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Abstract
Using a wavefront sensor, we have measured the temporal evolution of the lens induced in a Nd:YAG rod amplifier under side pumping by laser diode bars centered around 808 nm in a quasi-continuous wave regime. The evolution of the induced lens is drastically different when measured with a probe pulse centered at 532 nm or 1064.5 nm. To explain this evolution, we developed a model that accounts for both the excited state population of the Nd3+ ions and the thermal contribution to the refractive index of the amplifier. This model, which takes into account amplification and wavelength shift of the probe pulse at 1064.5 nm, makes it possible to quantitatively describe the spectral and temporal evolution of the amplifier focal length. It also shows that the excited state population contribution is more important around 1064.5 nm and can partly compensate for the thermal lens induced in the amplifier.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Master oscillator and power amplifier (MOPA) have attracted a lot of attention in recent years [1–6]. When high power is required, MOPA performs well compared with simple laser oscillator in terms of linewidth, wavelength tuning range, beam quality and pulse duration. To master the design of a MOPA, it is of crucial importance to evaluate and control the impact of the different amplification stages on the initial spatial beam profile. While the master oscillator can easily delivers laser pulses with a very good gaussian beam shape ($M^2 \leq 1.2$), power amplifier usually degrades the beam quality. The latter mainly results from an index gradient established perpendicularly to the propagation axis of the laser beam within the amplifying medium (AM). This index gradient is mainly due to the thermal gradient induced during the optical pumping of the AM. In a first approximation, this thermal gradient has generally a parabolic shape and the AM acts as a lens. This thermal lens often degrades the amplified beam profile, and in the worst case, it can damage the AM. The focal length of the thermal lens depends on the amplitude and shape of refractive index change (RIC) induced during the optical pumping of the AM. It writes $1/f_T\propto \frac {\delta n'(\lambda )}{\delta T}\nabla _r T$ where $\nabla _rT$ is the spatial thermal gradient and $\frac {\delta n'(\lambda )}{\delta T}$ is the thermal variation of the refractive index at laser wavelength $\lambda$ [7], which depends on the density and the polarizability of the AM [8]. The thermal gradient further generates deformations and stresses which modifies the focal length $f_T$ of the AM [9]. To improve the beam quality delivered by oscillators and laser amplifiers, different studies have reported the thermal lens in the AM [1,9–12]. It is worth mentioning that the heat equation drives the temporal evolution of the thermal lens effect in the AM.
However, besides thermal lens effect, another phenomenon also happens during the optical pumping of the amplifiers. This phenomenon, often overlooked, is linked to the change of the AM complex refractive index $n(\lambda )=n'(\lambda )+i n''(\lambda )$ during the population of the different electronic states of the doping ions [13–17]. As for the thermal effect and during the AM optical pumping, a spatial gradient of the density of excited state population $\nabla _r N_{ex}$ is established perpendicularly to the propagation axis of the amplified pulse. It also results in a spatial gradient of refractive index which makes the AM behave like a lens. The associated focal length writes: $1/f_{N_{ex}}\propto \frac {\delta n'(\lambda )}{\delta N_{ex}} \nabla _r N_{ex}$, where $\nabla _r N_{ex}$ is the spatial population gradient and $\frac {\delta n'(\lambda )}{\delta N_{ex}}$ is the population variation of the refractive index at laser wavelength $\lambda$. At variance with the thermal lens effect, the kinetic of this latter phenomenon is driven by the population and the relaxation time of $N_{ex}$. It was found to dominate over the thermal effects in Yb-doped crystals and glasses under pulse pumping [18–20]. It is comparable with the thermal effect in Nd-doped materials [21–24]. Hereafter, using a wavefront analysis method, we show that excited state population and thermal lenses can be easily revealed in a Nd:YAG rod amplifier under quasi-continuous side pumping by laser diode bars centered around 808 nm. Our technique makes it possible to record the temporal evolution of a probe pulse wavefront propagating in the AM. It is easily operated at various wavelengths, thanks to the broadband absorption spectra of the wavefront analyzer. Knowing the evolution of the curvature radius of this probe beam, we can quantitatively infer the evolution of the lens induced in the AM. In order to analyze our experimental results, we developed a model which accounts for the refractive index change (RIC) of the AM induced by both excited state population and thermal effects. We show that in our Nd:YAG rod amplifier, the excited state population contribution is more important at ${1064}\;\textrm{nm}$ and can partly compensate for the thermal lens.
2. Experimental setup
We use the setup displayed in Fig. 1(a). It measures the temporal evolution of the radius of curvature (ROC) of the probe beam at the exit of AM using a Schack-Hartmann wavefront sensor (WFS). This WFS (Thorlabs WFS40-5C), built around a 2048 x 2048 pixels CMOS camera, has an array of 73 x 73 microlens array. According to the deviation of each microlens beam spot, compared to the plane wave spot position, one can recalculate the local angle of incidence. The latter makes it possible to record the beam phase and to infer the ROC of the probe beam. Our WFS is able to measure the change of the ROC between ${0.005}\;\textrm{m}$ and ${100}\;\textrm{m}$ on a large spectral range (300 nm to 1100 nm). The AM used in this work, is 3 mm in diameter and 70 mm long Nd:YAG rod (Northrop Grumman RBAT34-1C2). It is side pumped by quasi-continuous-wave (QCW) optical pulses delivered by laser diode bars having central wavelength around 808 nm. Figure 1(b), represents the spectrum of spontaneous emission of the AM upon QCW excitation. A Nd:YAG microlaser (Standa STA-01-7-OEM ) delivers the probe beam at a frequency of 100 Hz. The probe pulse has a duration of $\sim\;{700}\;\textrm{ps}$, an energy of $\sim\;{200}\;\mathrm{\mu}\textrm{J}$ and it has a good Gaussian beam shape ($M^2\leq 1.3$). Figure 1(b) displays its spectrum. It is worth noting that its central wavelength is 1064.5 nm, slightly shifted from the main emission peak of the Nd:YAG at 1064.2 nm [25]. We send the linearly polarized probe pulse through a half-wave plate placed in front of a Faraday isolator. Afterward, it propagates through a half wave-plate and a polarizing beam splitter to finely adjust the energy of the pulse propagating in the AM. We measure the change of the ROC of the probe beam when it is either amplified (${1064.5}\;\textrm{nm}$) and non amplified (${532}\;\textrm{nm}$). To perform the experiment at 532 nm, we focus the probe beam and frequency double it in a BBO crystal, then a lens colimates it.
Fig. 1. (a) Experimental setup used to measure the temporal evolution of the focusing effect induced by AM pumping. (b) Spectrum of the laser source (red curve) and of the AM spontaneous emission (blue curve). (c) Chronogram of the different events of the experiment.
To provide reliable measurements of the ROC, we ensure that the beam size of the probe pulse remains constant on the WFS and that the spot size covers the whole CMOS sensor. In order to satisfy these two requirements, we conjugate the exit of the AM with the WFS using a pair of lenses ($L_3$ and $L_4$). As the size of the beam at the exit of the AM changes at 1064.5 nm and 532 nm, two different sets of lenses are used. Prior to our experiments, we measured the initial ROC of the probe beam reflected by the polarizing beam splitter and collected by a removable mirror. To check the magnification G of our conjugation system , we inserted a set of lenses with known focal length $f_{cal}$ with value between 50 and 2000 mm in the optical path of beam reflected by the beam splitter. Therefore, we were able to plot the ROC as a function of the focal length and so accurately calculate the magnification $G=f_4/f_3$ of our optical conjugation system. The ROC recorded by our WFS is directly linked to $f_{cal}$ and writes: $ROC= G^2 f_{cal}$. The linear fits of these ROCs indicate the measured magnifications $G_{{1064}\;\textrm{nm}}^{m}= 3.08\pm 0.10$ and $G_{{532}\;\textrm{nm}}^{m} = 6.63\pm 0.05$ at 1064.5 nm and 532 nm, respectively. The values are in good agreement with the focal length of the lenses $L_3$ and $L_4$ for which $G_{{1064}\;\textrm{nm}}^{e} = 3$ and $G_{{532}\;\textrm{nm}}^{e} = 6,67$ (Fig. 1(a)). We record the temporal evolution of the ROC upon optical excitation of the AM as follow. We synchronize the pulse delivered by the microlaser with the power supply controlling the laser diodes bars. Then, by delaying in time the electrical and microlaser pulses (Fig. 1(c)), we measure the temporal evolution of the ROC of the probe beam at the exit of the AM. The power supply of the laser diode is able to deliver electrical pulses of duration up to 600 µs and current up to 100 A at 100 Hz.
3. Experimental results
The temporal evolution of the focal lens $f(t)$ induced in the AM upon its excitation is directly related to the temporal evolution of the probe beam radius of curvature $ROC(t)$ at the exit of the AM. It writes $f(t)=ROC(t)/G^2$ where $G$ is the magnification of our conjugation system.
3.1 Evolution of the focal length of the AM recorded at $\lambda = {1064.5}\;\textrm{nm}$
Figure 2 displays the temporal evolution of the gain (Fig. 2(a)) and the focal length (Fig. 2(b)) of the AM upon its optical excitation. We record these data, driving the pump diodes with current of 100 A and a duration of 500 µs. As expected, during the excitation of the AM, the gain $g$ increases up to $g\sim 13$ at the end of the pump pulse. It then steadily relaxes back to $g=1$. When the pump pulse is on, the focal length $f$ of the AM starts from its quasi steady state regime value $f_{qs}= 0.53 \pm\;{0.02}\;\textrm{m}$ and rapidly increases up to $f= 0.61 \pm\;{0.02}\;\textrm{m}$ after $\sim\;{150}\;\mathrm{\mu}\textrm{s}$. Afterward, it decreases to $f= 0.53 \pm\;{0.02}\;\textrm{m}$ at the end of the pump pulse. As soon as the pump pulse is off, the focal length of the AM increases again and it is $f = 0.60 \pm\;{0.02}\;\textrm{m}$ after about 250 µs. Afterward, it relaxes back to its quasi-static value $f_{qs}$. We also record the evolution of the focal length of the AM while changing the current and the duration of the pump pulse. As displayed in Fig. 3(a), when the current $I$ increases, $f_{qs}$ decreases. It is $f_{qs} = 0.78 \pm\;{0.03}\;\textrm{m}$ for $I={80}\;\textrm{A}$ and $f_{qs}= 0.52 \pm\;{0.02}\;\textrm{m}$ for $I={100}\;\textrm{A}$. It is worth noticing that whatever the current, the focal length $f(t)$ of the AM is maximum after about 150 µs and then decreases. Similarly, when the pump current $I$ is off, $f(t)$ increases and reaches its maximum 250 µs later and then decreases back. The same trend is observed while keeping the current $I$ constant but increasing the duration of the pump pulse from 400 µs to 600 µs (Fig. 3(b)).
Fig. 2. (a) Temporal shape of the ${500}\;\mathrm{\mu}\textrm{s}$ laser diode electrical pulses (red curve). Evolution of the gain in the Nd:YAG rod for the probe pulse centered at $\lambda = {1064,5}\;\textrm{nm}$ (green curve). (b) Temporal evolution of the focal length $f(t)$ induced in the Nd:YAG rod recorded with a probe pulse centered at $\lambda = {1064,5}\;\textrm{nm}$.
Fig. 3. Temporal evolution of the focal length $f(t)$ of the Nd:YAG rod recorded with a probe pulse centered at $\lambda = {1064.5}\;\textrm{nm}$. The evolution of focal length $f(t)$ is plotted when the Nd:YAG rod is excited by laser diode bars (a) a duration of ${500}\;\mathrm{\mu}\textrm{s}$ with different diode pump currents, (b) with different pulse durations and a fixed current of ${100}\;\textrm{A}$.
3.2 Evolution of the focal length of the AM recorded at $\lambda = {532}\;\textrm{nm}$
We also record the focal length of the Nd:YAG rod at $\lambda ={532}\;\textrm{nm}$. For this experiment, due to an electrical problem that overheated the pump diode, its central wavelength shifted from 808 nm to 803.7 nm. The absorption coefficient of the AM at this wavelength is slightly lower. The evolution of the focal length of the AM recorded at $\lambda ={532}\;\textrm{nm}$ is displayed in Fig. 4(b) when a current of 100 A is applied during $\tau _{pump}={500}\;\mathrm{\mu}\textrm{s}$ at a repetition rate of 100 Hz to the laser diode bars. Under this condition, the quasi static focal length of the AM is $f_{qs} = 0.74 \pm {0.02}\;\textrm{m}$. At variance with the experiment operated at $\lambda = {1064.5}\;\textrm{nm}$, when the pump is on, the focal length $f(t)$ of the AM always decreases. At the end of diode pump pulse, it is about $f({500}\;\mathrm{\mu}\textrm{s}) = 0.68 \pm {0.02}\;\textrm{m}$. When the pump is off , $f(t)$ slowly increases back to its quasi-steady value. The increase of the focal length of the AM can be fitted by a sum of two increasing exponential functions, highlighting two characteristic times $\tau _1 = 236\pm {10}\;\mathrm{\mu}\textrm{s}$ and $\tau _2 = 8.0\pm\;{0.2}\;\textrm{ms}$. The result of this fit is plotted in red dash in Fig. 4(b). As discussed later, $\tau _1$ corresponds to the relaxation time of excited ${\rm Nd}^{3+}$ ions in the YAG matrix and $\tau _2$ is related to the diffusion of heat within the Nd:YAG rod. This phenomenon was already observed by Antipov [13].
Fig. 4. (a) Temporal evolution of the laser diode electrical pulses, for a current of 100 A and 500 $\mathrm {\mu }$s duration (red curve), and evolution of the laser probe gain when the its central wavelength is at $\lambda = {532}\;\textrm{nm}$ (green curve). (b) Evolution of the focal length of the Nd:YAG rod at $\lambda = {532}\;\textrm{nm}$ (blue dots). The red dashed line displays the relaxation of $f$ back to its quasi-steady state value $f_{qs}$.
4. Modification of the refractive index of the Nd:YAG rod induced by the thermal process
As already mentioned, the optical excitation of the ${\rm Nd}^{3+}$ ions in the ${\rm YAG}$ matrix increases the temperature of the AM which in turn modifies its refractive index. This is due to the quantum defect $\eta _h\sim 24\%$ of the AM induced by the difference in energy between the photon absorbed (around ${808}\;\textrm{nm}$) and emitted (around ${1064}\;\textrm{nm}$) by the ${\rm Nd}^{3+}$ ions (see Fig. 1(b)). The RIC $\delta n_{T}$ induced by temperature variation writes [7]:
where $\left (\frac {dn}{dT}\right )_{eff}$ is the effective thermo-optic coefficient. This coefficient accounts for the thermo-optic effect, the photo-elastic deformation, the stress and strain effects [14]. To model our experiment, we will consider that the radius $R$ of our Nd:YAG rod is small compared to its length $L$ ($R\ll L$) and behaves as an infinite cylinder. The temporal and spatial evolution of the temperature within the Nd:YAG rod can be computed by solving the heat equation. In cylindrical coordinates, the latter writes: where $Q(r,t)$ is the source of heat, $K_c$ the thermal conductivity, $\rho$ the density and $C_p$ the specific heat capacity of the YAG, $r$ is the radial coordinate and $t$ the time. This equation has to be supplemented by the boundaries conditions. The YAG rod being side-cooled by water, we assume that at its edge $T(r = R;t) = {298}\;\textrm{K}$. We also consider that prior to its optical pumping, the Nd:YAG rod is at ambient temperature ($T(r; t=0) = {298}\;\textrm{K}$). The temporal and spatial profile of the temperature within the Nd:YAG rod strongly depends on the way heat is deposited in the AM. The Nd:YAG rod is side pumped and the pump absorption within the rod has almost a Gaussian shape. Hence, the heat source writes: where $\sigma _{pump}$ is the pump cross section and $Q_0(t) = Q(r=0,t)$. The evolution of $Q_0(t)$ is driven by the number of ${\rm Nd}^{3+}$ ions excited in the $^4F_{3/2}$ level. When the pump intensity is small compared to the saturation intensity, the excited state population equation writes [26]:To solve the heat equation (Eq. (2)), we used the multiphysics software COMSOL. All the parameters used in the simulations are summarized in Table 1. The results of our simulations, computed considering that the AM is excited at a repetition rate of ${100}\;\textrm{Hz}$ by a pump pulse of ${500}\;\mathrm{\mu}\textrm{s}$ centered at $\lambda _{pump}={808}\;\textrm{nm}$ (for the experiment at 1064 nm) or $\lambda _{pump}={803.7}\;\textrm{nm}$ (for the experiment at 532 nm), are displayed in Fig. 5. It is important to mention that when the pump is switched on for the first time, the whole temperature of the Nd:YAG rod increases during $\sim {4}\;\textrm{s}$ before reaching this quasi-steady state regime (QSSR). Afterwards, the temperature gradient inside the rod periodically evolves as displayed in this figure. The evolution of the heat $Q_0(t)$ generated within the Nd:YAG rod and the temperature difference $\Delta T(t)$ between the center and the edge of the Nd:YAG rod in the QSSR are displayed in Fig. 5(a) in red and blue, respectively. In this QSSR, one can write:
According to the data displayed in Fig. 5(a), $\Delta T_{QSSR}\sim 4 K$ and $\delta T(t)\leq {0.10}\;\textrm{K}$. It is worth noting that in this QSSR, $\delta T(t)$ increases during $\sim {1}\;\textrm{ms}$ even if the pump pulse lasts only ${500}\;\mathrm{\mu}\textrm{s}$. Indeed when we switch the pump off, heat is still generated within the AM due to the relaxation of $N_{ex}$. According to Eq. (1), the RIC adopts the temporal distribution of $\Delta T(t)$ (Fig. 5(b)). In Fig. 5(c), we plotted the temperature distribution $T_{QSSR}(r)$ along the diameter of the Nd:YAG rod. This radial distribution is well fitted by a second order polynomial and properly accounts for the occurrence of a thermal lens. The thermal focal length induced by this temperature gradient writes [7]:
where,According to Fig. 5(b), the quasi-static RIC between the center and edges of the rod in the QSSR is $\Delta n_{QSSR} \sim\;2.9{\times}10^{-5}$. Hence, the quasi-static focal lens induced by $\Delta T_{QSSR}$ is $f_{T} \approx {0.54}\;\textrm{m}$ at $\lambda ={1064}\;\textrm{nm}$ (Eq. (11)). This value is in good agreement with our experimental data $f_T \approx f_{qs} = 0.53 \pm {0.02}\;\textrm{m}$. The computed temperature modulation $\delta T(t)$ between the center and the edge of the Nd:YAG rod in the QSSR increases by $\sim {0.13}\;\textrm{K}$. During $\sim {1}\;\textrm{ms}$, it should result in a decrease of the thermal focal length of about $\frac {\delta f_T}{f_{T}} \approx \frac {\delta T(t)}{\Delta T_{QSSR}}= - 3.2\%$ where $\delta f_T$ accounts for the change of thermal focal lens. This is at variance with our experimental data which indicate an increase of the focal length by about $\frac {\delta f}{f_{qs}} \sim + 10\%$ that lasts $\sim {500}\;\mathrm{\mu}\textrm{s}$. We also perform a similar numerical simulation at $\lambda ={532}\;\textrm{nm}$ (Fig. 4). As already mentioned, in this case we noticed that the central wavelength of the pump diode laser shifted by $\sim {4}\;\textrm{nm}$. Hence, we slightly lower the pump absorption coefficient (from 7.5% to 5.5% ). Under this condition $\Delta T_{QSSR}^{532 nm}\sim {3}\;\textrm{K}$ whereas at maximum $\delta T(t) \sim {0.1}\;\textrm{K}$. Accordingly, and in good agreement with our experimental data, $f_{T}\approx f_{qs} \approx {0.74}\;\textrm{m}$. The temperature in the center of the Nd:YAG in the QSSR induces a decrease of the focal length of the AM. However, whereas our simulation indicates $\frac {\delta T(t)}{\Delta T_{QSSR}} \sim -3\%$ at maximum, our data show that $\frac {\delta f}{f_{qs}}\sim -8\%$ at maximum. Besides, while experimentally the decrease of the focal length lasts ${500}\;\mathrm{\mu}\textrm{s}$, our thermal simulation indicates that it should decrease during $\sim {1}\;\textrm{ms}$ (see Fig. 4(b)). In conclusion, the model which only considers the heating of the Nd:YAG rod accounts well for the experimental quasi-static focal lens $f_{qs}$, but it fails to correctly describe the modulation of $f_{qs}$ in the QSSR. It is also worth mentioning that under slightly saturated absorption condition the kinetics of the thermal and excited population will be modified.
Fig. 5. (a) Temporal evolution of the heat generation coefficient $Q_0(t)$ (blue curve) and the change in temperature $\Delta T(t)$ (red curve) between the center and the edge of the rod during a pumping cycle. (b) Temporal evolution of the difference of refractive index between the center and the edge of the Nd:YAG rod (purple curve). (c) Computed quasi-steady state temperature distribution inside the rod (green curve). The black curve is a second order polynomial fit of the temperature distribution. The fitting function writes $T(r)= A_0 + A_2 r^2$ where $A_0 = {302.03}\;\textrm{K}$ and $A_2 = {-1.89}\;\textrm{K} \cdot \textrm{m}^{-2}$.
Table 1. Parameters used to simulate the Nd:YAG rod heating
5. Modification of the Nd:YAG rod refractive index via the Nd$^{3+}$ excited states population
The small modulation of the RIC of the AM in the QSSR has been previously reported [13–17]. Antipov et al. [13] have studied in detail the Nd:YAG RIC using a Jamin-Lebedev polarization interferometer. In this experiment, the Nd:YAG crystal was excited by pulsed laser diodes centered at $\lambda _p={808}\;\textrm{nm}$ and/or Nd:YAG laser pulses centered at $\lambda _p={266}\;\textrm{nm}$. The RIC was measured using a probe beam centered at $\lambda ={633}\;\textrm{nm}$. Under their experimental conditions, they were able to demonstrate that the phase shift experienced by the probe beam exhibits two response times. In good agreement with the fitting of our experimental results (Fig. 4(b)), the shortest was $\sim 238 \pm\;{9}\;\mathrm{\mu}\textrm{s}$ and the longest was $\sim 6.0 \pm \;{0.4}\;\textrm{ms}$. They attributed these two time constants to the relaxation time of excited ${\rm Nd}^{3+}$ ions and the thermal response of the Nd:YAG rod. Accordingly, in addition to the thermal effect, they have added to the RIC the Nd:YAG change of polarizability contribution resulting from the ${\rm Nd}^{3+}$ excited state population. This contribution writes:
where $F_L = \frac {n_0^2 + 2}{3}$ is the Lorentz factor, $n_0$ the refractive index of the unexcited Nd:YAG rod, $N_{ex}$ the density of excited ${\rm Nd}^{3+}$ ions and $\Delta p$ the polarizability variation between the ground $^4I_{11/2}$ and the excited $^4F_{3/2}$ states at the probe wavelength. The polarizability $p(\nu )$ of the Nd:YAG rod at a frequency $\nu = \frac {c}{\lambda }$ is given by:Table 2. Data used to compute the polarizability of the $^4F_{3/2}\;{\rightarrow }\;^4I_{9/2}$ transition [13].
According to equation (Eq. (13)) and far from any transition of the Nd$^{3+}$ ions, the polarizability $p(\nu )$ of the Nd$^{3+}$ remains almost constant. Moreover, we will neglect the small dispersion of the YAG matrix refractive index around $\lambda _p={1064}\;\textrm{nm}$ and $\lambda _p={532}\;\textrm{nm}$. At $\lambda _p={532}\;\textrm{nm}$ or $\lambda _p={633}\;\textrm{nm}$, the polarizability change is $\Delta{p}\;\sim\;4{\times}10^{-26}\;\textrm{cm}^3$ [13]. However, as soon as the probe pulse frequency is close to a transition of Nd$^{3+}$ ions, the Nd:YAG rod polarizability rapidly changes. In Fig. 6(a), we plotted the Nd$^{3+}$ polarizability variation around the transition between the $^4F_{3/2}$ and $^4I_{11/2}$ states centered at 1064.2 nm. It is worth noting that a small change of the probe wavelength from 1064.5 nm to 1064.0 nm results in a large change from $\Delta p\sim\;-1.7{\times}10^{-25}\;\textrm{cm}^3$ to $\Delta p\sim \;1.5{\times}10^{-25}\;\textrm{cm}^3$ of the Nd$^{3+}$ polarizability. This latter evolution is of importance for our experiment where the probe pulse central wavelength is initially about $\lambda = {1064.5}\;\textrm{nm}$. Upon its amplification in the Nd:YAG rod, the probe pulse central wavelength is slightly shifted towards the optimum amplification wavelength from 1064.5 nm to 1064.2 nm. We highlight this phenomenon in Fig. 6(b), which displays the probe pulse spectrum with and without amplification in the AM. In the latter experiment, the probe pulse experiences a central wavelength shift $\Delta \lambda \sim \;{0.26}\;\textrm{nm}$ and a decrease of its spectral bandwidth. To account for this shift, we simulate the evolution of the gain at small signal, $g(\lambda )$, of the Nd:YAG versus the wavelength. Considering that the probe pulse spectrum is Gaussian in shape, centered at $\lambda = {1064.5}\;\textrm{nm}$ with a half-width $\delta \lambda = {0.5}\;\textrm{nm}$, we compute the amplified probe pulse central wavelength variation during the AM pumping (Fig. 6(c)). At the end of the pump pulse, which lasts ${500}\;\mathrm{\mu}\textrm{s}$, the probe wavelength shift is about $\Delta \lambda \sim \;{0.26}\;\textrm{nm}$. This induces the AM polarizability change (Fig. 6(d)). According to this figure, the maximum polarizability variation is $\sim \;1.3{\times}10^{-25}\;\textrm{cm}^3$. In Fig. 6(e), we plot the RIC temporal evolution between the center and the edge of the Nd:YAG rod during a pumping cycle. At $\lambda \sim {1064}\;\textrm{nm}$, $\delta n_e$ drops to $\sim\;-3{\times}10^{-6}$, whereas, it rises to $\sim\;1{\times}10^{-6}$ at $\lambda ={532}\;\textrm{nm}$. In the simulations, the pump intensity distribution along the Nd:YAG rod diameter is Gaussian in shape and induces a Gaussian distribution of $N_{ex}$. The $N_{ex}$ distribution is therefore well approximated by a second order polynomial, making it possible to directly infer the induced lens from $\delta n_e$.
Fig. 6. (a) polarizability of the ${\rm Nd}^{3+}$ ions as a function of the wavelength. (b) Spectrum of the laser probe with (red curve) and without (blue curve) amplification in the Nd:YAG rod. (c) Temporal evolution of the laser diode electrical pulse (red curve) and simulation of the shift $\Delta \lambda$ experienced by the amplified probe pulse during a pumping cycle. (d) Temporal evolution of the laser diode electrical pulse (red curve) and the polarizability seen by the amplified laser pulse when its central wavelength shift (blue curve). (e) Temporal evolution of the RIC induced by the population of the ${\rm Nd}^{3+}$ excited state when the laser probe is centered at $\lambda = {532}\;\textrm{nm}$ (green curve) and at $\lambda = {1064.5}\;\textrm{nm}$ (red curve) during a pumping cycle.
6. Discussion
Taking into account the thermal and excited population contributions to the RIC, the Nd:YAG rod focal length evolution writes (Eq. (9)):
Fig. 7. Comparison between the simulated and the experimental data. (a) For a probe pulse centered at $\lambda = {532}\;\textrm{nm}$ and a pump laser diode current of $I = {100}\;\textrm{A}$ and a duration $\tau _{pump} = {500}\;\mathrm{\mu}\textrm{s}$. (b) For a probe pulse centered at $\lambda = {1064.5}\;\textrm{nm}$ and a pump laser diode pulse with $\tau _{pump} = {500}\;\mathrm{\mu}\textrm{s}$ duration, with different electrical currents. (c) For a probe pulse centered at $\lambda = {1064.5}\;\textrm{nm}$ and pump laser diode electrical pulse of different durations with a current $I = {100}\;\textrm{A}$.
When the probe pulse is centred at $\lambda = {532}\;\textrm{nm}$, Fig. 7(a) indicates the thermal and the excited state population contributions to the AM focal length add up. When the pump is on, both $\delta n_{T}(t)$ and $\delta n_{e}(t)$ increase. When the pump is off, $\delta n_{T}(t)$ still slowly increases during $\sim\;{500}\;\mathrm{\mu}\textrm{s}$ (Fig. 5(b)) whereas $\delta n_{e}(t)$ more rapidly decreases according to the relaxation of $N_{ex}$ (Fig. 6(e)). In consequence, the AM focal length decreases rapidly during $\sim\;{500}\;\mathrm{\mu}\textrm{s}$. Afterwards, the AM focal length decreases more slowly due to heat diffusion in the AM.
When the probe pulse is centred at $\lambda = {1064.5}\;\textrm{nm}$, Fig. 7(b)-(c)) indicate a very different behavior. At this wavelength, the Nd$^{3+}$ polarizability contribution to $\delta n_{e}$ is large and negative. Since $\delta n_{e}\sim -3 \delta n_{T}$ at maximum, we simplify the discussion and assume that the behavior displayed these figures mainly results from $\delta n_{e}(t)+\Delta n_{QSSR}$ which kinetics is displayed in Fig. 6(e). When the pump is switched on (resp. off), $N_{ex}$ rapidly increases (resp. decreases), $\delta n_{e}(t)+\Delta n_{QSSR}$ decreases (resp. increase) inducing an increase (resp decrease) of the AM focal length. As $N_{ex}$ further increases, the AM gain increases resulting in a shift backward (resp. forward) of the probe central wavelength. The latter experiences a reduction (resp. an increase) of the Nd$^{3+}$ polarizability (Fig. 6(d)) which induces an increase (resp. decrease) of $\delta n_{e}(t)+\Delta n_{QSSR}$. Hence, the AM focal length decreases (resp. increases). As soon as the pump pulse is switched off, the reverse process takes place.
7. Conclusions
In summary, we evaluated the focusing effect induced in a Nd:YAG rod amplifier. The experiment we built makes it possible to measure at different wavelengths the temporal evolution of the focal lens induced in a AM rod submitted to periodic pump pulses of a few hundred microseconds. We also develop a model which well accounts for the evolution recorded experimentally. This model is valid in the unsaturated regime and low pump duty cycle. These experiments and the simulations indicate that two different mechanisms are responsible for a quasi-static focal lens $f_{qs}$ and its periodic modulation $\delta f(t)$. When the AM is optically pumped, its temperature increases and a quasi-static temperature gradient $\Delta T_{QSSR}$ is induced along the rod diameter. It results in a gradient of refractive index $\Delta n_{QSSR}$ which makes the AM acts as a lens with focal length $f_{qs}$. It is the latter contribution which is usually compensated in a conventional amplifier system. However, when the steady state regime is established, a small periodic temperature modulation $\delta T(t)$ of the AM remains. The latter induces a small periodic modulation of thermal AM focal length $\delta f(t)$. Besides the thermal effect, we have shown that another phenomenon related to the ${\rm Nd}^{3+}$ excited state population must be added. When the probe pulse wavelength is far from any ${\rm Nd}^{3+}$ resonances, for instance at $\lambda = {532}\;\textrm{nm}$, these two contributions add and result in a periodic modulation of $f_{qs}$. The different temporal evolutions of these two phenomena make it possible to easily discriminate them. When the probe beam wavelength is at $\lambda = {1064.5}\;\textrm{nm}$, close to a resonance of the ${\rm Nd}^{3+}$, it is amplified in the AM and experiences a drastic change in AM polarizability induced by the population excited state. This latter phenomenon results from the probe pulse amplification which induces a shift of its central wavelength which in turn results in an original evolution of periodic modulations of the $f_{qs}$. Our experiment clearly evidences this phenomenon and indicates that this effect may partly compensate for the thermal lens effect. This effect should be more pronounced reducing the quasi steady steady state lens. This can be achieved by reducing the pump repetition rate while keeping the pump peak power constant.
Funding
Conseil Régional Nouvelle-Aquitaine; Agence Nationale de la Recherche (LAMA ANR-18-ASMA-0004-01).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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