1 Introduction

Due to the physical size, flexibility and ease of fabrication, microchip lasers have been extremely characterized for desirable operating in different areas. They can be designed in either pulsed or continuous wave (cw) regimes possessing single frequency and single transverse mode [1, 2]. By the feasibility of using broad gain bandwidth materials fabricating a microchip laser providing very wide tunable near-infrared radiation have been implemented [3]. Recently, Q-switched microchip lasers are employed for nano- and picosecond generation of pulses with the peak power of several megawatts [4]. Owing to their small size and generating short pulses of several nanoseconds, they are becoming replacement for other laser sources such as CO2 laser for the variety industrial applications like micro drilling and machining [5]. On the other hand, an intensive and narrow bandwidth microchip laser can be also utilized as a pump source for optical nonlinear interactions such as second- and third-harmonic generation of solid-state and semiconductor lasers as well as optical parametric oscillation and difference-frequency generation [6]. The most common form of a microchip laser generally uses a short cavity containing sub-centimeter sized gain medium embraced by a Q-switching slab. Such a small structure provides large cavity mode spacing and consequently, only one mode can be amplified under the gain envelope.

In addition to the single frequency operation, dual-mode operation is also possible along an adjustable frequency spacing. One example is the generation of almost stable dual-frequency radiation from a 2 mm diode-pumped microchip Nd:YAG laser with a variable frequency spacing from 14 MHz to 1.5 GHz [7]. The fluctuation of the frequency beating was reported less than 10 kHz over a period of 10 minutes. It was achieved by the variation of orthogonally linearly polarized modes of the cavity.

Multimode hopping behavior of a microchip Nd:YAG laser is also improved by use of an adjustable external cavity around the laser [8]. Therefore, the output spectral properties of a microchip laser can be modulated through the length of the laser cavity. However, technical limitation of the crystal growth may impose a further limitation on the laser crystal length which in turn affects the performances of the laser. Depending on the crystal length the absorption of the pump light in the cavity medium determines the output power of the laser. The realization of simultaneous oscillation of two or more modes can be performed by exploiting large cavity mode spacing associated with a few millimeters of a Nd:YAG crystal.

Since the cavity space of a microchip laser is completely filled by the coated gain material, the optical path difference between the traveling rays inside the resonator associated with the refractive index variation may change the operating characteristics of the laser [9].

It is partly due to the non-uniform distribution of the heat generated in the laser crystal which affects the laser performances such as stability, beam quality as well as single-frequency operation of the laser [10–12]. Moreover, thermal constraints are capable of rotating the polarization state of the output laser beam which limits its application to the laboratory and field measurements [13]. Depending on the thermal stabilization technique, geometry of the device and material properties, heat flow in the laser cavity can be investigated in one, two or three dimensions.

In this paper, three-dimensional heat distribution in the cavity of a Q-switched microchip Nd:YAG laser is modeled for different cooling conditions and for various number of pulses propagating throughout the chip. It is shown that the relaxation of the medium to the steady-state is strongly dependent to the pulse repetition rate, laser pump power and temperature of the Nd:YAG chip which can be changed and controlled within a specific range. Computer-generated diagrams are also presented schematically to illustrate the relationship between those variable parameters and the relaxation time scale of the medium. The effects of the generated heat on the output spectral properties of the emerging laser beam are also numerically simulated for actual practical values. In the experimental part of this investigation, the mode behavior of a picoseconds passively Q-switched Nd:YAG microchip laser is accurately characterized by careful temperature control of the Nd:YAG chip which was performed by an electronically controlled Peltier device. It is observed that the number of the oscillating output modes is quiet temperature dependent and single-longitudinal mode operation can be obtained by temperature tuning of the Nd:YAG crystal. Obtained results show good agreement between the described numerical model and the experimental values.

2 Thermal modeling of a microchip laser

A typical microchip laser experiences heating of the gain medium through several kinds of process. Those include non-radiative transitions of the excited laser levels and the excitation of undesirable quantum states. The latter is a consequence of broad band pump sources which can be avoided by alternative narrow spectral bandwidth diode lasers in the efficient end-pumped scheme. Owing to the small physical size of the gain material in a typical microchip laser, its cavity is very sensitive to the generated heat and therefore understanding of the heating mechanism and its distribution is essential for stability and long time laser operation. On the other hand the advantage of utilizing small sized gain medium is the relative ease and speed of heat conduction to the provided surrounding medium. In this section thermal effects in an end-pumped microchip Q-Switched Nd:YAG laser is modeled analytically to obtain the optimum oscillation condition of the laser modes.

The standard differential equation that determines the heat distribution and flow in the laser crystal is [9]:

$$ \rho c \frac{\partial T(x,y,z,t)}{\partial t} =Q (x,y,z,t )+K\nabla^{2}T(x,y,z,t) $$
(1)

where constant parameters ρ, c and K are density, specific heat and thermal conductivity of the Nd:YAG crystal, respectively, and Q(x,y,z,t) is the thermal load per unit volume inside the medium. Since heat generation in the laser material is a time dependent process, therefore, for a pulsed laser left-hand side term of Eq. (1) is appreciable.

Pulse width of a Q-switched microchip laser is typically in the order of a few nanosecond [7] which it is quite small compared to the thermal response time of the gain medium. Thus it can be assumed that the generated heat is not too sensitive to the traveling pulse shape. Eventually, the associated function with the thermal source in the crystal can be defined as

$$ Q =\left \{ \begin{array}{l@{\quad}l} \frac{\eta P_{\mathrm{pump}}}{\pi w_{p}^{2}(z)}\alpha e^{-\alpha z}, & r \leq w_{p}\\ 0, & r > w_{p} \end{array} \right . $$
(2)

where r is the transverse distance from the cavity axis, P pump is the input pump power, α is the pump-dependent absorption coefficient of the Nd:YAG crystal, w p is the beam waist of the pump beam inside the medium and η is the quantum efficiency of the lasing transitions which is defined as the ratio of the laser photon energy to the absorbed pump photon energy. For an efficient Nd:YAG laser η is approximately equal to 24 %.

Heat generation mechanism in solid-state lasers is mostly due to non-radiation transitions from the pump level to the upper laser level and to the transitions from the lower laser level to the ground state.

According to the Beer–Lambert law non-uniform heat distribution in the Nd:YAG chip is induced by non-uniform absorption of the pump intensity across the medium. Additional non-uniformity can be attributed to the variation of the spot size of the focused pump beam while traveling throughout the z-axis of the cavity.

Furthermore, consecutive traveling laser pulses inside the cavity are capable of changing the heat distribution too until the temperature relaxes to the steady-state value. Depending on the input pump power and repetition rate of the laser pulses, three-dimensional heat equation (1) must be continuously solved for a certain number of pulses. Here calculation is performed by using finite-difference approach for more than 104 pulses. Figure 1 shows schematically the microchip Nd:YAG laser geometry which is modeled in this investigation.

Fig. 1
figure 1

The z-cut schematic modeling geometry of the diode-pumped microchip Nd:YAG laser. The direction of the pump light is normal to the Nd:YAG aperture as shown by concentric red circles

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The required boundary conditions are associated with the Nd:YAG laser structure used in this investigation. Those are determined by the following practical assumptions:

(1) As can be seen in Fig. 1, Nd:YAG crystal is wrapped in a very thin indium foil inside a bulk copper housing. By this arrangement, heat is transferred to the copper via convection (Eq. (3)). Therefore, heat convection between the crystal and its surrounding environment can be expressed as

$$ -K\frac{\partial T(x,y,z)}{\partial z}\bigg|_{z=0}=\eta \bigl[T (x,y,z=0 )-T_{0}\bigr] $$
(3)

where h is assumed either the convection coefficient of the copper block or surrounding air, T 0 is the constant temperature of the environment and K is the conduction coefficient of the Nd:YAG crystal [14].

(2) The bottom of the massive copper housing is in contact with a Peltier device for temperature variation and control of the Nd:YAG crystal within the accuracy of ±0.1 °C.

The most important consequence of the non-uniform heat distribution is the formation of different optical paths associated with the bunches of rays within the beam cross-section which experience a distribution of temperature while propagating across the cavity. It can be treated by OPD approach the same way as can be used for thermal lensing description of the solid-state materials [9]. In the media possessing smaller OPD values, thermal effects are quite negligible. Thermally induced OPD is either because of the non-uniform refractive index formation inside the medium or inhomogeneous thermal expansion of the laser crystal length. For a very thin slab of the gain medium the following differential equation can be used as in [9]:

(4)

The constant parameters v,α T ,n and C r are the Poisson ratio, thermal expansion coefficient, refractive index and photoelastic coefficient of the Nd:YAG crystal, respectively, where T(a,b,z,t) is the temperature of its side faces which are in contact with the bulk copper block.

In the first bracket of Eq. (4), the first term is responsible for temperature variation of the refractive index, while the second one is related to the thermal expansion of the crystal. The third term shows the contribution of photoelastic effects in the OPD value and, however, it can be neglected because of its much smaller value compared to the others. The last consequence of the thermal effects is the spectral bandwidth broadening of the laser which leads to the variation of the number of oscillating modes under the gain envelope. The most significant broadening mechanism, Δv b , here can be associated with the heating process as can be expressed by [15]

$$ \Delta v_{b}=v_{0}\biggl(\frac{8k_{B}T\ln 2}{CV} \biggr)^{\frac{1}{2}} $$
(5)

where

$$ C=\frac{k_{B}T}{V}\biggl(\frac{\Delta l}{l}\biggr)^{-2} $$
(6)

is the longitudinal elastic constant of the Nd:YAG crystal, k B is the Boltzmann’s constant, Δl is the change in the cavity length and V is the lasing mode volume. By simultaneous calculation of Eqs. (1) and (5), the broadening of the laser spectral bandwidth due to the non-uniform thermal distribution and hence the number of the oscillating modes can be evaluated.

3 Thermal simulation

Following the heat distribution in the Nd:YAG crystal, Eq. (1) is solved by use of finite-difference method in a provided FORTRAN code. For further comparison between theoretical and experimental results, it was performed for actual practical values associated with the microchip Nd:YAG slab. Those values are listed in Table 1.

Table 1 Characteristics of the Nd:YAG chip which are utilized in the experimental part of this investigation
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As specified previously, laser crystal interchange the heat with the surrounding copper block via convection with the coefficient of h=400 W/m2 °C. The same mechanism is true for the thermal interchange between two end facet of the Nd:YAG crystal with the air. The related convection coefficient can be described by h=10 W/m2 °C. By taking into account that the diode pump of the Nd:YAG laser provides output power of about 2 W and is concentrated onto a 200 μm beam waist, Eq. (1) can be solved while the effects of the Peltier cooling are ignored. As depicted in Fig. 2, heat is non-uniformly distributed in the lateral directions.

Fig. 2
figure 2

Heat distribution across the Nd:YAG crystal while the Peltier is off. The temperature of the environment is assumed 25 °C and repetition rate of the laser is assumed 4 kHz

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It is further clarified in the figure that most of the heat is distributed at the input of the crystal where the photon population is intensive. Shown in Fig. 3, while the Peltier is adjusted for a certain temperature, heat distribution mechanism is remained the same but local temperatures are decreased. Instead, the difference between higher and lower temperature points is increased. One physical explanation is that the flow of the confined heat can easily escape from the device by the Peltier cooling effect.

Fig. 3
figure 3

Heat distribution across the Nd:YAG crystal while the Peltier is on and set for 20 °C. The temperature of the environment is assumed 25 °C and repetition rate of the laser is assumed 4 kHz

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The time scale of the temperature relaxation to the steady-state value can be estimated with respect to the two most effective variable parameters as pump power and repetition rate of the utilized Nd:YAG laser. By assuming a fixed repetition rate at 4 kHz, the relaxation time can be computed for various pump powers. Figure 4 illustrates that, depending on the diode pump power and cooling condition, the point with higher temperature inside the crystal tends to the steady-state value in a different manner. Although the Nd:YAG microchip laser used in the experiment utilizes a 2 W diode pump, additional powers are also included to investigate the thermal relaxation dependence of the Nd:YAG crystal to different input pumps.

Fig. 4
figure 4

Thermal relaxation with time for the point possessing higher temperature inside the crystal, while in (a) Peltier is off and in (b) Peltier is set for 20 °C. Repetition rate of microchip laser is kept constant at 4 kHz equal to the practical value. The pump beam waist was assumed 200 μm and the temperature of the environment is assumed 25 °C. With increasing the pump power the required time for obtaining the steady-state condition is increased

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Indicated in the figure when pump power exceeds 2 W, it takes much longer until crystal reaches to the steady state temperature. This is because of the low thermal conductivity of the Nd:YAG crystal which lags the heat flow to the surrounding environment. Figure 5 shows the thermal relaxation of the crystal for different repetition rates of the pulsed microchip laser.

Fig. 5
figure 5

Thermal relaxation with time for the point possessing higher temperature inside the crystal, while in (a) Peltier is off and in (b) Peltier is set for 20 °C. The power of the pump is kept constant at 2 W equal to the practical value. The pump beam waist was assumed 200 μm and the temperature of the environment is assumed 25 °C. The steady-state condition is reached much later for lower pulse repetition rates

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As characterized in the figure as long as the repetition rate of the pulsed microchip laser increased, steady-state condition is reached in a shorter time scales. As a result it is found that the contribution of the repetition rate to the thermal relaxation is more significant in comparison with the pump power contribution.

With the thermal distribution characterization inside the Nd:YAG crystal, the OPD values of individual traveling pulses can be calculated by taking into account the cooling effect of the Peltier. The result of this calculation is performed in Fig. 6.

Fig. 6
figure 6

Calculated OPD values of the traveling pulses across the medium in the vicinity of the crystal width. It is performed with two conditions, with and without Peltier. The power of the pump is 2 W, pump beam waist is 200 μm, and repetition rate is assumed 4 kHz. The temperature of the environment is assumed 25 °C and Peltier is set for 20 °C while cooling

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The effect of the cooling on the OPD reduction is quite remarkable, as expected. It is decreased by a factor of about 100. Therefore, it can be concluded that by cooling of the Nd:YAG crystal the linewidth of the laser can be significantly reduced too. This conclusion is verified in the given results depicted in Fig. 7.

Fig. 7
figure 7

Variation of the fundamental bandwidth of the Nd:YAG gain medium versus pump power with and without (□) Peltier cooling. Cooling of the crystal is implemented for (▲) T=30 °C, (▽) T=25 °C, (■) T=20 °C, (∘) T=15 °C. Calculation is performed for 4 kHz repetition rate and 200 μm pump beam. The temperature of the environment is assumed 25 °C while cooling

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As specified in the above illustrated figure, whereas the Nd:YAG crystal is cooled down to about 15 °C, the bandwidth of the laser is significantly reduced. However, the laser bandwidth is not significantly affected by the pump power. As a result OPD as well as laser bandwidth reduction by heat extraction from the crystal may result to the change of mode behavior of the laser output.

4 Experimental arrangement

As illustrated schematically in Fig. 8, an optical arrangement is provided to verify the computed results described in the above.

Fig. 8
figure 8

Experimental arrangement of a passively Q-switched microchip laser for the characterization of its output modes with the variation of the Nd:YAG temperature utilized by a Peltier device. The laser cavity consists of a 1.5×1.5×2 mm3 Nd:YAG crystal including Q-switching Cr+4:YAG slab

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Laser pumping is provided by the output beam of a 2 W diode laser operating at 808 nm. It is focused into a Nd:YAG crystal by using a graded index lens. Nd:YAG crystal is embraced by a Cr+4:YAG slab as Q-switching element to form a 2±0.05 mm cavity length which provide a mode spacing of approximately Δλ=0.154 nm. After optimization and accurate alignment an average output power of about 170 mW was obtained from the microchip laser setup. Output power characterization of the fabricated microchip laser is verified in Fig. 9.

Fig. 9
figure 9

Variation of the Nd:YAG microchip laser output power with electrical input power of the diode pump for three different temperatures. It is performed by a Peltier which was mechanically connected to the pump diode

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By utilization of the knife-edge technique, the beam size of the Nd:YAG laser beam was measured with w=1 mm while its beam quality factor was obtained M 2=1.1. Those apertures of the provided Nd:YAG chip where coated to form required feedback for the lasing. The repetition rate of the laser was measured 4 kHz.

Shown in Fig. 8, the Nd:YAG chip is mounted on an intermediate air-cooled Peltier device the same way as it is pictured in Fig. 1. This arrangement is potentially capable of directing the heat flow from the cavity to the surrounding environment and hence facilitates the thermal relaxation of the gain medium. As predicted in the simulated model the mode behavior of such microchip device is very temperature sensitive and it can be modified through temperature control of the Nd:YAG chip. While the temperature of the chip was accurately changed within a specific range the laser beam is directed toward a commercial 7.5 GHz Fabry–Pérot analyzer. Figure 10 shows trace spectrum of the microchip laser while the Peltier changes the Nd:YAG crystal temperature from T=31 °C to T=19 °C with ΔT=±0.1 °C of accuracy.

Fig. 10
figure 10

Output spectrum of the microchip laser traced by a 7.5 GHz Fabry–Pérot analyzer while the connected Peltier to the Nd:YAG crystal is set for three different temperatures. The experiment is performed at room-temperature and variable pump power. The pump beam waist is approximately equal to 200 μm and repetition rate is measured 4 kHz. In (a) T crystal=31 °C, in (b) T crystal=20 °C and in (c) T crystal=19 °C. The fluctuating spikes superimposed on the feature of the laser modes are associated with the interferometric nature of the utilized Fabry–Pérot analyzer itself. They are regularly distributed around the center of the interference pattern

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Wavelength calibration was carefully performed using the well-known atomic transition lines of a standard discharge argon lamp. This results to the average linewidth of Δλ=0.05 nm for modes which is quite comparable with the value reported in the literature for a typical Nd:YAG laser [17].

As long as the temperature of the Nd:YAG crystal is raised up to T=31 °C, due to the low thermal response time of the crystal the generated heat is confined inside the cavity and the heat flow to the outside is decreased. Therefore, individual pulses experience their own OPD values, and simultaneous dual-mode operation becomes possible as shown in Fig. 10(a). This behavior is advantageous for many applications such as THz generation in either semiconductor devices or nonlinear interactions based on difference-frequency generation.

When the Peltier starts to cool the Nd:YAG crystal down to a lower temperature as T=20 °C they will now compete in the energy and one is attenuated while another become stronger. This new situation is depicted in Fig. 10(b). As indicated in Fig. 10(c), further Nd:YAG cooling results to the quenching of the unable mode and the fundamental single-mode operation of the laser is recovered. Mode fluctuation can be alternatively remounted by temperature variation of the Nd:YAG chip through Peltier switching between single-mode and dual-mode temperatures. Referred to Fig. 7, this special behavior which is observed for the first time, can be interpreted physically as the non-uniform heating distribution changes the bandwidth of the Nd:YAG gain envelope and gain switching between the oscillating modes become possible. In the previous work [16] by using a thin slab etalon inside a cavity of a continuous wave Nd:YAG laser, oscillating of two simultaneous longitudinal modes was performed at room-temperature. However, the significant reduction intensity of those modes was because of the induced insertion loss of the etalon to the cavity.

5 Conclusion

The formation of heat distribution in a cavity of a passively Q-switched microchip Nd:YAG laser is modeled. By numerical investigation of the given differential heat equation together with the real boundary condition of a laboratory microchip laser it was shown that the heat is distributed non-uniformly inside the Nd:YAG chip. The effects of the input parameters such as pulse repetition rate and pump power on this distribution was modeled and schematic diagrams were generated from the results. Dependence of the temperature relaxation time on the pulse repetition rate was deduced and OPD values for individual traveling pulses experiencing different local temperatures was computed and illustrated. Most important effect of the non-uniform heat distribution in the Nd:YAG chip was shown as gain bandwidth variation which is capable of changing the number of laser output modes. Those investigations were performed when the Peltier was set for single-mode and dual-mode temperatures. In the experimental part of this work a laboratory michrochip laser was fabricated and its spectral properties were characterized precisely. It was observed that due to the much heat confined in the cavity and hence formation of two different OPD values along the propagation direction two strong longitudinal modes could be oscillated simultaneously. The thermal-broadened gain envelope is also capable of oscillating for those two longitudinal modes. For a certain temperature of the Nd:YAG chip which was performed by a Peltier element connected to the Nd:YAG chip, single-mode operation became possible. To our knowledge, gain switching between single- and dual-mode operation of a microchip Nd:YAG laser owing to the thermal effects has been observed for the first time in this investigation.