Study of operation dynamics for crystal temperature measurement in a diode end-pumped monolithic Yb:YAG laser.

Study of operation dynamics for crystal temperature measurement in a diode endpumped monolithic Yb:YAG laser Hee-Jong Moon,1,* Changhwan Lim,2 Guang-Hoon Kim,3 and Uk Kang3 2

1 Department of Optical Engineering, Sejong University, Seoul 143-747, South Korea Quantum Optics Division, Korea Atomic Energy Research Institute, Daejeon 303-353, South Korea 3 RSS Center, Korea Electrotechnology Research Institute, Seoul, 121-835, South Korea * [email protected]

Abstract: A temperature measurement scheme was proposed in a diode end-pumped thin monolithic Yb:YAG laser by analyzing the red-shifting behaviors of each lasing peak. The amount of peak shift was measured on the basis of the threshold lasing spectrum by using a chopped pump beam. In order to determine the effective scale factor, the ratio between the peak shift and the temperature rise, the dynamics of the spectral shift, the output beam profile, and the output power were investigated. The effective scale factor was determined to be about 0.0114 nm/°C in the case of the crystal sandwiched by copper bocks with a hole, wherein the plane stress approximation is valid. On the other hand, the effective scale factor significantly decreased in the case of the crystal sandwiched by sapphire plates. ©2013 Optical Society of America OCIS codes: (140.3615) Lasers, ytterbium; (140.3480) Lasers, diode-pumped; (140.6810) Thermal effects.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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#199699 - $15.00 USD Received 17 Oct 2013; revised 1 Dec 2013; accepted 4 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031506 | OPTICS EXPRESS 31506

14. S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG,” Appl. Phys. B 79(2), 221–224 (2004). 15. S. Chenais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89–153 (2006). 16. J. Petit, B. Viana, and Ph. Goldner, “Internal temperature measurement of an ytterbium doped material under laser operation,” Opt. Express 19(2), 1138–1146 (2011). 17. H. J. Moon, Y. T. Chough, and K. An, “Cylindrical microcavity laser based on the evanescent-wave-coupled gain,” Phys. Rev. Lett. 85(15), 3161–3164 (2000). 18. J. Dong, A. Shirakawa, K.-I. Ueda, and A. A. Kaminskii, “Effect of ytterbium concentration on cw Yb:YAG microchip laser performance at ambient temperature-Part I: Experiments,” Appl. Phys. B 89(2–3), 359–365 (2007). 19. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley, 1991). Chap 14. 20. O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31(6), 763–765 (2006). 21. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). 22. A. E. Siegman, Laser (Oxford University, 1986). Chap 25. 23. R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAlO3, LiYF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80-300 K temperature range,” J. Appl. Phys. 98, 103514 (2005). 24. S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers- Part II: Evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004). 25. A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28(4), 1057–1069 (1992). 26. J. T. Verdeyen, Laser Electronics (Prentice-Hall, 1995), Chaps. 3 and 6. 27. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1970). Chap 13. 28. X. Xu, Z. Zhao, J. Xu, and P. Deng, “Thermal diffusivity, conductivity and expansion of Yb3xY3(1-x)Al5O12 (x = 0.05, 0.1 and 0.25) single crystals,” Solid State Commun. 130(8), 529–532 (2004). 29. D. C. Brown, R. L. Cone, Y. Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).

1. Introduction Yb-doped crystals, such as Yb:YAG, have attracted a great deal of interest because they have a number of interesting properties for efficient diode-pumped solid state lasers. Due to the simple electronic structure (excited state manifold 2F5/2 and ground state manifold 2F7/2) of Yb3+, excited-state absorption, concentration quenching can be avoided [1,2]. The quantum efficiency can exceed 0.9 by pumping with a 940 nm laser diode (LD), which reduces the thermal load compared to the conventional Nd:YAG laser. The doping concentration can be high, which is beneficial in diode-end-pumping geometries with small crystal thickness [3–6]. Due to the quasi-three level nature of Yb:YAG at room temperature, the sublevels of 2F7/2 have considerable populations [7]. Because the reabsorption loss at 1048 nm is smaller than that at 1030 nm, the laser prefers oscillation around 1048 nm if the doping concentration is high or the reflectivity of the output coupler is high, even though the emission cross section at 1030 nm is larger than that at 1048 nm. To achieve an efficient laser, cooling configurations should be adapted to remove the generated heat to the surrounding heat sink. In end-pumping cases, various cooling geometries were studied such as edge cooling with a heat sink, face cooling, and conductive cooling [8–11]. In order to evaluate the efficiency of the cooling geometry, it is essential to measure the crystal temperature, particularly in the illuminated region. The radial gradient of the temperature profile within the pumped region can be inferred from measurements of the thermal lens strength [12,13]. Measurement of absolute temperature is also important because the laser properties are dependent not only on the temperature gradient but also the absolute temperature itself. Few noble measurement schemes for absolute temperature were reported. Absolute temperature mapping on the surface of the crystal was reported using an infrared camera with Yb:YAG crystals [14,15]. Internal temperature measurement in a Yb:GaGdAlO4 under laser action was reported by analyzing the fluorescence spectra of impurities [16], in which the signal was collected orthogonal to the beam axis of the exposed crystal.


In this paper, we propose a new measurement scheme for the average temperature of the crystal along the beam axis by investigating the dynamics of the spectral shift, the thermal lens, and the output power using a chopped pump beam. The temperature and strain profiles in a monolithic crystal produce phase change on the cavity modes, resulting in the spectral shift of each lasing peak. The amount of peak shift is linearly related to the temperature change. The scheme was applied to two cooling geometries. 2. Experimental A schematic diagram of the experimental setup is shown in Fig. 1. The laser crystal was 20 at.% Yb:YAG with a 0.75 mm thickness and 5 mm × 5 mm cross section. The entrance face of the crystal had a dichromatic coating with a high reflectivity at around 1050 nm and with a high transmission at 930 nm. The other output coupling face also had a dichromatic coating with a high reflectivity at 930 nm for uniform pumping along the crystal and with ~ 92% reflectivity at around 1050 nm. The heat removal was conducted by two contacting copper blocks having a central hole at both faces of the crystal as shown in Fig. 1. The diameter D of the hole was chosen to be 3 mm, which is 4 times larger than the crystal thickness. The crystal used was temperature controlled in a crystal oven, in which a thermistor sensed the copper temperature. A fiber-coupled cw laser diode (FC-LD) was used for pumping, wherein the core size was 50 μm with NA of 0.12. The LD beam from the fiber passed through an optical isolator consisting of a Faraday rotator and two prism polarizer in order to block reflected light to the fiber. Three aspheric lenses (L1 ~ L3) were used for collimation, focusing, and recollimation of the pump beam. The pump beam finally focused by L4 on the crystal through the center of the copper hole. The maximum pump power at the crystal was ~ 1.05 W with a wavelength of 932 nm. In order to investigate operation dynamics, an optical chopper was inserted between L2 and L3 for producing repetitive pump pulses. A specifically fabricated chopper blade had a small single slot for one pulse per turn. The pump beam can be operated in cw mode or pulse mode by translating the chopper. The timing of the chopped pulse was monitored by a digital oscilloscope with a photo diode PD1 by detecting the reflected beam from the crystal. The cw or pulsed lasing spectra were measured with an optical spectrum analyzer (OSA; ANDO AQ6315A) coupled with a single mode optical fiber. The instantaneous intensity profile of the output beam was measured by a translatable photo diode PD2 with a small pinhole (size of 50 μm) attached on the front face of its window. The dynamic of the output power was measured with a photo diode PD3 where the output beam was collected on the detection area by a lens L5. All photo diodes were operated in the level of the linear regime.

Fig. 1. Experimental setup.


3. Characteristics of continuous wave (cw) operation The intensity profile and divergence property of the pump beam are important parameters on the temperature profile, and lensing effect. Figure 2(a) shows the measured beam profile at the focal point in free space by using the knife edge method. The profile was slightly asymmetric and irregular. The beam size (the distance between tail points of the profile) was about 68 μm . The measured beam sizes along the propagation distance were shown in Fig. 2(b). The beam size varied in the range of 68 ~ 78 μm along the crystal. Because LD wavelengths were about 10 nm away from the absorption peak 941nm, the absorption efficiency during the double pass in the crystal was measured as relatively low as ~ 82% and was insensitive to the LD current or copper temperature, which reduced the temperature gradient along the crystal. The threshold pump power in cw mode was about 0.17W. As the pump power Pin increased, the output power of Yb:YAG laser increased linearly up to a maximum power of about 0.34 W. The slope efficiency with respect to the absorbed power was about 48%. 120 110

Pump Beam Size (μm)

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90

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Fig. 2. (a) Measured pump beam profile at the focal spot in free space, (b) Dependence of pump beam size along the propagation distance. The corresponding locations of two crystal surfaces are indicated accounting for the refraction effect of pump beam in the crystal.

4. Spectral behaviors of lasing modes Figure 3(a) shows the dependence of cw lasing spectra with Pin at a copper temperature Tc = 30 °C. The lasing spectra appeared around 1050 nm and consisted of several periodic peaks with a spacing of 0.4 nm. The mode spacing can be approximated as λ 2 / (2n 0 L) and calculated to be about 0.404 nm by using n0 = 1.82, which agrees well with the measured value. These lasing peaks should correspond to different longitudinal modes with the same transverse mode of TEM00. As Pin increased, two kinds of wavelength shift stemming from the crystal temperature increase were observed. First of them, the envelope of the lasing spectrum showed a red-shift. The center wavelength of the envelope in a broad gain medium can be determined from the threshold condition for laser oscillation [17]. Due to the increased reabsorption loss in Yb:YAG, the center wavelength tends to shift to lower absorption region (longer wavelength side), resulting in the red shift of the envelope with the increased pump power [18]. Second, each peak of longitudinal modes shifted also to longer wavelength side. This red shift of each peak is due to the round-trip phase change experienced by the cavity modes. In Fig. 3(a), the amount of envelope shift was fairly larger than that of the peak shift. It is notable that the crystal temperature rise can be inferred from the peak shift regardless of the envelope shift because the mechanisms of the both shifts are entirely different from each other. The frequency pulling effect during lasing cannot also affect the shift of each peak because it is negligibly small in this high finesse (~ 70) cavity [19]. As the crystal temperature increases, the crystal strain also increases. The crystal strain field produces an increased crystal length and bulging surface [15]. The thermally induced refractive index profile and bulging surface induce a thermal lens. There might be other


sources of lensing effect, such as the refractive index variation by change in populations and the aperture guiding effect [20,21]. In order to find a dominant lensing mechanism and to exactly measure the amount of peak shift, the dynamics of the laser performances in pulse mode is required to be surveyed. (b)

Intensity(a.u.)

Pin = 1.05W

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Fig. 3. (a) CW lasing spectra with Pin at Tc = 30 °C Shifting behaviors of each peak were indicated as horizontal arrows starting from a peak whose position was indicated as a vertical line. The center wavelength (vertical arrow) of the lasing envelope (dotted line) also showed a red-shift, (b) Single shot output power around the threshold (A & B) in pulse mode. (C) ; averaged signal of multiple shots, (c) Behavior of Δλ(τ) with a fixed Pin, measured in a very low chopping frequency.

Figure 3(b) shows a typical output behavior in PD3 around the lasing threshold in pulse mode. The delay time τ is defined as the time elapsed from the on time of the pump pulse. When the pump pulse is on (τ = 0), the population of the upper level increases until the threshold condition is satisfied. Above the threshold, laser spiking occurs (A and B) [22]. The threshold delay time τ th (or the laser buildup time) decreased as Pin increased. The shifting dynamic of a lasing peak in pulse mode (the duty cycle was as low as ~ 2% for preventing temperature change on the whole crystal) could be obtained from the external trigger mode operation of OSA by adjusting the delayed trigger time from the oscilloscope. Typical behavior of a peak shift Δλ( τ) (from the threshold peak) was measured, as shown in Fig. 3(c) with Pin ~ 0.85 W and Tc = 30 °C. While τ th became shorten with Pin, the spectral peak position measured at τ th was not changed with Pin, irrespective of the change of τ th . The peak shift Δλ ( τ) increased fast, up to a few ms (~ 3 ms) as fitted as an exponential decay line and then more slowly. The dynamics of the crystal temperature rise should resemble to that of Δλ( τ) . Figure 4(a) shows the shifting behavior of a threshold peak (relative to the one at Tc = 30 °C) measured by varying Tc. The threshold peak shifted linearly with the slope of 0.0112( ± 0.0003) nm/°C in the range of Tc = 30 ~ 50 °C. If the aforementioned other lensing effects are negligible, the shift of a threshold peak must be induced by thermal origin. Hence,


the slope or corresponding scale factor SLu of a cavity mode in this case of uniform temperature change can be described as

SL u ≡

dλ λ   dn   dL   λ  dn  λ L   + n0  γ. = + n 0 αT  ≡  = dT n 0 L   dT   dT   n 0  dT  n0

(1)

dn is the thermal coefficient of refractive index, α T is the linear thermal expansion dT coefficient, and γ is the thermal change of the optical path length. Measured value of SLu ~ 0.0112 nm/°C corresponds to γ ~ 19.3 × 10−6[K−1], which is similar to the reported value ~ 18.6 × 10−6[K−1] measured at λ = 1064 nm near room temperature with undoped YAG [23]. The overall peak shift Δλ in cw mode, defined as the peak shift from τ = τ th to τ = ∞ , was measured as the spectral difference between a peak position in cw mode (Fig. 3(a)) and that at threshold in pulse mode. An example is shown in the inset of Fig. 4(b). The measured Δλ was nearly linear to Pin, as shown in Fig. 4(b). If the crystal was pumped with no laser oscillation, Δλ will follow approximately the green line passing the cw threshold point because the thermal load under no laser action would be enhanced compared to the case of laser oscillation [24]. The measured peak shifts Δλ and Δλ(τ) are important parameters to estimate the temperature rise in cw mode and pulse mode, respectively. 0.25

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Fig. 4. (a) Temperature dependent peak shift at threshold in pulse mode, (b) Dependence of overall peak shift Δλ on Pin at Tc = 30 °C in cw mode. Inset: A peak in cw spectrum (red line) was red-shifted ( Δλ ) from that in the threshold spectrum (blue line) in pulse mode (Pin ~ 0.75 W).

5. Temperature profile in the crystal and scale factors

Figure 5(a) shows a schematic average temperature profile of the Yb:YAG crystal in Fig. 1 expected when heat was generated by the absorbed cw pump beam. The temperature profile T(r, z) is a function of the radial distance r and of the longitudinal distance z. In a steady state case, the heat equation describing T(r, z) is given as [15] ∇ 2 T( r, z ) =

Q th ( r, z ) . Kc

(2)

where K c is the thermal conductivity of the crystal and Q th ( r, z ) ( = ηΡ abs ( r, z ) ) is the thermal load per unit volume, and where Ρ abs ( r, z ) is the absorbed power per unit volume and η is the fractional thermal load. T(r, z) can be determined by the first boundary condition (continuity of the thermal flux) and second boundary condition of the crystal at z = 0, L described as


 ∂T  j ⋅ n = K c   = H( Ts (r) - Tc ).  ∂n 

(3)

where j is the thermal flux density, Ts (r) is the surface temperature of the crystal and H is the heat transfer coefficient. Ts (r ≥ b = D / 2) might be different from Tc because the coppercrystal interface should not be an ideal contact due to the existence of the coating layer on the crystal, the thin oxide layer and surface roughness on the copper. The determination of T(r, z) is very difficult because H is generally unknown and the fractional thermal load η under laser action depends on the laser extraction efficiency as previously reported [24]. Region A in Fig. 5(a) is hottest due to the existence of heating sources. In region C, heat transfers to copper through the interface with a temperature gap profile. The measured overall peak shift Δλ in Fig. 4(b) reflects the average temperature rise (T(0) − Tc ) along the beam axis. The goals of following analysis are to find the relationship between the peak shift induced by phase change and the temperature rise, as well as to introduce an effective scale factor relating both parameters.

Copper

L

Yb:YAG

(a)

Tc

(b)

r

C

L+ΔL(r)

b = D/2 0

A

2a

2a z

T(r) T(0)

2w0

T(b) Tc

Fig. 5. (a) Schematic temperature profile in the crystal, (b) Mode distribution of TEM 00 , w0 ; waist radius, 2a: pump beam size.

Consider the optical path of a round-trip ray in a crystal (length L, temperature Tc, refractive index n0(Tc)) parallel to the beam axis with a radial distance r. Without a pump beam, the optical path length (OPL) δoff (r) of the ray is 2n 0 (Tc )L . The resonance wavelengths λ q0 are determined by δoff (r) = qλ 0q ( q is an integer). With the cw pump on, induced temperature field T(r, z) and strain field ε(r, z) changes the OPL as [15,25] L

δon (r) = 2n 0 ΔL(r) + 2  n(T, ε)dz. 0

(4)

The relative OPL δrel (r) (= δon (r) − δoff (r)) in suitable assumptions can be expanded as [25] δ rel (r) = 2n 0 < ε z > +2

∂n ∂n < T(r) − Tc > +2 j=r,θ,z < εj > . ∂T ∂ε j

(5)

where < > represents integral along z. The first term represents the overall length change ( ΔL(r) = < ε z > ) of the crystal, which induces bulging surfaces similar to that shown in Fig. 5(b), the second is a result of the refractive index change with temperature, and the third represents strain induced birefringence. The difference of the relative OPL ( = δ rel (0) - δ rel (r)) is the origin of the thermal lens, which permits TEM 00 mode as shown in Fig. 5(b).


The round-trip phase change experienced by the TEM 00 mode with the pump on corresponds to the longitudinal phase change along the beam axis (r = 0), which can be divided into two parts: main contribution by the above OPL change δ rel (0) and small contribution by the phase retardation effect experienced by the Gaussian beam [26]. Thus, it is reasonable to calculate first the peak shift due to the dominant OPL change effect alone (uncorrected case), and then small correction term due to the phase retardation effect. In the uncorrected case, the resonance wavelength will be red-shifted from λ 0q to a wavelength denoted by λ qun , when the pump is on. The peak shift Δλ qun ( = λ qun - λ 0q ) is δ rel (0) 0 λ q . For a uniform pump profile, the first and third term in Eq. (5) could 2n 0 L be derived under the plane stress approximation (PSA), which is valid for small aspect ratio L/2b. Our case (L/2b = 0.25) is in the valid regime of PSA [25]. < ε z (0) > can be approximated under PSA and traction free rod boundaries by using stress-strain relations as [25,27]

calculated as

< ε z (0) > 2v b ≅ (1 + v) < T(0) − T(b) > − 2  rdr + < T(b) − Tc > . (6) αT b 0

where v ( ≅ 0.25) is Poisson’s ratio. The second term in Eq. (6) is very small compared to the first in case of small pump beam size (2a) as was as our setup ( a / b ~ 0.026 ). The third term represents the contribution by temperature gap at the interface. By assuming the second term in Eq. (6) is 0, Δλ qun can be approximately calculated as  1 ∂n   1 ∂n  0.05   0 Δλ qun ≅  + (1 + v ) − + αT  λ 0q (T(b) − Tc ) ≡  αT  λ q T(0) − T(b) +   n ∂T  n0   (7)   n 0 ∂T   0

(

(

)

(

)

)

(

)

SL nu T(0) − T(b) + SL u T(b) − Tc ≡ SLun T(0) − Tc .

where T ( ≡ < T > / L ) represents average temperature along the crystal. In Eq. (7), SL nu is the calculated scale factor involved to the non-uniform temperature and strain fields. In our knowledge, the formula of SL nu is firstly derived here including the birefringence term. By choosing dn/dT = 8.6 ×10−6 °C−1  and α T = 6 × 10−6 °C−1  for the measured SL u , of which the value of α T was taken from the data of 25 at. % Yb:YAG [28], SL nu is calculated as about 0.0127 nm/ °C . If the second term in Eq. (6) is included for the case of a / b ~ 0.026, SL nu is calculated as a slightly decreased value of 0.0124 nm/ °C . In Eq. (7), Δλ qun is linearly related to the temperature rise ( T(0) − Tc ) with a scale factor SLun , which depends on the ratio between the temperature difference ( T(0) − T(b) ) and temperature gap ( T(b) − Tc ). SLun depends on the interface condition. For instance, it is equal to SL nu if no temperature gap exists at the interface. The pump profile in Fig. 2 was not the uniform hat top profile. However, SL nu in this case should be nearly the same as that of the uniform case because Eq. (6) is valid for any symmetric temperature profile and the birefringence term in Eq. (5) is much smaller than other terms.


The actual resonance wavelength denoted by λ q of the TEM 00 mode in the symmetric resonator in Fig. 5(b) should be corrected by the small phase retardation contribution and be determined by [26] L 2 tan −1 ( ) 2z on 0 (8) δ (0) = [q + ]λ q . π where z 0 is the Rayleigh range ( = πn 0 w0 2 / λ q ) of the TEM 00 mode. The positive second term accounts for the phase retardation effect of Gaussian beam relative to a plane wave along the beam axis (r = 0). Due to λ q < λ qun , the actual peak shift Δλ q ( ≡ λ q − λ q0 ) is slightly smaller than Δλ qun . The correction term will increase with the temperature rise, but slightly deviate from the linear dependency. However, the actual peak shift Δλ q could be assumed to be nearly linear to the temperature rise, like as Δλ qun , because Δλ qun is very larger than the phase retardation counterpart. The measured Δλ should be nearly equal to Δλ q on the assumption that the measured threshold peak position would be close to λ 0q . If then, Δλ can be related as Δλ ≅ SLeff (T(0) − Tc ).

(9)

where SLeff is the effective scale factor, the ratio between Δλ and the temperature rise. SLeff should be slightly smaller than SLun due to the phase retardation effect. In order to evaluate this effect, Rayleigh range z0 should be determined by measuring the waist radius w0 . 6. Lensing dynamics in the crystal

Typical signals of PD2 at several transverse position x with Pin = 0.85 W and Tc = 30 °C are shown in Fig. 6(a). From the data at various x, the instantaneous output beam profile can be obtained as illustrated in Fig. 6(b). For τ = 0.18 ms (near the threshold), the profile was slightly asymmetric and its center deviated slightly from the cw output beam center (x ~ 0). This may be attributed to the slight asymmetric pump profile shown in Fig. 2. For τ = 0.33 ms and 3 ms, the center positions appeared around x ~ 0 and its intensity profiles were well fitted with the Gaussian function depicted as lines. This implied that TEM 00 mode became stabilized in the interval of τ = 0.18 ~ 0.33 ms due to the increased lensing effect.

Fig. 6. (a) PD2 signals at several position x transverse to the beam propagation, and the pump signal from PD1. The distance d from the crystal to PD2 was 52 cm, (b) Intensity profile at various τ, w ; beam radius at d = 52 cm.


The waist radius w0 in the crystal can be determined by w0 ≅ λ / [π(θ / 2)] , where θ / 2(≈ w / d) is the divergence angle of TEM 00 mode [26]. Based on the data in Fig. 6, the temporal evolution of w0 with τ could be deduced as shown in Fig. 7(a). With Pin = 0.85 W and Τ c = 30 °C , w0 measured at about 36 μm in cw mode. In pulse mode, w0 decreased from about 53 μm at the threshold to about 38 μm around τ ~ 6 ms. It is notable that 2 w0 ~ 106 μm at the threshold is fairly larger than the pump beam size (70 ~ 80 μm) in Fig. 2. It was reported that aperture guiding (or the effect of soft aperture in the pump region) is the predominant mechanism near the threshold for stabilizing the cavity mode in Yb:YAG lasing at 1030 nm [21]. If this is also true in our setup, 2 w0 at the threshold should be comparable to the pump beam size contrary to the observation. Because the lasing wavelength in our setup was ~ 1050 nm, at which the absorption cross section is ~ 9 times smaller than that at 1030 nm [29], the aperture guiding effect must be negligibly smaller than the thermal one. As τ exceeded ~ 3 ms, w0 changed very slowly because the thermal lens was established. That is, the thermal lens can be treated as a steady state after τ ~ 3 ms. There is an equivalent symmetric concave resonator consisting of two spherical mirrors with an effective curvature radius R eff , in which w0 of TEM 00 mode is equal to the measured one. The effective dioptric power (DP) of one mirror could be approximated as DP = 2n 0 / R eff . The dynamics of the deduced DP from Fig. 7(a) are shown in Fig. 7(b). Magnified behavior (up to τ = 1 ms) of DP with Τ c = 30 °C is shown in the inset. It is notable that DP slowly increased just after the threshold (from about 6 [ m −1 ]) and then rapidly after τ ~ 0.2 ms. The data after τ ~ 0.2 ms were well fitted with an exponential decay curve (red line). The difference of DP at the threshold compared to the red line was about 2.5 [ m −1 ]. This behavior is likely attributed to an additional lensing mechanism through population changes of the 2F5/2 and 2F7/2 by pump pulse [20]. The spatial population profile will induce a population lens because the two levels have different polarizability. Thus, DP can be divided into DPT of the thermal lens and DPP of the population lens. Both DPP and DPT will increase before laser threshold. 54

30 o

(a)

50

TC= 30 C

o

o

TC= 44 C

48 46

o

TC= 45 C

44

o

TC= 46.5 C

42 40 38 36

o

CW, TC= 30 C

34 0

1

2

3

4

5

6

Delay Time τ (ms)

Dioptric Power (1/m)

Waist Radius w0 (μm)

52

CW,30 C

25 20 15

o

10

TC= 30 C

5 0

(b) 0

1

2

16 14 12 10 8 6 4 2 0 0.0 0.2

3

0.4

0.8 1.0 ms

0.6

4

5

6

Delay Time τ (ms)

Fig. 7. (a) Dynamic of waist radius w0 with delay time τ at various Τ c , (b) Corresponding effective dioptric power (DP) with delay time τ .

After the threshold, DPP will not increase further but decrease with τ because the threshold population of the upper level would decrease due to the rapid enhancement of the mode coupling efficiency between the lasing mode and pump profile. On the other hand, DPT will increase continuously until the thermal lens is established. Therefore, DPP could be


negligible above τ ~ 0.2 ms, resulting in a stabilized center position of the output, as observed. The measured peak shift Δλ(τ) in Fig. 3(c) was based on the threshold peak position in pulse mode. In Fig. 8, the behaviors of the actual resonance wavelength λ q (red) and the uncorrected one λ qun (blue) with τ in pulse mode were schematically shown. The difference δλ r (τ) ( ≡ λ qun (τ) − λ q (τ) ) corresponds to the phase retardation effect and could be estimated

from the experimental results in Fig. 7 by using Eq. (8). It is worthy to estimate the offset ( δλ ) between the measured threshold peak position λ q (τ th ) and λ 0q . Careful analysis of the behaviors of Figs. 3(c) and 7(b) revealed Δλ qun (τ th ) ~ 0.008 nm and δλ r (τ th ) ~ 0.006 nm. That is, δλ was estimated as ~ 0.002 nm,

which is negligibly small compared to Δλ in Fig. 4(b). Therefore, the measurement of peak shift based on the threshold peak position is a reliable method and the previous assumption ( Δλ ≅ Δλ q ) is reasonable. By analyzing the cw data in Fig. 7, we found that δλ r by the phase retardation effect was very small (~ 5% of λ qun ). This supports the validity of the assumption, applied in Eq. (9), that Δλ is nearly linear to the temperature rise. Wavelength un

λ q ( τ)

δλr(τ)

λq(τ) un

Δλ (τ) Δλunq (τ)

0

λq

Δλ(τ)

Measured peak shift Threshold position Pump off

δλ τ

0 τth

Delay Time

un

Fig. 8. Schematic behaviors of λ q and λ q with delay time in pulse mode.

7. Dynamics of output power

In order to determine the temperature rise and SLeff , it is desirable to find a laser operation parameter that is sensitive to the absolute temperature. Here, the output power is considered to be a relevant parameter for this requirement. As the crystal temperature increases with a fixed Pin , the output power decreases due to the increase of threshold pump power and decrease of the slope efficiency. Figure 9 shows the dynamics of output power, at various Τ c with Pin = 0.85 W. It is notable that the output power in pulse mode rapidly increased up to τ ~ 1.5 ms. This is the result of the enhanced mode coupling efficiency due to the rapid decrease in w0 . Enhanced mode coupling efficiency leads to the increased (effective small signal) gain coefficient seen by the lasing mode, resulted in increased output power. After τ ~ 1.5 ms, the mode coupling efficiency very slowly varied. Thus, the characteristic of temperature dependent output power appeared as a gradual decrease. As Τ c increased further from 30 °C , the levels of the output power decreased as expected. There existed a configuration in pulse mode where the output power became the same as that of the cw mode. For example, the output power at τ = 6 ms with Τ c ~ 44.5 °C was nearly the same as that in the cw mode with Τ c = 30 °C . In both cases, the absolute temperature T(0) should be equal.


The dynamics of DP at various Τ c were also investigated from the analysis of PD2 signals, as shown together in Fig. 7. It is noteworthy that DP at τ = 6 ms and Τ c ~ 44.5 °C was nearly the same as that of the cw data at Τ c = 30 °C within the error range. This means that not only the absolute temperature T(0) but also the thermal lens strength are all the same in both cases. This supports that Eq. (7) can be also applicable to the pulse mode after the thermal lens is established ( τ ~ 3 ms).

Fig. 9. Dynamics of the output power (linear to the voltage of PD3) at various Τ c with Pin = 0.85 W.

The heat flow from the pumped region to the interface would take very long τ (>> 10 ms) in pulse mode. Hence, no temperature gap is expected in the small range of τ (3 ms ~ 10 ms) because the temperature profile change will occur within a few pump beam size. Thus, the temperature rise in pulsed mode can be determined by Eq. (7), where SLun = SL nu Table 1 summarizes measured parameters (bold) and estimated ones (italic) calculated from the experimental data in the two configurations. The temperature rise at τ = 6 ms with Τ c = 44.5 °C is approximated as Δλ un (τ) / SL nu ~ 5.3 °C . Thus, T(0) at τ = 6 ms in pulse mode is ~ 49.8 °C , which is the same as for cw mode with Τ c = 30 °C . Hence, the temperature rise in cw mode is determined as ~ 19.8 °C . Thus, SLeff in Eq. (9) can be determined as 0.225 nm/19.8 °C ~ 0.0114 nm/ °C . The value of SLun in Eq. (7) in cw mode is estimated as Δλ un /19.8 °C ~ 0.0120 nm/ °C , which is slightly small than the calculated value of SL nu ~ 0.0124 nm/ °C due to the temperature gap in the cw setup. In Fig. 4(b), the temperature scale, calibrated by the determined SLeff ~ 0.0114 nm/ °C , is also plotted on the right axis. The temperature rise was ~ 6.1 °C around the cw threshold and ~ 22.8 °C at maximum pump power. We estimated the error in the temperature rise to be about ± 1 °C at maximum pump power. Table 1. Measured (bold) and Estimated (italic) Parameters

Τ c ( °C ) 30 44.5

τ = ∞ (cw)

τ = 6 ms Δλ(τ)

δλr (τ )

Δλ (τ )

0.055 nm

0.013 nm

0.068 nm

un

Δλ

δλr

0.225 nm

0.013 nm

Δλ 0.238 nm un


8. Conductive cooling geometry with sandwiched sapphires

The proposed scheme for temperature measurement can be expanded to various other cooling geometries. In order to demonstrate this, we applied the scheme to a conductive cooling geometry. The experimental setup is the same as Fig. 1 except that the crystal is sandwiched by two sapphire plates (thickness of 1 mm) as shown in Fig. 10(a). Because sapphire has a higher thermal conductivity than those of air or Yb:YAG, a reduced temperature rise is expected due to the effective heat removal from the crystal. In this geometry, the generated heat flows both radially and longitudinally. In order to guarantee that sapphire and crystal are well contacted, copper blocks pressed the sapphires in both directions. The sapphire temperature is assumed to be equal to the copper temperature Τ c . One surface of the sapphires was AR coated on the pump wavelength and the other side contacting the crystal was uncoated. The dependence of the overall peak shift Δλ with Pin in cw mode is shown in Fig. 10(b). Compared to Fig. 4(b), Δλ was fairly reduced. For example, Δλ was only ~ 0.094 nm at Pin = 1.05 W, which is about 3 times smaller than that in Fig. 4(b).

Fig. 10. (a) Conductive cooling geometry with sapphires, (b) Measured overall peak shift Δλ with Pin at Τ c = 30 °C .

A larger value of Δλ is desirable for more precise estimation of SLeff with a high Τ c setup. The dynamics of output power was measured at a higher Τ c = 60 °C with Pin = 0.85 W, as shown in Fig. 11. The output showed a gradual decrease after τ ~ 1.5 ms similar to Fig. 9. As Τ c increased further from 60 °C , the power level decreased also. At a fixed τ = 5 ms, the output with Τ c = 70 °C was nearly equal to that of cw mode with Τ c = 60 °C . The corresponding peak shifts were measured as Δλ(τ = 5ms) ~ 0.055 nm, Δλ ~ 0.13 nm, respectively. Because this cooling geometry cannot apply PSA, T(0) in the small range of τ should be obtained differently from the previous procedure. In Fig. 11, the output power at τ = 3 ms with Τ c = 71 °C is nearly equal to that at τ = 5 ms with Τ c = 69 °C . The difference in Δλ(τ) between τ = 5 ms and 3 ms was measured at about 0.015 nm at Τ c = 70 °C . The output power will decrease linearly with a small increase of crystal temperature. Hence, the ratio 0.015 nm/2 °C = 0.0075 nm/ °C is estimated as the experimental scale factor relating the measured Δλ(τ) to the temperature rise in the small range of τ . Thus, the temperature rise at

τ = 5 ms with Τ c = 70 °C is estimated as ~ 7.3 °C or T(0) in cw mode with Τ c = 60 °C is about ~ 77.3 °C . Thus, SLeff can be estimated as ~ 0.13 nm/17.3 °C = 0.0075 nm/ °C .


Fig. 11. Dynamics of the output power at various Τ c .

The estimated value of SLeff ~ 0.0075 nm/ °C is substantially smaller than that of ~ 0.0114 nm/ °C in the first geometry. This is attributable to the reduced contribution of the length change term in Eq. (5). The most plausible cause is the effect of prevention of crystal expansion by cold sapphires. The presence of the sapphires changes the mechanical boundary condition thereby pressing against the crystal from free expanding in both directions. Another possibility is that the sapphire plates changes the temperature field as the surface region of the crystal cooler than the central region due to the longitudinal heat flow, thereby causing less contribution of the surface expansion. The temperature dependent peak shift at the threshold was measured in the conductive cooling geometry as in Fig. 4(a). The resultant slope SL u in this case was nearly equal to that in Fig. 4(a). This is because not only the crystal but also the sapphire plates simultaneously expand as Τ c increases, and the expansion coefficient of sapphire is similar to that of Yb:YAG. We measured SL u around Τ c = 60 °C and found it to be slightly higher (~ 5%) than that, at around Τ c = 30 °C ~ 50 °C . Thus, SLeff at Τ c = 30 °C should be similar to about 0.0075 nm/ °C in experimental error. In Fig. 10(b), the temperature rise corresponding to Δλ are also plotted by using the estimated SLeff . At maximum Pin , the temperature rise was ~ 12.5 °C , which is about half of that in the first geometry. The reduced temperature rise should be due to the effective heat removal by the sapphires. 9. Conclusions

Operation dynamics were investigated using a chopped pump beam in order to measure the internal crystal temperature in diode end-pumped thin monolithic solid state lasers, wherein plane stress approximation was valid. The lasing peak showed a continuous red shift with the pump power due to the temperature increase. The peak shift could be exactly measured based on the threshold spectrum in pulse mode. The lensing dynamics was investigated and found that the thermal origin is predominant to other lensing effects. By analyzing the dependence of output power dynamics with temperature, we could determine the effective scale factor, the ratio between the measured peak shift and the average temperature rise along the beam axis. The proposed scheme was applied to a conductive cooling geometry where the crystal was sandwiched by sapphire plates. The effective scale factor was significantly reduced due to the reduced contribution of the length change. The proposed scheme for the temperature measurement of the crystal can be applied to various other crystals and cooling geometries in the diode end pumped lasers. Though the scheme was investigated in the pump power level of ~ 1 W with a very small pump size, it is scalable to a higher power level with a larger pump size. Our approach for absolute


temperature measurement under laser action directly utilized the operating output beam and has strong advantages in the case where the crystal is enclosed in the heat sink. Acknowledgment

This work was funded by the Seoul Metropolitan Government, Korea, under contract of R&BD Program WR100001.


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