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We demonstrate the differences in the excited state transmission (EST) for different modes in 8 μm core diameter, Er3+- doped silica fiber. The S2 (Spatially and Spectrally resolved) imaging method was used to determine the modal composition of the transmitted beam and to analyze the group delays of the higher order modes. We register the up-converted emission under two beam excitation (980 nm + 850 nm or 790 nm) and propose the numerical model for the anti-Stokes emission analysis. Taking additionally into account the interference of the beating fiber modes, one can expect the inhomogeneous spatial distribution of the excited ions. This was predicted by numerical calculations. The obtained results have been confirmed by taking photo of the up-converted emission as seen from the side of the fiber.
© 2015 Optical Society of America
1. Introduction
The mode imaging connected with evaluation of the modal gain becomes more and more important in the context of application in the broadband fiber amplifiers, fiber lasers and fiber communication techniques. There are several mode imaging methods, among them one can distinguish two methods: C2 (Cross-Correlated) [1] and S2 (Spatially and Spectrally resolved) [2] imaging. First requires to use two arms of an interferometer (e.g. Mach-Zehnder), probe and reference, the second is more natural and simple in application to evaluation of a given fiber, since as the reference arm the fundamental mode (of fixed phase on a whole core crossection) can serve. In our work we focused our attention on the interaction between excited area of the active fiber and different modes of propagated beam, and we used S2 method to evaluate its modal content. In order to obtain the probe beam consisting of single mode we controlled modal content of the propagated beam adjusting collimation optics and verified modal beam purity using S2 method. Then we measured the excited state transmission (EST) spectra for consecutive modes.
The S2 mode imaging method has been introduced [2] and successfully used in several works [3–6] to the passive as well as active fibers. Dealing with the active fiber, (Er3+ doped) in the present work we used S2 method combined with the excited state absorption/gain/bleaching spectroscopy [7–11] using one experimental setup. Additionally we focused on the observed distinct up-converted emission in green under the presence of 790 nm, 840 nm and 980 nm beams. Facing the mode beating we expected the inhomogeneous spatial distribution of the excited ions [12] along the fiber. After recognition of the modal content in the fiber we have shown that there is a distinct difference between the EST spectra measured for two different modes; at given wavelengths we have shown the dependence of the ESA and gain/bleaching on the excitation power, and, photographed the specific distribution of the intensity of the up-converted green emission due to the mode beating. All the observed phenomena have been interpreted by means of simulation with use of numerical calculations.
Taking into account the popularity of the EDFA (Erbium Doped Fiber Amplifier) systems, we focused on commercially available Er3+- doped silica fiber. Fiber under test was core pumped LIEKKI 16-8/125 with 8 μm core diameter and NA of 0.13.
2. Experimental setups
To measure the modal composition of the propagated beam we used the setup similar to used in [4] and composed of broadband probe (signal) light sources (superluminescent diodes, Superlum series), 500 mm monochromator (Andor Shamrock SR 500i) with 1200 lines/mm grating and a CMOS camera (ThorLabs DCC 1645M, 5.2 μm pixel size) serving as detector. The spectral resolution of such a system was e.g. around 0.03 nm in the region of 900-1000 nm (where we could expect e.g. an optical gain due to 4I11/2 → 4I15/2 transitions). As a sources of excitation we used an Ar+ laser (Spectra Physics BeamLok 2065 Laser System, for EST measurements). Experimental setup is depicted in Fig. 1.
Fig. 1 Experimental setup for S2 mode imaging method (a), experimental setup for EST measurements (b). Switching between measurements is realized without any action on collimation optics of the probe beam. M - monochromator, C - camera, SLD - superluminescent diode, PD - photodiode.
The EST measurements have been performed with use of the lock-in technique [13–18]. We used Signal Recovery 7265 DSP lock-in amplifiers.
Semiconductor lasers (ThorLabs TCLDM9 kit, working at ~980 nm) or Ti:Sapphire laser working at ~798 nm and Optical Parametric Oscillator (OPO) (790 nm or 840 nm) served as sources for upconverted excitations in another experimental setup addressed to taking picture of the up-converted luminescence from the side of the fiber. We used the Ti:Sapphire system in part of our investigations because it can be coupled more effectively into the gain fiber than the diode which has worse beam quality than the output of the Ti:Sapphire laser
3. Mode imaging, dispersion and phase in the LP11 mode
The modal analysis was performed by means of the S2 (Spatially and Spectrally resolved) imaging [2,3]. A technique, described in detail in [19], has been applied. In the S2 method, when registering transmission spectrum of high resolution, one can see a “sinus-like” interference structure which is a result of beating of those different modes which take part in the propagation process. Taking Fourier transform of such a spectrum we enter into the time domain. Figure 2 shows the sum of the modules of Fourier transforms of the spectra taken from every camera pixel (a), the intensity distribution of the fundamental mode LP01 (b), and the intensity and electric field distributions of the mode LP11 (c,d).
Fig. 2 Fourier transform of the collected spectra (a), the LP01 mode intensity distribution (b), intensity (c) and electric field (d) distributions of the LP11 mode. The 99.8% of the whole beam intensity is propagated in the fundamental mode.
The Fig. 2(a) can actually be treated as a distribution spectrum of the group delays Δτg of higher order modes (here: LP11) with respect to the fundamental mode. Performing, for a given mode, the measurements of the group delays in several consecutive spectral ranges we can obtain a spectrum of these delays being an illustration of the mode dispersion. The result of such a procedure applied to the LP11 mode is shown in Fig. 3. The solid line is the result of numerical simulation performed with use of the analytical solutions of the characteristic equation for the step index cylindrical core fiber [19,20].
Going back to Fig. 2(d) and focusing an attention on the “left hand” or “right hand” side of the electric field distribution (mode LP11) (Fig. 4(a)) one can measure an interference spectrum of the electric field intensity (Fig. 4(b)), and it is seen that spectra for both sides are reversed in phase each other. And there is an additional effect: one can see a decrease of modulation frequency with the wavelength, which is connected with the above mentioned dispersion (nonlinear dependence between wave vector and wavelength leads to much smaller, than observed in Fig. 4(b), decrease of the modulation frequency). It is taken into account in data analysis.
Fig. 4 Electric field distribution of the LP11 mode (a), and part of the spectrum showing inversed phases of the electric field taken from opposite peaks (from single camera pixels) of the electric field in the LP11 mode (b).
4. EST measurements in two different modes
To see how the EST characteristics depend on the propagation mode we have examined a core pumped fiber of 1.7 m length, core diameter 8 μm and numerical aperture 0.13. Due to the simple modal structure of the beam propagated in this fiber (only LP01 and LP11 are present) in the analyzed wavelength range we could use simpler, than previously used [19], procedure to obtain the EST for a given mode. By adjusting the optics at the fiber input we were able to get almost pure fundamental mode (99.8% of the transmitted energy) or pure LP11 mode (97%). We have used the same setup as in S2 measurement. This time we removed the camera and inserted instead a multi-mode fiber, which merges probe beam radiation from all the output beam image. This allows to get rid of mode interference fringes (fringes are weak due to the high modal purity of the probe beam) from the transmission spectrum. This is due to averaging the signals coming from the regions where electric field of higher order modes is in opposite phase. The probe beam intensity is analyzed by the avalanche photodiode connected to the lock-in amplifier. Since our experimental setup allows for reconfiguration from S2 measurement mode to the EST measurement mode, without changing modal composition of the probe beam (switching is realized only by changing connections between single mode fibers, and collimation optics stands intact), we can measure the EST of a given mode directly. In this case the procedure is as follows: first we use S2 to monitor modal composition of the beam during adjustment of the collimation optics, once we get almost pure beam propagation in a given mode, we switch to the EST measurement and then we collect transmission spectra with, and without presence of the pump beam and divide them by each other. In contrast to previously described method [16,21] we remained at getting the EST spectra by direct comparison of transmission spectra. This procedure allows us to obtain the EST spectra without losing spectral resolution, which was inherent feature of previously used method [19], where the signal must be collected over given spectral region covering at least few mode beating cycles to distinguish the modes.
Owing to the lock-in detection in the EST measurements only the signals of the probe beam modulation frequency (~1 kHz) were amplified, whereas excitation beam intensity remained constant. Thus we eliminated the amplified spontaneous emission (ASE) signal, which was constant under constant excitation. Dividing the spectrum of the probe beam passing through the excited fiber by that for unexcited fiber, we obtained the EST spectrum consisted of the ESA, optical gain and GSA bleaching [11,18,22]. Such a procedure can be successful only if one uses a very stable source of the probe beam. This was the case, since we used stabilized superluminescent diodes.
Launching the probe beam into the fiber, and using micromanipulation, we paid a special attention to obtain the most of the propagated energy in a given mode. Thus we have achieved 99.8% energy in the LP01 mode and 0.2% energy in the LP11 mode in the first case, and, in the second: 97% in the LP11 mode and 3% in the LP01. The EST characteristics for both cases are presented in Fig. 5, where Ip is the signal beam intensity passing through the pumped fiber and Iu is the same for the unpumped fiber. Taking ln [Ip(λ)/Iu(λ)] we thus obtain the EST spectrum in terms of the excited state optical density.
One can see a distinct dependence of the EST spectra on the mode of propagation. It means that in different modes population in different excited area of the core takes part in the process.
Having the EST spectra for two modes we can focus our attention on three regions of the most distinct differences: ~790 nm, 850 nm and 980 nm. It is interesting how at these wavelengths the ESA and gain-bleaching alter with increasing excitation power. The results are shown in Fig. 6. The solid lines are the results of numerical simulations described below.
Fig. 6 Measured (points) and calculated (lines) values of the EST under excited beam of growing power for the LP01 and LP11 modes.
For numerical simulations we had to assume some spatial distribution of the pump beam across the fiber core. Since we were not able to perform the S2 imaging at 488 nm, we used the beam propagation method (BPM) (e.g [23,24].) and simulated the propagation of the 488 nm beam in the 20 mm section of the fiber. We assumed central position of the focal spot of the pump beam with respect to the fiber core. According to the geometry of collimation optics and diameter of the pump laser beam we assumed the Gaussian beam of 11 μm diameter at the fiber input. Then for each layer of BPM simulation we calculated the factor of correlation between electric field of the input beam and electric field distribution at this layer. Thus, layer by layer, we created the correlation function. Taking the Fourier transform of this function we estimated the modal composition of the pump beam [25]. This allowed us to determine a content of each individual mode in the propagated pump beam. Thus it was established that 98% of the pump beam will be propagated in the fundamental mode, whereas remaining 2% in the LP02 mode. In the next step, using BPM, the spatial distributions of the LP01 and LP11 modes of the 790 nm, 850 nm and 980 nm beams were calculated. Then, using the simplest model of deexcitation in the Er3+ ions we constructed the following set of rate equations describing the system kinetics:
where N0, N1, N2 correspond to the 4I15/2, 4I13/2 and 4F7/2 states populations respectively, Ip is the excitation beam intensity, IASE is the signal beam intensity, σGSA and σSE are the ground state absorption and stimulated emission crossections respectively, and Wr1,0 and Wnr2 are the radiative and nonradiative decay rates corresponding to the states indexed in subscripts. The set of Eqs. (1), in stationary conditions (dNi/dt = 0), can be solved analytically and the obtained steady state populations can then be used in the set of differential equations describing the powers of beams propagated in different modes along the fiber:where PpM, Ps1, Ps2, Ps3, and PASE± are the powers of given modes (M is the mode number) of the 488 nm, 790 nm, 850 nm, 980 nm beams and 1550 nm spontaneous emission power propagating in one or another direction. E are normalized intensities of electric fields of these beams, PpT is total power of the 488 nm beam, PASET is total power of the ASE beam, and N0(PpT,PASET,x,y) and N1(PpT,PASET,x,y) are relative populations of the ground and first excited state. All the populations are normalized to unity so that Nd represents the dopant concentration. The sign ( + ) corresponds to the propagation direction opposite to excitation direction, (-) corresponds to the propagation direction consistent with excitation direction. In the above sets of equations the influence of the probe beam on the ground and excited state populations has been neglected, because of low probe beam intensity in comparison with the excited beam intensity. The set (2) has been solved numerically with use of the Runge-Kutta routine of fourth order. In Table 1 the set of used parameters is presented.
Table 1. Parameters used in numerical solution of the set of Eqs. (2)
Wnr2 has been determined with use of the familiar relation: , where B = 1.4 × 10−12s−1 and α = 4.7 × 10−3cm [29].
It seems that the simulation results seen in Fig. 6 quite reasonably reflect the experimental data. A possible source of relatively poor fit, seen for gain-bleaching of the LP11 mode, can be an insufficiently precise determination of the input parameters of the exciting beam in the BPM simulation, which can lead to inaccurate calculation of spatial distribution of excitation in the active core.
5. Up-converted emission and mode beating, calculations and observation
We have observed the upconverted emission in green (525 nm, 545 nm) using either one wavelength of excitation (798 nm) or combination of two (790 nm and 980 nm). The respective energy diagrams along with excitation channels (a) and (b) are presented in Fig. 7.
Fig. 7 Energy diagrams used for interpretations of the observed up-converted emissions. Excitation by 798 nm line of the Ti Sapphire laser (channel a), and by a pair of two lasers working at 980 nm and 790 nm or 980 nm and 840 nm (channel b). As sources of the 790 nm and 840 nm beams we used superluminescent diodes when we used them both together or Optical Parametric Oscillator (OPO) during experiments described in this section.
In the present work, to analyze the mode interference impact on spatial distribution of the upconverted luminescence, we use experimental results for the combination of 980 nm and 790 nm, but similar model might be applied for combination of 980 nm and 840 nm. Hence we provide description for both combinations of the beams.
To prove that the upconversion occurs with participation of two photons, as illustrated, we have first exploited the situation seen in Fig. 7(a) and measured the dependence of the emission intensity I versus excitation power P and fitted the obtained courses to the familiar formula:
(where n is a number of photons taking part in the process). Using the procedure similar to that described in previous chapter, applied to the processes seen in Fig. 7(a), we have theoretically reproduced the experimental I(P) dependencies for 525 nm and 545 nm emissions and intensity ratio: I(525)/I(545). Interpreting the fitting lines by means of the formula (3) we have obtained n ≅ 1.8 (for 525 nm) or n ≅ 1.6 (for 545 nm), which, concerning some extent of saturation [30] and possible contribution of other transitions, makes plausible an assumption of two photons taking part in the excitation process.The light of the upconverted emission just like the probe beam (signal) light undergoes modal propagation. Because we have simultaneously: excitation beams and upconverted emission beams (of two wavelengths) a situation becomes extremely complex in face of interfering modes. Especially complex is the deexcitation process seen in Fig. 7(b), where we use two sources of beams leading to the ESA transitions and addressed to 525, or 545 nm upconverted emissions. To figure out how this complexity reflects in the excitation distribution along the fiber and in the overall light distribution, we have first performed detailed numerical simulation.
We started with situation where the 980 nm beam is an excitation beam at the first stage, leading to the first excited state and another, separately entered beam: 790 nm or 840 nm provides a source of the second stage excitation, from the first to higher excited state (ESA transitions). This beam is entered and propagates in the same direction as the 980 nm beam. Additionally, with simplifying assumption of relatively small intensities of the 790 nm or 840 nm beams and weak influence of the ESA transitions on the ground and first excited state populations, we could calculate spatial distribution of the 4I13/2 state under exciting 980 nm beam of a given modal structure. Having the distribution of the ions in the first excited state we can then calculate the populations of the 2H11/2 and 4S3/2 states, that are excited by 790 nm or 840 nm. To determine the population of the 4I13/2 state we solved analytically the following set of rate equations:
where N0, N1 and N2 are, respectively, lowest three consecutive states of the erbium ion seen in Fig. 7(b). The populations of the 4S3/2 and 2H11/2 states, defined below as N5 and N6 can be calculated by solution of the next set of rate equations:where Wnr5,6 are nonradiative decay rates for the states as indexed. In the above sets of rate equations we took into account only the states of substantial populations.Having the transversal distributions of the populations in the considered states we could simultaneously calculate the longitudinal development (layer by layer) of the mode-dependent powers of pumping beams propagating in a given, and opposite direction:
where EM(x,y) is the electric field distribution of the M-th mode and Nd, as previously, is the dopant concentration. At this point we can calculate the intensities of the beams necessary to populate the erbium energy states:where nefM is an effective refractive index for the M-th mode. Hence the set (7) describes the dependence of light intensity on the phases of modes. These intensities will be proportional to the excitation extent (population) of a given state and to the intensity of the emitted light.The procedure described above served us to determine the spatial distribution of emitting state populations in situation of interfering and beating modes. Among various results we can show one, concerning the population of the first excited state 4I13/2 under 980 nm excitation, distributed along the fiber, for 2 mm section (Fig. 8(a)), and spatial distribution of the intensity of the 840 nm beam in dependence on phase difference between its modes (LP01 and LP11): for a given phase difference (b) and opposite to (b) phase difference (c), for the same section. The colors between contour lines, going from violet (for zero values), through dark blue, blue, green, yellow, up to red, correspond to higher and higher population (a) or higher and higher light intensity (b), (c).
Fig. 8 Calculated spatial distribution of the 4I13/2 state population under 980 nm excitation in face of beating modes (a), and spatial distribution of the 840 nm beam in dependence on mode phase difference (for LP01 and LP11 modes) at the distance of 2 mm along the fiber.
Using the same method as before, we have also calculated the distribution of the Er3+ ions in the 4S3/2 state (responsible for the upconverted emission) due to the interference of the LP01 and LP11 modes for excitation lines: 980 nm and 790 nm or 980 nm and 850 nm.
The results of the distribution simulations for the pair 980 nm and 790 nm are presented in Fig. 9(a) in a grey scale format. Then, we have utilized a custom optics and ThorLabs DCC 1645M camera and tried to take a photo of the luminescing fiber as seen from a side. The photographed upconverted luminescence is shown in Fig. 9(b).
Fig. 9 Calculated spatial distribution of the 4S3/2 state population (proportional to the upconverted green luminescence) under 980 nm excitation and 790 nm beams (causing the ESA transitions) in face of beating modes (a), and spatial distribution of the intensity of the upconverted green luminescence, as seen from a side (b).
We have managed to take a readable picture on the length of 2.3 mm. It seems that we have obtained a reasonable agreement between simulated and photographed luminescence distribution. Similar effects were analyzed experimentally and theoretically by other groups [31,32].
6. Conclusion
Using the S2 method we have determined the modal composition of the beam propagated in the fiber and measured the mode dependent excited state transmission (EST: ESA, gain, bleaching) spectra. The S2 imaging gave us the mode group delays and the modes intensity distribution. This allowed us to determine two paths of propagation corresponding to two different modes and to measure the EST spectrum for each of them. It appears that the EST spectrum depends distinctly on the mode of propagation. The same concerns the EST dependence on the excitation power. We have also developed a method of simulation of complex cases when we deal with beams propagating in different directions, of different wavelengths, and exciting the ions in face of interfering and beating modes. Taking into account the mode beating with contribution of the green upconverted luminescence, we were capable to predict, by the numerical simulations, the inhomogeneous spatial distribution of the erbium ions in the excited states. The predicted effects have been confirmed by observation, especially the last case of beating modes with contribution of the upconverted emission. This has been proved by taking direct photo from the side of the fiber.
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