd3+ shown in Fig. 1. Calculations performed in similar scheme for different dopants are described in [5,6,8,] Relatively poor fit for the LP11 mode in Fig. 7(b) is connected with the fact that the calculations were performed for the unbent fiber whereas the experimental data were taken for the bent one (only in the case 2 - Fig. 7(b) the fiber was coiled).
The calculations, whose results are depicted in Fig. 7, were performed in the following steps. According to Fig. 1, we have constructed the set of Eq. (3) for the energy levels occupation:
where: N0, N1 i N2 stand for 4I9/2, 4F3/2, 4G7/2 respectively, σse900 = 11x10−21 cm2 is the stimulated emission cross sections at 900 nm, σse1060 = 20x10−21 cm2 is the stimulated emission cross sections at 1060 nm [23], σgsa514 = 4,5x10−21 cm2 is the ground state absorption cross sections at 514 nm [23], Wnr2 = 2,3x108 s−1 is nonradiative decay rate, Wr1 = 2050 s−1 is the total radiative decay rate of the 4F3/2 level [23], Ip is the excitation beam intensity, and Is1 and Is2 are the intensities of the signals at 900 nm and 1060 nm. We assumed, that the reabsorption of the ASE signal (emission is centered at 900 nm, and the absorption is centered at 890 nm) has a small impact on the N0 and N1 levels occupation and we omitted it in Eq. (3), to simplify the calculations. Another simplification is to assume that the ground state is fed by the stimulated transitions at 1060 nm immediately hence we omitted the occupation of the 4I11/2 energy level. In a similar way we assumed, that the each energy level between 4G5/2 and 4F3/2 is depopulated instantly by the nonradiative transitions to the next, lower lying, level. This simplification lets us to take into account the only one nonradiative decay rate Wnr2 (equal to the decay rate of the 4G7/2). Calculation of the nonradiative decay rate was performed with respect to the energy gap between given energy level and the closest lower lying level, and according to the formula: Wnr = Be-αΔE, where B = 1,4x10−12 s−1, α = 4,7x10−3 cm [24]. Stimulated emission cross section σse900 was calculated with the McCumber formula [25] and the measured absorption spectrum as in [26,27]. Solving the set of Eqs. (3) we determined the analytical formula for the occupations of the 4I9/2 and 4F3/2 energy levels in conditions of the 514 nm (pump beam), 900 nm (the signal beam and the ASE), and 1060 nm (ASE) presence: N0(PpT,Ps1T,Ps2T,x,y) = N0(x,y) and N1(PpT,Ps1T,Ps2T, x, y) = N1(x,y).In the next step the analytical formulas for N0(x,y) and N1(x,y) were inserted into the set of differential Eqs. for the propagation of the 900 nm, 1060 nm, and 514 nm beams:
where: PpM is the power of the M-th mode of the pump beam, Ps1M is the power of the M-th mode of the 900 nm beam, Ps2± is the power of 1060 nm ASE propagating in one or another direction, PpT, Ps1T, Ps2T - are the incoherent sum of power over modes or directions, Ep/s1/s2M(x,y) is the electric field distribution of the M-th mode, Nd = 3.1x1018 cm−3 is the dopant concentration, and σgsa900 = 3x10−21 cm2 is the ground state absorption cross sections at 900 nm . Electric field distribution of each mode and each wavelength was calculated with beam propagation method (BPM) by the Fourier transform of correlation function of the initial beam and the simulated electric field distribution along the propagation direction. With the same method we performed modal decomposition of the pump beam in the case 2. Result obtained with parameters of the optics we used and with assumed central position of collimated pump beam on the fiber core indicates that 80% of the pump beam power is propagated in the fundamental mode, almost 20% in LP02, and negligible power in the higher order modes.The numerical solution of Eq. (4) provides us the transmission of the fiber for the probe beam in the presence or in the absence of the pump beam. By dividing the calculated transmission of the pumped fiber by the transmission of the unexcited fiber we obtain the calculated analogue of the measured gain. To be consistent with the experiment, we take into account both, the gain and the bleaching at 900 nm (see Eq. (4)). In Eq. (4) the terms σse900 and σgsa900 stand for the cross sections of the stimulated emission and absorption averaged over the 890 nm - 905 nm wavelength range (the exact center of the wavelength range is 897.5 nm, and it is approximated to 900 nm in the subscripts and in the text). It reflects the conditions of the experiment, where the intensities of the modes and gain were measured and calculated within the wavelength range from 890 nm to 905 nm and averaged over this range, since the Fourier transform requires the finite spectrum of the frequencies to distinguish between fiber modes. In the cases 1 and 3 we assumed the uniform pump distribution in the core cross section. The BPM calculations performed for the signal wavelength in the case of the bent fiber suggest that even as large bent radius as 10 cm leads to the significant LP02 losses. The large discrepancy between the experimental data and the calculated gain for the LP11, presented in Fig. 7b is probably the result of the fiber modes intensity deformation in the case of the bent fiber [28]. We tried to improve the accuracy of our calculations for this case, by replacing in Eq. (4) the electric field distributions of the modes of the unbent fiber by the distributions of the electric fields of the modes calculated for the bent fiber. Electric field distribution of the modes of the bent fiber were calculated with the BPM and the effective refractive index profile described in [29, 30]. This correction improved the result of the calculations slightly, and we decided to keep the primary solution. It is well known that the higher order modes experience much higher losses than the fundamental mode, when the beam is propagated in the coiled fiber [28–30], and this rule might contribute to the gain. But, the probe beam undergoes the same losses when the pump beam is on or off, and in the simple approach, the ratio of the probe beam intensities after propagation in the pumped or unpumped fiber should not depend on the losses. At this moment we expect, that the discrepancy between the experiment and the calculations in Fig. 7(b) is the sum of the modes deformation and the modal losses distribution due to the fiber coiling, which are in fact strictly related each other.
5. Conclusions
The measurements of the fiber modes gain and bleaching were performed using the combination of S2 fiber mode imaging method and lock-in detection. Method proposed in this paper allows for determining the fiber modes interaction with excited area of the active fiber core not only in the case of high gain, but in the spectral regions of any optical transitions, where the excited states are involved (low gain, ESA or bleaching). Moreover this method does not require a high power of the probe beam, because signals related to the ASE are eliminated from the measured signal by use of lock-in detection. This characteristic of the method extends the dynamical range of the used setup, and allows for small signal gain measurements. Distinct dependence of the gain on the excitation geometry was demonstrated and confirmed by numerical calculations, and it is consistent with the results of the results described in other publications [6,31].
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