Abstract

Temperature dependencies of stimulated emission cross section for Nd:YAG, Nd:YVO4, and Nd:GdVO4 was carefully evaluated. Our spectral evaluations with fine spectral resolution were carried out under the condition that the population inversion was induced into samples by a weak pumping field. Within the temperature range from 15°C to 65°C, the variation of emission cross section at 1.06 μm in Nd:YAG was −0.20%/°C, while those in Nd:YVO4 and Nd:GdVO4 for π-polarization were −0.50%/°C and −0.48%/°C, respectively. Consideration of measured temperature dependence gave the numerical model for temperature dependent emission cross sections of Nd-doped solid-state laser materials. We have also presented numerical approximations of this model for our samples by a simple polynomial, which can be applicable within the temperature range from 15°C to 350°C.

© 2012 OSA

1. Introduction

Thanks to the recent advance in giant-micro photonics [1], it has become clear that the extremely high power optical output can be extracted even from microchip laser configurations by means of thoroughly investigated optimal cavity designs [2]. The near infrared giant pulse generated from high power microchip lasers can offer high power visible, ultraviolet, and terahertz-wave by wavelength conversion [35], which are very useful not only for science fields but also for various industrial fields [6,7].

It is important for generating intense laser pulses with high repetition rates to manage the temperature distribution inside the solid-state laser gain media, because thermal problems often cause the deterioration of laser performances. Since the pump energy concentrates on the small laser volume inside laser media, thermal effects are easily enhanced even under the low power operation in the case of microchip lasers [8]. Especially thermal lensing and thermal birefringence of laser gain media have been well studied among various thermal problems in high power laser cavity [9]. However, only few attentions have been paid for the temperature dependencies of the spectroscopic characteristics in laser gain media traditionally. If these untraditional thermal problems exist, they must cause the deterioration of laser performances immediately.

The existence of temperature-dependent spectroscopic characteristics in Nd:YVO4 was proved by laser performances where output pulse energies of high power Q-switched Nd:YVO4 lasers were severely dependent on temperature [10]. Therefore, we must consider the variation of spectroscopic properties due to temperature change in laser gain media besides its thermal quenching of luminescence. On the contrary, the input-output characteristics of high power Q-switched Nd:YAG lasers were hardly dependent on their operation temperature [11]. Because the energy of Q-switched pulse depends not on fluorescent lifetime but the stimulated emission cross section (σem), the precise numerical models of σem depending on temperature T for Nd-doped laser gain media have been strongly desired.

There are two important evaluations about the temperature dependencies of spectroscopic characteristics for Nd:YAG [12]. One is the temperature independent fluorescent lifetime. This fact indicates the invariant decay rate of the spontaneous emission emitted from Nd-ions, which gives the basic assumption that the temperature dependency of σem is caused by the variation in the spectral profile in fluorescence I(ν) at frequency ν. Another shows that σem of Nd:YAG has a temperature dependence of −0.12%/K. Although σem of Nd:YAG reported in [12] is independent on Nd-doping concentration (CNd), we recently found that σem of Nd:YAG depended on CNd and its fabricating process from the measurement of the fluorescence with high spectral resolution [13]. Compared to our finer spectral resolution of 0.05 nm, the resolution of 0.4 nm in [12] could bring less sensitivity for the variation of the emission intensity with narrow bandwidth from rare-earth trivalent. From these situations, we consider that the −0.12%/K-dependence of σem in Nd:YAG is doubtful.

About the problem of the accuracy in the evaluation of the temperature-dependent stimulated emission cross section σem(T), very pregnant results are shown in [14] and [15], where reported dependences of σem in Nd:YVO4 were −0.18%/K and −0.68%/K, respectively. While finer resolution of 0.07 nm in [15] compared to 0.5 nm in [14] can gives more accurate result in general, we must take care that an amplified spontaneous emission occurs easily in the Nd:YVO4 with high population inversion. Since high pump density over 2.5 kW/cm2 was comparable to 4.4 kW/cm2 of pump saturation intensity of Nd:YVO4 [16], −0.68%/K in [15] could be an enhanced result affected by the amplified spontaneous emission. Thus, comparative researches for various laser media and fine evaluations with the high spectral resolution under the weak pump field are desired in order to establish the numerical model for σem(T) in solid-state laser media.

In this work, σem(T) of Nd:YAG, Nd:YVO4, and Nd:GdVO4 were comparatively evaluated. By using measured data, we developed the experimentally proved general model for the spectroscopic properties of Nd-doped solid-state laser media. We also presented the numerical approximation for σem(T) in Nd:YAG, Nd:YVO4, and Nd:GdVO4, where differences between σem(T) of Nd:YAG and Nd:orthovanadates were discussed.

2. Experimental setup

We evaluated a 1.0at.% Nd:YAG single crystal (Scientific materials Co.), a 1.0at.% Nd:YVO4 single crystal (ITI Electro Optics Co.), and a 1.0at.% Nd:GdVO4 single crystal (Shandong Newphotons Science and Technology Co.). These samples had 1-mm thickness and were mirror polished on both surfaces.

During the measurement of fluorescence spectra samples were pumped by 808-nm radiation from a fiber coupled LD (LIMO40-F400-DL808, LIMO GmbH) with 100Hz repetition rate and 10% duty ratio. The averaged pump power was limited below 4 mW in order to prevent the amplified spontaneous emission. Emitted pump beam was collimated and focused onto sample surfaces by lenses with focal length of 100 mm and 80 mm, respectively. Fluorescence from samples was analyzed by a monochrometer (TRIAX-550, JOBIN YVON) with less than 0.05-nm resolution and detected by linear-InGaAs-array detector (IGA512-1x1, JOBIN YVON).

Samples were sandwiched by the copper plates of which temperature were controlled between 15°C and 65°C by a pertier device and a thermo-electric controller (LDT-5948, ILX Lightwave Co.). Similarly temperature of samples was tuned within the range from 25°C to 145°C by using of heater for the purpose of examining the developed numerical model for the spectroscopic properties of Nd-doped solid-state laser media.

The calibration of the wavelength was done within an error of 0.2 cm−1 by Ne lamp (Pencil style calibration lamp 6032, Oriel), Xe lamp (Pencil style calibration lamp 6033, Oriel) and Hg lamp (Pen-Ray Mercury lamp, UVP). We referred 80 lines in these lamps within the range from 1033nm to 1150nm.

3. Results

Figure 1 shows fluorescent intensities emitted from Nd:YAG, Nd:YVO4 and Nd:GdVO4 at 20°C accompanied with transitions from 4F3/2 to 4I11/2. These intensities were normalized, and factors for normalization was proportional to η defined by Eq. (B8) where the maximum intensity at 20°C was set to 1.0. While the line-bandwidth of the main peak in fluorescence from Nd:YAG in Fig. 1 was 0.78 nm that was comparable to 0.80 nm of π-polarized fluorescence from Nd:YVO4, π-polarized fluorescence from Nd:GdVO4 had wider line-width of 0.94 mn. Temperature dependence of these maximum emission intensity of Nd:YAG, Nd:YVO4 and Nd:GdVO4 are shown in Fig. 2 .Temperature dependence of σem is almost the same as that of the intensity in fluorescence as discussed in Appendix B. From the dependence of peak intensities on T shown in Fig. 3 , we evaluated thermal reductions of σem for Nd:YAG, Nd:YVO4 (π- and σ-polarization), and Nd:GdVO4 (π- and σ-polarization) to be −0.20%/°C, −0.50%/°C, −0.37%/°C, −0.48%/°C, and −0.37%/°C within the temperature range between 15°C and 65°C, respectively.

 

Fig. 1 Normalized emission intensities in fluorescence at 1.06 μm under 25°C: Nd:YAG (a), Nd:YVO4 π- and σ-polarization (b) and (c), and Nd:GdVO4 π- and σ-polarization (d) and (e).

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Fig. 2 Temperature dependence of fluorescences from Nd-doped laser crystals: Nd:YAG (a), Nd:YVO4 π- and σ-polarization (b) and (c), and Nd:GdVO4 π- and σ-polarization (d) and (e).

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Fig. 3 Temperature dependence of the normalized intensities at the emission peak.

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4. Discussions

4.1 Population inversion generated under measured condition

The effect of gain narrowing causes the underestimation of measured line-bandwidth [17], therefore it can bring fatal estimation errors in σem(T). In order to reduce these estimation errors, the optical gain in samples is required to be low. Optical gain coefficient g in pumped laser media is given by [16]

g=γ1+γNσem,
where N, α, and γ are number density of Nd trivalent, absorption coefficient, and the ratio between pump intensity and saturation intensity. From spectroscopic parameters in our previous report [18], γ and g at the pumping surface in [15] were 0.57 and 6.4 cm−1, respectively. Oppositely those in our experimental setup were only 0.0095 and 1.7 cm−1, respectively. By disregarding the saturation effect in pump absorption efficiency, an averaged optical gain Gtra for fluorescence detected collinear with transmitted pump is given by
Gtra=0ldkexp(αk)exp[klγexp(αz)1+γexp(αz)Nσemdz]/0ldkexp(αk),
where l is the thickness of the pumped sample, and pump beam was irradiated along z-axis. Similarly, averaged optical gain Gref for fluorescence detected collinear with reflected pump is expressed by
Gref=0ldkexp[α(lk)]exp[0kγexp(αz)1+γexp(αz)Nσemdz]/0ldkexp[α(lk)].
While Gref in [15] is 1.69, Gtra in our setup is only 1.01. As a result of this difference between the induced optical gain of pumped samples, the effect of gain narrowing in our work is much smaller than that in [15]. This should be the reason why temperature variation evaluated in [15] is larger than our result.

4.2 General model for σem(ν) and lorentzian decomposition of I(ν)

We propose a general model for σem(T) in Appendix B, where σem(T) is given by

σem(ν,T)=λ216π2n2τjijfkaibiffiΔνif(Δνif/2)2+(ννif)2,
where λ, n, τj, fi, ai, bif, νif, and Δνif are wavelength, refractive index, radiative lifetime of the emitting manifold j, the fractional population of the Stark-level i in manifold j, relative decay rate of level i, branching ratio between the transition from i to the Stark-level f in the terminating manifold k, the center frequency of the transition from level i to level f, and its spectral bandwidth of FWHM, respectively. In Eq. (4) temperature dependent components are ai, bif, fi, νif, and Δνif.

In order to use above model, it is necessary to determine ai, bif, νif, Δνif experimentally. These parameters can be evaluated from measured I(ν) at various temperature by the fitting to following expression:

I(ν)ν3ijfkaibiffiΔνif(Δνif/2)2+(ννif)2,
which is derived at Eq. (B7) in Appendix B. With fitting parameters of cif and dif, dependences of νif and Δνif on T are given by [19]
νif(T)=νif(0)cif(TΘD)40ΘDTx3ex1dx,
Δνif(T)=Δνif(0)+dif(TΘD)70ΘDTx6ex(ex1)2dx,
where ΘD is Debye temperature. We can also determine cif and dif by means of least square fitting to experimentally measured line-shift and line-broadening.

Russell-Saunders terms relating to optical transition around 1-μm region in Nd-doped laser gain media are explained at Appendix A. While the emitting manifold is 4F3/2 term consists of Stark-levels of R1 and R2, the terminating manifold is 4I11/2 term made of Stark-levels of Yn (n is from 1 to 6). Therefore, our fluorescence contains 12 emission peaks. Figure 4 shows experimentally obtained bif, νif, Δνif in Nd:YAG, and least square fitting to these values. Detected temperature dependences in ai and bif are less than error level, thus these values are treated as a constant in following discussions. νif and Δνif were well fitted by using of Debye temperatures of YAG, YVO4, GdVO4 that were reported in [20] and were 795K, 718K, and 677K, respectively. The fitting results in Nd:YAG and Nd:orthovanadates are summarized in Tables 1 , 2 and 3 .

 

Fig. 4 Temperature dependence of bif (a), νif (b), and Δνif (c) in Nd:YAG. In these figures markers show experimental results and lines show the fitting.

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Tables Icon

Table 1. Spectral parameters of representative transitions in Nd:YAG

Tables Icon

Table 2. Spectral parameters of representative transitions in Nd:YVO4 in π-polarization

Tables Icon

Table 3. Spectral parameters of representative transitions in Nd:GdVO4 in π-polarization

4.3 Simulation of the fluorescence dependent on temperature

Equations (4) and (5) and parameters shown in Tables 1-3 can give the prediction for I(ν) and σem(ν) at any required temperature. Figure 5 shows simulations of I(ν) emitted from Nd:YAG at 25°C, 85°C, and 145°C. Although these simulations are calculated by using of parameters derived from experiments carried within the range under 65°C, they coincide to additional experiments with tuned temperature from 25°C to 145°C by heater. Therefore, this model was experimentally proved within the range from 15°C to 145°C. Since the electron-phonon interaction assumes the simple density of state defined by Debye model in order to develop this model, this model should be correct within the temperature range without phase transition under Debye temperature.Figure 6 shows the prediction for I(ν) emitted from Nd:YAG, Nd:YVO4, and Nd:GdVO4 by means of Eq. (5) and Tables 1-3. These predictions directly indicate the difference of the temperature-stability in fluorescence between Nd:YAG and Nd:orthovanadates, which is discussed in detail at section 4.5.

 

Fig. 5 I(ν) emitted from Nd:YAG at various temperatures. Solid lines are simulations calculated from Eq. (5) with parameters in Table 1, and dashed lines are experimentally measured under temperature tuning by heater.

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Fig. 6 Predictions for the temperature-dependent I(ν) emitted from Nd:YAG (a), Nd:YVO4 and Nd:GdVO4 (b) by means of Eq. (5) and Tables 1-3.

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4.4 Numerical models for σem

It is not convenient for laser cavity design involving thermal problems to use Eq. (4), because there are too many spectral parameters to be managed. Since σem at only the maximum emission peak is important for laser designs, Eq. (4) can be approximated by following polynomial:

σem(T)=σem(T0)(e0e1T+e2T2e3T3+),
where ei is the coefficient for i-th term in Taylor expansion and dependent of materials. T0 is a temperature as a standard of σem. Although σem of Nd-doped laser media depends on the nature of host materials such as CNd and fabrication processes as discussed in [13], the value reported by past reports [21] will give a useful candidate for σem(T0). More higher order polynomials will produce the approximation of σem(T) at more wider temperature range (or higher accuracy). Here ei under the condition T0 is 20°C are summarized in Table 4 .

Tables Icon

Table 4. ei for Nd:YAG, Nd:YVO4, and Nd:GdVO4 under T0 is 20°C

Simulations calculated by Eq. (5) and approximations expressed by Eq. (8) with parameters in Table 4 within the range from 15°C to 300°C compared to experimental peak intensity are shown in Fig. 7 . Figure 7 indicates that Eq. (8) is enough applicable for the cavity design of high-power microchip lasers.

 

Fig. 7 Temperature dependence of emission intensity of various Nd-doped laser media.

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4.5 Difference between Nd:YAG and Nd:orthovanadates

Similarly to previous reports [12,14,15], obtained temperature dependencies in Nd:YAG and Nd:orthovanadates seem to be linear and non-linear, respectively. However, it is not correct to consider that either will not show a behavior interpreted by the model such as Appendix B in Nd:YAG and Nd:orthovanadates. It is because the behavior of other emission peaks of Nd:YAG except the main peak is equivalent to Nd:orthovanadates.

Though temperature dependence of the emission bandwidth in the main emission peak of Nd:YAG seems to be quite small, the emission bandwidth of two emission peaks that composed the main peak shows similar dependencies to other emission peaks in Nd:YAG, and Nd:orthovanadete as shown in Fig. 8 . However, the splitting between these two peaks becomes narrower under higher temperature. This is the reason why the main emission peak in Nd:YAG shows the different temperature-dependences from other emission peaks.

 

Fig. 8 Temperature dependence of the line-bandwidth of emission peaks in Nd-doped materials.

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Our experimental results indicated the novel design rule for the selection of the laser medium: Nd:YAG is suitable for highly stable operations, and Nd:orthovanadate is useful for temperature tuning. In both case our experimentally proved numerical model of σem(T) in Eq. (8) will be eminently useful for developing high power lasers by use of Nd-doped laser gain media.

5. Conclusion

We evaluated σem(T) of Nd:YAG, Nd:YVO4, and Nd:GdVO4 from temperature dependent I(ν). Under our evaluation the population inversion was induced into samples by a weak pumping field, and fluorescence was detected with fine spectral resolution. At the temperature range from 15°C to 65°C, the variation of σem in Nd:YAG was estimated to be −0.20%/°C at 1.06 μm, while those in Nd:YVO4 for π- and σ-polarization were 2.5 times and 1.8 times compared to Nd:YAG, respectively. Temperature dependency of fluorescence emitted from Nd:GdVO4 is quite similar to that of Nd:YVO4.

By using of temperature dependence in the electron-phonon interaction, we developed a general numerical model of σem(T) for Nd-doped solid-state laser materials. Since this model is based on the density of state defined by Debye model, it should be correct in the temperature range without phase transition under Debye temperature. We have also presented numerical approximations of this model for our samples by a simple polynomial. This approximation should be applicable from 15°C to 350°C, and was experimentally confirmed at the temperature range from 25°C to 145°C.

Appendix A: Stark levels in Nd-doped laser media

Energy levels of the lower terms of Nd trivalent are composed of three 4f electrons. They are electrically shielded from the external field by the octet of perfectly filled 5s- and 5p-orbital. Therefore, these lower terms of Nd trivalent form Russell-Saunders coupling due to Coulomb interaction between three 4f electrons. This Russell-Saunders term can be described by 2S+1L, where S and L are spin angular momentum quantum number and aztimuthal quantum number of three 4f electrons. Spin-orbit interaction inside 4f electron makes 2S+1L split to the terms 2S+1LJ that have different total angular momentum quantum number J.

When 2S+1LJ terms in Nd:YAG are arranged in low-energy order, they are 4I9/2, 4I11/2, 4I13/2, 4I15/2, 4F3/2, 4F5/2, 2H9/2, 4F7/2, 4S3/2, 4F9/2, .... . Therefore the energy level of ground state in Nd:YAG is 4I9/2. Excitation by 808 nm pump source excite Nd trivalent from 4I9/2 to 4F5/2, then they relaxed to the metastable state of 4F3/2, which is the emitting level of fluorescence. Via 1-μm fluorescence Nd trivalent in 4F3/2 decays to 4I11/2, which is the terminating level of optical transition in 1-μm region. Finally Nd trivalent in 4I11/2 non-radiatively relaxes to 4I9/2.

Although 2S+1LJ has multiplicity of 2J+1, 2S+1LJ of Nd trivalent shows splitting to utmost J+1/2 levels due to Kramers' degeneration under zero magnetic field condition. In the case of the emitting level 4F3/2, it splits into 2 levels named R1 and R2. Similarly, the terminating level 4I11/2, it splits into 6 levels named Yi (i is an integer within the range from 1 to 6) by crystal field splitting. The schematic diagram of energy levels in Nd trivalent is shown in Fig. 9 .

 

Fig. 9 Energy levels of the lowest terms of Nd trivalent. Only 4F3/2 become the emitting level under 808-nm pumping, and only 4I11/2 become the terminating level for 1-μm fluorescence.

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Appendix B: Numerical model for σem(T)

In this work the fluorescence from Nd-doped materials is a composition of the emission between Stark level-i and level-f that are belonging to different Russell-Saunders manifolds as described in Appendix A. In ordered crystalline materials each emission peak can be assumed to have a lorentzian spectral profile, where the line-shape function gif(ν) of the transition from the Stark level-i to Stark level-f is expressed by

gif(ν)=Δνif2π1(Δνif/2)2+(ννif)2,
Thus the line-shape function gik(ν) of the transition from i to Russell-Saunders manifold k is expressed by
gik(ν)=fkbifgif(ν),
and bif should be normalized by
fkbif=1.
Here we can obtain the line-shape function gjk(ν) of the transition from manifold-j to manifold-k that is given by
gjk(ν)=ijaifigik(ν),
where fi is the fractional population of level-i in the manifold-j, and ai is defined from the decay rate of level-i (Ai) by

ai=Ai/ijAifi.

By using of gjk(ν), we can obtain σjk(ν) and the spectral profile of fluorescent intensity I jk(ν) as

σjk(ν)=λ28πn2gjk(ν)ijAi,
Ijk(ν)=ην3c2gjk(ν)=ην32πc2ijfkaibiffiΔνif(Δνif/2)2+(ννif)2,
where c and η are the photon speed, and a normalization factor that is defined by
η=λIjk(ν)dλ,
and proportional to a total intensity of fluorescence, respectively. Equation (B6) is well-known as Füchtbauer-Ladenburg relation.

By using of radiative lifetime of manifold j τj that is equal to the inverse of the total decay ratio of manifold j, σjk(ν) is given by

σjk(ν)=λ216π2n2τjijfkaibiffiΔνif(Δνif/2)2+(ννif)2.
From Eqs. (B7) and (B9) it is shown that temperature dependence of σjk(ν) is almost equal to I jk(ν) at each emission peak, because line-shift of emission peaks due to temperature change are negligible compared to wavenumber itself. Temperature dependent components in the expression of σjkem(ν) are only νif, Δνif, fi and bif. Dependences of fi on T is given by Boltzmann distribution as following:
fi(T)=exp[hνif(T)kT]/ijexp[hνif(T)kT],
where h and k are Plank's constant, and Boltzmann's constant, respectively.

If the emission peak in the fluorescence consists of one transition between Stark level-i and -f, and not overlapped with other transitions, σem(T) are expressed by [13]

σem(T)=σjk(νif(T))=λ24π2n2aibiffiΔνifijAiaibiffiΔνif.

In this work, we assumed that Ai is independent on temperature as mentioned in introduction. The temperature dependence of bif for the transition from 4F3/2 to 4I11/2 in Nd:YAG, Nd:YVO4, and Nd:GdVO4 was not able to be experimentally confirmed. Even if this temperature variation existed, it is too small to detect under experiments with accuracy of our spectroscopic measurements.

Acknowledgments

This work was partially supported by Genesis Research Institute and by the Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

References and links

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2. H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008). [CrossRef]   [PubMed]  

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5. S. Hayashi, K. Nawata, H. Sakai, T. Taira, H. Minamide, and K. Kawase, “High-power, single-longitudinal-mode terahertz-wave generation pumped by a microchip Nd:YAG laser [Invited],” Opt. Express 20(3), 2881–2886 (2012). [CrossRef]   [PubMed]  

6. M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009). [CrossRef]   [PubMed]  

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9. W. Koechner, Solid-State Laser Engineering, 6th revised and updated edition (Springer Science + Business Media, Inc., 2006), Chap. 7.

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11. M. Tsunekane and T. Taira, “High temperature operation of passively Q-switched, Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.P.30, Munich, Germany, June 14–19 (2009).

12. J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005). [CrossRef]  

13. Y. Sato and T. Taira, “Variation of the stimulated emission cross section in Nd:YAG caused by the structural changes of Russell-Saunders manifolds,” Opt. Mater. Express 1(3), 514–522 (2011). [CrossRef]  

14. G. Turri, H. P. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009). [CrossRef]  

15. X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011). [CrossRef]  

16. Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004). [CrossRef]  

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18. Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005). [CrossRef]  

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References

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  1. T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics,” Opt. Mater. Express 1(5), 1040–1050 (2011).
    [CrossRef]
  2. H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
    [CrossRef] [PubMed]
  3. R. Bhandari and T. Taira, “> 6 MW peak power at 532 nm from passively Q-switched Nd:YAG/ Cr4+:YAG microchip laser,” Opt. Express 19(20), 19135–19141 (2011).
    [CrossRef] [PubMed]
  4. R. Bhandari and T. Taira, “Megawatt level UV output from [110] Cr4+:YAG passively Q-switched microchip laser,” Opt. Express 19(23), 22510–22514 (2011).
    [CrossRef] [PubMed]
  5. S. Hayashi, K. Nawata, H. Sakai, T. Taira, H. Minamide, and K. Kawase, “High-power, single-longitudinal-mode terahertz-wave generation pumped by a microchip Nd:YAG laser [Invited],” Opt. Express 20(3), 2881–2886 (2012).
    [CrossRef] [PubMed]
  6. M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
    [CrossRef] [PubMed]
  7. M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
    [CrossRef]
  8. T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
    [CrossRef] [PubMed]
  9. W. Koechner, Solid-State Laser Engineering, 6th revised and updated edition (Springer Science + Business Media, Inc., 2006), Chap. 7.
  10. S. Joly and T. Taira, “Novel method for pulse control in Nd:YVO4/Cr4+:YAG passively Q-switched microchip laser,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.8.1, Munich, Germany, May 22–26 (2011).
  11. M. Tsunekane and T. Taira, “High temperature operation of passively Q-switched, Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.P.30, Munich, Germany, June 14–19 (2009).
  12. J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
    [CrossRef]
  13. Y. Sato and T. Taira, “Variation of the stimulated emission cross section in Nd:YAG caused by the structural changes of Russell-Saunders manifolds,” Opt. Mater. Express 1(3), 514–522 (2011).
    [CrossRef]
  14. G. Turri, H. P. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
    [CrossRef]
  15. X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011).
    [CrossRef]
  16. Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
    [CrossRef]
  17. P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
    [CrossRef]
  18. Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
    [CrossRef]
  19. T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
    [CrossRef]
  20. Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
    [CrossRef]
  21. T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
    [CrossRef]

2012 (1)

2011 (5)

2010 (1)

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

2009 (2)

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[CrossRef] [PubMed]

G. Turri, H. P. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
[CrossRef]

2008 (2)

H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
[CrossRef] [PubMed]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[CrossRef]

2007 (1)

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[CrossRef]

2005 (3)

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[CrossRef]

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[CrossRef]

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[CrossRef]

2004 (1)

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[CrossRef]

1991 (1)

1969 (1)

T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[CrossRef]

Ando, A.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Balembois, F.

X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011).
[CrossRef]

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[CrossRef]

Bass, M.

G. Turri, H. P. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
[CrossRef]

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[CrossRef]

Bhandari, R.

Cornacchia, F.

Délen, X.

Dong, J.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[CrossRef]

Druon, F.

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[CrossRef]

Fujii, M.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[CrossRef] [PubMed]

Georges, P.

X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011).
[CrossRef]

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[CrossRef]

Hayashi, S.

Inohara, T.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Ishizuki, H.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[CrossRef] [PubMed]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[CrossRef]

Jenssen, H. P.

Kan, H.

Kanehara, K.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Kawase, K.

Kido, N.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Kobayashi, T.

Kushida, T.

T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[CrossRef]

Minamide, H.

Miyazaki, M.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[CrossRef] [PubMed]

Mukai, A.

Nawata, K.

Nozawa, Y.

Rapaport, A.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[CrossRef]

Raybaut, P.

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[CrossRef]

Saikawa, J.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[CrossRef] [PubMed]

Sakai, H.

Sato, Y.

Y. Sato and T. Taira, “Variation of the stimulated emission cross section in Nd:YAG caused by the structural changes of Russell-Saunders manifolds,” Opt. Mater. Express 1(3), 514–522 (2011).
[CrossRef]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[CrossRef]

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[CrossRef]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[CrossRef]

Szipocs, F.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[CrossRef]

Taira, T.

S. Hayashi, K. Nawata, H. Sakai, T. Taira, H. Minamide, and K. Kawase, “High-power, single-longitudinal-mode terahertz-wave generation pumped by a microchip Nd:YAG laser [Invited],” Opt. Express 20(3), 2881–2886 (2012).
[CrossRef] [PubMed]

R. Bhandari and T. Taira, “Megawatt level UV output from [110] Cr4+:YAG passively Q-switched microchip laser,” Opt. Express 19(23), 22510–22514 (2011).
[CrossRef] [PubMed]

T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics,” Opt. Mater. Express 1(5), 1040–1050 (2011).
[CrossRef]

Y. Sato and T. Taira, “Variation of the stimulated emission cross section in Nd:YAG caused by the structural changes of Russell-Saunders manifolds,” Opt. Mater. Express 1(3), 514–522 (2011).
[CrossRef]

R. Bhandari and T. Taira, “> 6 MW peak power at 532 nm from passively Q-switched Nd:YAG/ Cr4+:YAG microchip laser,” Opt. Express 19(20), 19135–19141 (2011).
[CrossRef] [PubMed]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[CrossRef] [PubMed]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[CrossRef]

H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
[CrossRef] [PubMed]

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[CrossRef]

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[CrossRef]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[CrossRef]

T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
[CrossRef] [PubMed]

Tonelli, M.

Tsunekane, M.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Turri, G.

Ueda, K.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[CrossRef]

IEEE J. Quantum Electron. (3)

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[CrossRef]

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[CrossRef]

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (4)

Opt. Lett. (1)

Opt. Mater. Express (2)

Phys. Chem. Chem. Phys. (1)

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[CrossRef] [PubMed]

Phys. Rev. (1)

T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[CrossRef]

Phys. Status Solidi A (1)

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[CrossRef]

Rev. Laser Eng. (1)

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[CrossRef]

Other (3)

W. Koechner, Solid-State Laser Engineering, 6th revised and updated edition (Springer Science + Business Media, Inc., 2006), Chap. 7.

S. Joly and T. Taira, “Novel method for pulse control in Nd:YVO4/Cr4+:YAG passively Q-switched microchip laser,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.8.1, Munich, Germany, May 22–26 (2011).

M. Tsunekane and T. Taira, “High temperature operation of passively Q-switched, Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.P.30, Munich, Germany, June 14–19 (2009).

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Figures (9)

Fig. 1
Fig. 1

Normalized emission intensities in fluorescence at 1.06 μm under 25°C: Nd:YAG (a), Nd:YVO4 π- and σ-polarization (b) and (c), and Nd:GdVO4 π- and σ-polarization (d) and (e).

Fig. 2
Fig. 2

Temperature dependence of fluorescences from Nd-doped laser crystals: Nd:YAG (a), Nd:YVO4 π- and σ-polarization (b) and (c), and Nd:GdVO4 π- and σ-polarization (d) and (e).

Fig. 3
Fig. 3

Temperature dependence of the normalized intensities at the emission peak.

Fig. 4
Fig. 4

Temperature dependence of bif (a), νif (b), and Δνif (c) in Nd:YAG. In these figures markers show experimental results and lines show the fitting.

Fig. 5
Fig. 5

I(ν) emitted from Nd:YAG at various temperatures. Solid lines are simulations calculated from Eq. (5) with parameters in Table 1, and dashed lines are experimentally measured under temperature tuning by heater.

Fig. 6
Fig. 6

Predictions for the temperature-dependent I(ν) emitted from Nd:YAG (a), Nd:YVO4 and Nd:GdVO4 (b) by means of Eq. (5) and Tables 1-3.

Fig. 7
Fig. 7

Temperature dependence of emission intensity of various Nd-doped laser media.

Fig. 8
Fig. 8

Temperature dependence of the line-bandwidth of emission peaks in Nd-doped materials.

Fig. 9
Fig. 9

Energy levels of the lowest terms of Nd trivalent. Only 4F3/2 become the emitting level under 808-nm pumping, and only 4I11/2 become the terminating level for 1-μm fluorescence.

Tables (4)

Tables Icon

Table 1 Spectral parameters of representative transitions in Nd:YAG

Tables Icon

Table 2 Spectral parameters of representative transitions in Nd:YVO4 in π-polarization

Tables Icon

Table 3 Spectral parameters of representative transitions in Nd:GdVO4 in π-polarization

Tables Icon

Table 4 ei for Nd:YAG, Nd:YVO4, and Nd:GdVO4 under T0 is 20°C

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

g= γ 1+γ N σ em ,
G tra = 0 l dkexp( αk )exp[ k l γexp( αz ) 1+γexp( αz ) N σ em dz ] / 0 l dkexp( αk ) ,
G ref = 0 l dkexp[ α( lk ) ]exp[ 0 k γexp( αz ) 1+γexp( αz ) N σ em dz ] / 0 l dkexp[ α( lk ) ] .
σ em ( ν,T )= λ 2 16 π 2 n 2 τ j ij fk a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν ν if ) 2 ,
I( ν ) ν 3 ij fk a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν ν if ) 2 ,
ν if ( T )= ν if ( 0 ) c if ( T Θ D ) 4 0 Θ D T x 3 e x 1 dx ,
Δ ν if ( T )=Δ ν if ( 0 )+ d if ( T Θ D ) 7 0 Θ D T x 6 e x ( e x 1 ) 2 dx ,
σ em ( T )= σ em ( T 0 )( e 0 e 1 T+ e 2 T 2 e 3 T 3 + ),
g if ( ν )= Δ ν if 2π 1 ( Δ ν if /2 ) 2 + ( ν ν if ) 2 ,
g i k ( ν )= fk b if g if ( ν ) ,
fk b if =1.
g jk ( ν )= ij a i f i g i k ( ν ) ,
a i = A i / ij A i f i .
σ jk ( ν )= λ 2 8π n 2 g jk ( ν ) ij A i ,
I jk ( ν )= η ν 3 c 2 g jk ( ν )= η ν 3 2π c 2 ij fk a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν ν if ) 2 ,
η= λ I jk ( ν )dλ ,
σ jk ( ν )= λ 2 16 π 2 n 2 τ j ij fk a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν ν if ) 2 .
f i ( T )= exp[ h ν if ( T ) kT ] / ij exp[ h ν if ( T ) kT ] ,
σ em ( T )= σ jk ( ν if ( T ) )= λ 2 4 π 2 n 2 a i b if f i Δ ν if ij A i a i b if f i Δ ν if .

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