5. Numerical simulation of optimized resonator transients for generating ultrashort Ce:LiCAF laser pulses

Sub-nanosecond (0.7 ns), single-pulse, ~290 nm laser emission from a Ce:LiCAF crystal using an oscillator with a 25 mm cavity length and 25% output coupler reflection has been demonstrated experimentally. In this report, the end-pumping configuration was used to excite the crystal with an Nd:YAG laser emitting 266-nm wavelength (fourth harmonics), 5-ns pulses at 10 Hz repetition rate [70]. Another study reported the generation of 150-ps laser pulses from a 10-mm long Ce:LiCAF crystal. In this report, the laser oscillator was established using a cavity length of 15 mm, output coupler reflection of 30%, and 75 ps excitation pulses [71]. Despite the short pulse durations reported in these studies, the inherent properties of Ce:LiCAF provides the capability for this solid-state laser gain medium to generate even shorter UV pulse durations, which up to this point has not been fully realized. Numerical calculations play an important role in determining the influence of optical parameters, such as pumping energy, Q-value and cavity length on the output energy and pulse duration of a Ce:LiCAF UV laser, and hence in optimizing the laser oscillator design.

The Ce:LiCAF crystal used in the numerical calculations has 1 mol% Ce3+ ion doping and is 1 cm long. This crystal is placed inside a Fabry-Perot laser cavity of length L. It is end-pumped by the fourth harmonics (266 nm) of a ps Nd:YAG laser. The pump pulse is a Gaussian beam with 75 ps (FWHM) pulse duration. The end mirror of the laser cavity is flat with reflectivity denoted by R1. The output coupler of the laser cavity is also flat with reflectivity denoted by R2. It is assumed that both mirrors have uniform reflectivity within the emission bandwidth of the laser crystal. The optical properties of the Ce:LiCAF crystal given in Table 2, which is detailed in several papers, are used as calculation parameters [2, 20–22, 24–30].

Laser emission is approximated using a system of two homogeneous broadened singlet states. Eqs. (1)–(4) show the modified rate equations as it applies to multiple wavelengths, which accurately simulates the broad emission bandwidth of the Ce:LiCAF UV laser [72–74]:

$$\frac{\partial N\_1}{\partial t} = P(t) + \left[\sum\_{1}^{n} \sigma\_{di} I\_i\right] N\_o - \left[\sum\_{1}^{n} \sigma\_{ei} I\_i + \frac{1}{\tau}\right] N\_1 \tag{1}$$

$$P(t) = \frac{P\_{in} \left[1 - \exp\left(-\alpha l\right)\right] \lambda\_p}{hc\pi r^2 l} \exp\left[\frac{-4\ln\left(2\right)\left(t - t\_o\right)^2}{\Delta l^2}\right] \tag{2}$$

$$\frac{\partial I\_1}{\partial t} = \left[ 2(\sigma\_{ei} N\_1 - \sigma\_{ai} N\_0)l - \beta \right] \frac{I\_i}{T} + A\_i N\_1 \tag{3}$$

$$T = \frac{2[L + l(n - 1)]}{c} \tag{4}$$

time. P is the rate of pumping which is further described by Eq. (2) where λ<sup>p</sup> is the pump laser's wavelength, Pin is the power of the pump, l is the Ce:LiCAF crystal's length, r is the pump beam's radius inside the laser medium, <sup>h</sup> is Planck's constant which is 6.62606957 <sup>10</sup><sup>34</sup> J s, <sup>c</sup> is

Table 2. Optical properties of the Ce:LiCAF crystal and values of parameters that were kept constant in numerical

(pump laser's wavelength), t is the duration of pumping, Δt is the laser pump's pulse duration,

as a function of time is given by Eq. (3) where β is the round-trip loss defined by β = ln(R1R2) in which case R<sup>1</sup> is the end mirror's reflectivity and R<sup>2</sup> is the output coupler's reflectivity, σei is the emission cross-section at wavelength λi, σai is the absorption cross-section at wavelength λ<sup>i</sup> [24], Ai is a constant that simulates spontaneous emission at wavelength λ<sup>i</sup> and its value is considered equal for all wavelengths since the duration of pumping is much greater than the memory time of the system of equations [72–74]. In this work, the value of Ai is about three times longer than the decay time of fluorescent dyes as estimated using the fluorescence decay time of Ce:LiCAF, which us about 25 ns. T is the cavity round-trip time as defined by Eq. (4) where L is the laser cavity's length and n is the Ce:LiCAF crystal's refractive index. The values of the constants used

Rate Eqs. (1)–(4) were used to model the experimental results [71, 75] by using the same parameters that were used in the experiment [71]. The Ce:LiCAF crystal length, l = 10 mm; <sup>N</sup> = 5 1017 cm<sup>3</sup> for a doping concentration of 1 mol%; refractive index, <sup>n</sup> = 1.41; length of cavity, L = 15 mm; reflectivity of end mirror, R<sup>1</sup> = 100%; reflectivity of output coupler, R<sup>2</sup> = 30%; radius of pump beam, r = 100 μm; energy of pump, Epump = 94 μJ; and the laser pump's pulse duration, Δt = 75 ps. These values are given in Tables 2 and 3. The spectro-temporal plot of the simulated broadband emission from 286 to 290 nm is shown in Figure 7. The corresponding

, α is the absorption coefficient of Ce:LiCAF at 266 nm

Ultrashort Pulse Generation in Ce:LiCAF Ultraviolet Laser http://dx.doi.org/10.5772/intechopen.73501 143

. The laser intensity inside the cavity

the speed of light which is 3 108 m s<sup>1</sup>

simulations.

in the simulations are given in Table 2.

and t<sup>o</sup> is the time when Pin is maximum. P has a unit of s<sup>1</sup>

Ce3+ doping concentration 1 mol% Doping density, <sup>N</sup> <sup>5</sup> 1017 cm<sup>3</sup> Absorption coefficient at 266 nm (wavelength of pump laser) 4 cm<sup>1</sup>

Refractive index, n 1.41 Fluorescence lifetime, τ 25 ns

Wavelength of pump laser, λ<sup>p</sup> 266 nm Radius of pump beam inside the crystal 100 μm Pulse duration of pump pulse 75 ps

Spontaneous emission constant, Ai 0.2 <sup>10</sup><sup>10</sup> cm s<sup>2</sup>

Planck's constant, <sup>h</sup> 6.62606957 <sup>10</sup><sup>34</sup> J s

Speed of light, <sup>c</sup> <sup>3</sup> <sup>10</sup><sup>8</sup> m s<sup>2</sup>

Reflectivity of end mirror, R<sup>1</sup> 100%

Absorption cross-section, <sup>σ</sup>ai 2.606 <sup>10</sup><sup>19</sup> cm<sup>2</sup> at 290 nm Emission cross-section, <sup>σ</sup>ei 9.6 <sup>10</sup><sup>18</sup> cm<sup>2</sup> at 290 nm

The population density in the upper laser state as a function of time is given by Eq. (1) where N<sup>0</sup> is the lower-state population density, N<sup>1</sup> is the upper-state population density, N = N<sup>0</sup> + N<sup>1</sup> is the total doping density, Ii is the intensity of the laser with wavelength λi, τ is the fluorescence decay


5. Numerical simulation of optimized resonator transients for generating

Sub-nanosecond (0.7 ns), single-pulse, ~290 nm laser emission from a Ce:LiCAF crystal using an oscillator with a 25 mm cavity length and 25% output coupler reflection has been demonstrated experimentally. In this report, the end-pumping configuration was used to excite the crystal with an Nd:YAG laser emitting 266-nm wavelength (fourth harmonics), 5-ns pulses at 10 Hz repetition rate [70]. Another study reported the generation of 150-ps laser pulses from a 10-mm long Ce:LiCAF crystal. In this report, the laser oscillator was established using a cavity length of 15 mm, output coupler reflection of 30%, and 75 ps excitation pulses [71]. Despite the short pulse durations reported in these studies, the inherent properties of Ce:LiCAF provides the capability for this solid-state laser gain medium to generate even shorter UV pulse durations, which up to this point has not been fully realized. Numerical calculations play an important role in determining the influence of optical parameters, such as pumping energy, Q-value and cavity length on the output energy and pulse duration of a Ce:LiCAF UV laser,

The Ce:LiCAF crystal used in the numerical calculations has 1 mol% Ce3+ ion doping and is 1 cm long. This crystal is placed inside a Fabry-Perot laser cavity of length L. It is end-pumped by the fourth harmonics (266 nm) of a ps Nd:YAG laser. The pump pulse is a Gaussian beam with 75 ps (FWHM) pulse duration. The end mirror of the laser cavity is flat with reflectivity denoted by R1. The output coupler of the laser cavity is also flat with reflectivity denoted by R2. It is assumed that both mirrors have uniform reflectivity within the emission bandwidth of the laser crystal. The optical properties of the Ce:LiCAF crystal given in Table 2, which is detailed

Laser emission is approximated using a system of two homogeneous broadened singlet states. Eqs. (1)–(4) show the modified rate equations as it applies to multiple wavelengths, which

No � <sup>X</sup><sup>n</sup>

hcπr<sup>2</sup><sup>l</sup> exp �4ln 2ð Þð Þ <sup>t</sup> � to <sup>2</sup>

1

σeiIi þ 1 τ

Δt 2 " #

N<sup>1</sup> (1)

<sup>T</sup> <sup>þ</sup> AiN<sup>1</sup> (3)

<sup>c</sup> (4)

(2)

" #

accurately simulates the broad emission bandwidth of the Ce:LiCAF UV laser [72–74]:

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup>ð Þ <sup>σ</sup>eiN<sup>1</sup> � <sup>σ</sup>aiN<sup>0</sup> <sup>l</sup> � <sup>β</sup> � � Ii

<sup>T</sup> <sup>¼</sup> <sup>2</sup>½ � <sup>L</sup> <sup>þ</sup> l nð Þ � <sup>1</sup>

The population density in the upper laser state as a function of time is given by Eq. (1) where N<sup>0</sup> is the lower-state population density, N<sup>1</sup> is the upper-state population density, N = N<sup>0</sup> + N<sup>1</sup> is the total doping density, Ii is the intensity of the laser with wavelength λi, τ is the fluorescence decay

1 σaiIi " #

ultrashort Ce:LiCAF laser pulses

142 Numerical Simulations in Engineering and Science

and hence in optimizing the laser oscillator design.

∂N<sup>1</sup>

∂I<sup>1</sup>

in several papers, are used as calculation parameters [2, 20–22, 24–30].

<sup>∂</sup><sup>t</sup> <sup>¼</sup> P tðÞþ <sup>X</sup><sup>n</sup>

P tðÞ¼ Pin <sup>1</sup> � exp ð Þ �α<sup>l</sup> � �λ<sup>p</sup>

Table 2. Optical properties of the Ce:LiCAF crystal and values of parameters that were kept constant in numerical simulations.

time. P is the rate of pumping which is further described by Eq. (2) where λ<sup>p</sup> is the pump laser's wavelength, Pin is the power of the pump, l is the Ce:LiCAF crystal's length, r is the pump beam's radius inside the laser medium, <sup>h</sup> is Planck's constant which is 6.62606957 <sup>10</sup><sup>34</sup> J s, <sup>c</sup> is the speed of light which is 3 108 m s<sup>1</sup> , α is the absorption coefficient of Ce:LiCAF at 266 nm (pump laser's wavelength), t is the duration of pumping, Δt is the laser pump's pulse duration, and t<sup>o</sup> is the time when Pin is maximum. P has a unit of s<sup>1</sup> . The laser intensity inside the cavity as a function of time is given by Eq. (3) where β is the round-trip loss defined by β = ln(R1R2) in which case R<sup>1</sup> is the end mirror's reflectivity and R<sup>2</sup> is the output coupler's reflectivity, σei is the emission cross-section at wavelength λi, σai is the absorption cross-section at wavelength λ<sup>i</sup> [24], Ai is a constant that simulates spontaneous emission at wavelength λ<sup>i</sup> and its value is considered equal for all wavelengths since the duration of pumping is much greater than the memory time of the system of equations [72–74]. In this work, the value of Ai is about three times longer than the decay time of fluorescent dyes as estimated using the fluorescence decay time of Ce:LiCAF, which us about 25 ns. T is the cavity round-trip time as defined by Eq. (4) where L is the laser cavity's length and n is the Ce:LiCAF crystal's refractive index. The values of the constants used in the simulations are given in Table 2.

Rate Eqs. (1)–(4) were used to model the experimental results [71, 75] by using the same parameters that were used in the experiment [71]. The Ce:LiCAF crystal length, l = 10 mm; <sup>N</sup> = 5 1017 cm<sup>3</sup> for a doping concentration of 1 mol%; refractive index, <sup>n</sup> = 1.41; length of cavity, L = 15 mm; reflectivity of end mirror, R<sup>1</sup> = 100%; reflectivity of output coupler, R<sup>2</sup> = 30%; radius of pump beam, r = 100 μm; energy of pump, Epump = 94 μJ; and the laser pump's pulse duration, Δt = 75 ps. These values are given in Tables 2 and 3. The spectro-temporal plot of the simulated broadband emission from 286 to 290 nm is shown in Figure 7. The corresponding spectral dynamics derived from integrating along the horizontal axis of Figure 7 is shown in Figure 8. The maximum laser intensity is observed at around 288.5 nm. This corresponds to the wavelength where the gain coefficient is also maximum. The distinct feature observed at around 289.5 nm is consistent with experimental observation [20–30] and is reported to be present for


any amount of doping. Figure 9 shows the temporal dynamics of the laser emission, which was derived by integrating along the vertical axis of Figure 7. Electron density builds up in the upper laser state of Ce:LiCAF until population inversion is achieved. Lasing threshold is reached around 0.3 ns after the onset of the pump pulse (not shown). The lasing threshold was measured when laser emission is observed. Peak emission is achieved about 0.5 ns after the onset of the pump pulse. The pulse duration is estimated from the full-width-at-half-maximum (FWHM) to be around 154 ps. Results of the temporal dynamics simulation are comparable, within the limits of jitter, to experimental results estimated from the streak camera image that was obtained experimentally [71, 75]. The experimental pulse duration is about 150 ps [71]. The good agreement between the numerical and experimental results indicate that the system of two homogeneous broadened singlet states and the modified rate equations as it applies to multiple wavelengths provide good approximations for predicting the experimental outcome. In succeeding calculations, the model described above was then used to optimize the optical parameters in

Ultrashort Pulse Generation in Ce:LiCAF Ultraviolet Laser http://dx.doi.org/10.5772/intechopen.73501 145

Eqs. (1) and (2) clearly show that the rate of change of excited state population density and laser intensity strongly depend on the pump energy. Therefore, we solved the rate equations for varying pump energies from 25 to 600 μJ in order to determine its effect on the temporal evolution of the Ce:LiCAF laser emission. Results are shown in Figure 10a–f. The parameters used in the numerical simulation were kept constant except for the pump energy. These parameters were the same as the experimental parameters, i.e. crystal length, <sup>l</sup> = 10 mm; <sup>N</sup> = 5 1017 cm<sup>3</sup> for 1 mol% doping concentration; cavity length, <sup>L</sup> = 15 mm; end mirror reflectivity, R<sup>1</sup> = 100%; output coupler reflectivity, R<sup>2</sup> = 30%; pump beam radius, r = 100 μm; and pulse duration of the laser pump, Δt = 75 ps. Lasing is achieved when the pump energy is around 37.5 μJ (Figure 10b). Figure 10c–f clearly shows that the laser pulse duration shortens as the pump energy is increased up to 360 μJ (Figure 10d). As the pump energy is increased, the time delay between the onset of laser emission and the pump also decreases due to the earlier occurrence of population inversion. The laser emission at different pump energies (Figure 11) shows that the spectral bandwidth becomes broader when the

order to achieve ultrashort pulse emission from a Ce:LiCAF solid-state UV laser.

Figure 9. Temporal dynamics of the numerically calculated spectro-temporal profile in Figure 7.

Figure 7. Spectro-temporal evolution of the low-Q, short cavity Ce:LiCAF laser pulse emission obtained numerically using the rate equations. The oscillator parameters are <sup>N</sup> = 5 <sup>10</sup><sup>17</sup> cm<sup>3</sup> , l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, Epump = 94 μJ, and n = 1.41.

Figure 8. Spectral dynamics of the numerically calculated spectro-temporal profile in Figure 7.

Figure 9. Temporal dynamics of the numerically calculated spectro-temporal profile in Figure 7.

spectral dynamics derived from integrating along the horizontal axis of Figure 7 is shown in Figure 8. The maximum laser intensity is observed at around 288.5 nm. This corresponds to the wavelength where the gain coefficient is also maximum. The distinct feature observed at around 289.5 nm is consistent with experimental observation [20–30] and is reported to be present for

Figure 7. Spectro-temporal evolution of the low-Q, short cavity Ce:LiCAF laser pulse emission obtained numerically

, l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%,

Crystal length, l 10 mm Cavity length, L 15 mm Reflectivity of output coupler, R<sup>2</sup> 30% Energy of pump beam 94 μJ

Table 3. Ce:LiCAF resonator parameters used to reproduce experimental results.

using the rate equations. The oscillator parameters are <sup>N</sup> = 5 <sup>10</sup><sup>17</sup> cm<sup>3</sup>

Figure 8. Spectral dynamics of the numerically calculated spectro-temporal profile in Figure 7.

r = 100 μm, Δt = 75 ps, Epump = 94 μJ, and n = 1.41.

144 Numerical Simulations in Engineering and Science

any amount of doping. Figure 9 shows the temporal dynamics of the laser emission, which was derived by integrating along the vertical axis of Figure 7. Electron density builds up in the upper laser state of Ce:LiCAF until population inversion is achieved. Lasing threshold is reached around 0.3 ns after the onset of the pump pulse (not shown). The lasing threshold was measured when laser emission is observed. Peak emission is achieved about 0.5 ns after the onset of the pump pulse. The pulse duration is estimated from the full-width-at-half-maximum (FWHM) to be around 154 ps. Results of the temporal dynamics simulation are comparable, within the limits of jitter, to experimental results estimated from the streak camera image that was obtained experimentally [71, 75]. The experimental pulse duration is about 150 ps [71]. The good agreement between the numerical and experimental results indicate that the system of two homogeneous broadened singlet states and the modified rate equations as it applies to multiple wavelengths provide good approximations for predicting the experimental outcome. In succeeding calculations, the model described above was then used to optimize the optical parameters in order to achieve ultrashort pulse emission from a Ce:LiCAF solid-state UV laser.

Eqs. (1) and (2) clearly show that the rate of change of excited state population density and laser intensity strongly depend on the pump energy. Therefore, we solved the rate equations for varying pump energies from 25 to 600 μJ in order to determine its effect on the temporal evolution of the Ce:LiCAF laser emission. Results are shown in Figure 10a–f. The parameters used in the numerical simulation were kept constant except for the pump energy. These parameters were the same as the experimental parameters, i.e. crystal length, <sup>l</sup> = 10 mm; <sup>N</sup> = 5 1017 cm<sup>3</sup> for 1 mol% doping concentration; cavity length, <sup>L</sup> = 15 mm; end mirror reflectivity, R<sup>1</sup> = 100%; output coupler reflectivity, R<sup>2</sup> = 30%; pump beam radius, r = 100 μm; and pulse duration of the laser pump, Δt = 75 ps. Lasing is achieved when the pump energy is around 37.5 μJ (Figure 10b). Figure 10c–f clearly shows that the laser pulse duration shortens as the pump energy is increased up to 360 μJ (Figure 10d). As the pump energy is increased, the time delay between the onset of laser emission and the pump also decreases due to the earlier occurrence of population inversion. The laser emission at different pump energies (Figure 11) shows that the spectral bandwidth becomes broader when the

Figure 10. Temporal evolution of the Ce:LiCAF laser emission for different pump energies: (a) 25 μJ, (b) 37.5 μJ, (c) 150 μJ, (d) 360 μJ, (e) 450 μJ, and (f) 600 μJ. The dashed plot represents N1(t) while the solid plot represents I(t). The following parameters were kept constant: <sup>N</sup> = 5 � <sup>10</sup><sup>17</sup> cm�<sup>3</sup> , l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, and n = 1.41.

pump energy is increased. This is expected since wavelengths close to 288 nm have sufficient energy to achieve considerable gain.

The laser output energy was calculated using

$$E\_{out} = \int\_0^t \int\_{\lambda\_1}^{\lambda\_n} (1 - R\_2) \pi r^2 h c \,\Big|\, I(\lambda, t) \tag{5}$$

used in the simulations is 600 μJ as the damage threshold is reached at this amount of energy for a 100 μm-beam radius. The laser cavity that was used in Ref. [71] amounts to a laser emission efficiency of about 23% as indicated by Figure 12. The same figure (Figure 12) also shows that a 360 μJ—pump energy could generate 96 ps pulses. This would be the shortest pulse duration. From this observation, the spectro-temporal evolution of the laser pulse emis-

Figure 12. Output energy and pulse duration for different pump energies. Calculation parameters used simulate exper-

, l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, and n = 1.41. The

Ultrashort Pulse Generation in Ce:LiCAF Ultraviolet Laser http://dx.doi.org/10.5772/intechopen.73501 147

Figure 11. Spectral profiles of the laser emissions for the different pump energies shown in Figure 10.

On the other hand, the suppression of the laser emission is evident where the Ce:LiCAF's gain coefficient is lower, particularly on either side of the 288.5-nm wavelength. Because of this, the maximum intensity of the broadband UV laser emission is expected at about 288.5 nm. Figure 14 confirms that the gain coefficient is still maximum at around 288.5 nm and hence, a shift in the position of the laser peak is not expected. Figure 10d shows the temporal evolution of the laser pulse from 284 to 293 nm. This wavelength range spans the whole bandwidth of the UV laser

sion at 360 μJ—pump energy is calculated and shown in Figure 13.

imental conditions: <sup>N</sup> = 5 1017 cm<sup>3</sup>

slope efficiency is about 23%.

where I(λ, t) is the laser intensity at wavelength λ and time t. As expected, the output energy increases as the pump energy is increased. This trend is shown in Figure 12.

Figures 10 and 12 indicate that in theory, increasing the pump energy would result to an ultrashort (ps pulse duration) laser pulse with micro Joule energy. However, absorption saturation and the damage threshold of the crystal limit the choice of pump energy in experiments. Eq. (4) is therefore integrated in order to determine where absorption saturation of the 266-nm pump begins for the crystal used in experiments. The crystal is 1-cm long and the total number of Ce3+ ions is 6.3 � 1014. The absorption saturation is determined to begin at 1.4 mJ pump energy when the beam spot radius is 100-μm. However, the damage threshold of Ce:LiCAF at 266 nm (wavelength of pump) is about 2 J/cm<sup>2</sup> [76]. Therefore, the maximum pump energy

Figure 11. Spectral profiles of the laser emissions for the different pump energies shown in Figure 10.

pump energy is increased. This is expected since wavelengths close to 288 nm have sufficient

Figure 10. Temporal evolution of the Ce:LiCAF laser emission for different pump energies: (a) 25 μJ, (b) 37.5 μJ, (c) 150 μJ, (d) 360 μJ, (e) 450 μJ, and (f) 600 μJ. The dashed plot represents N1(t) while the solid plot represents I(t). The following

ð Þ <sup>1</sup> � <sup>R</sup><sup>2</sup> <sup>π</sup>r<sup>2</sup>hc

where I(λ, t) is the laser intensity at wavelength λ and time t. As expected, the output energy

Figures 10 and 12 indicate that in theory, increasing the pump energy would result to an ultrashort (ps pulse duration) laser pulse with micro Joule energy. However, absorption saturation and the damage threshold of the crystal limit the choice of pump energy in experiments. Eq. (4) is therefore integrated in order to determine where absorption saturation of the 266-nm pump begins for the crystal used in experiments. The crystal is 1-cm long and the total number of Ce3+ ions is 6.3 � 1014. The absorption saturation is determined to begin at 1.4 mJ pump energy when the beam spot radius is 100-μm. However, the damage threshold of Ce:LiCAF at 266 nm (wavelength of pump) is about 2 J/cm<sup>2</sup> [76]. Therefore, the maximum pump energy

<sup>λ</sup> <sup>I</sup>ð Þ <sup>λ</sup>; <sup>t</sup> (5)

, l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, and

energy to achieve considerable gain.

parameters were kept constant: <sup>N</sup> = 5 � <sup>10</sup><sup>17</sup> cm�<sup>3</sup>

146 Numerical Simulations in Engineering and Science

n = 1.41.

The laser output energy was calculated using

Eout ¼

ðt

λ ðn

λ1

0

increases as the pump energy is increased. This trend is shown in Figure 12.

Figure 12. Output energy and pulse duration for different pump energies. Calculation parameters used simulate experimental conditions: <sup>N</sup> = 5 1017 cm<sup>3</sup> , l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, and n = 1.41. The slope efficiency is about 23%.

used in the simulations is 600 μJ as the damage threshold is reached at this amount of energy for a 100 μm-beam radius. The laser cavity that was used in Ref. [71] amounts to a laser emission efficiency of about 23% as indicated by Figure 12. The same figure (Figure 12) also shows that a 360 μJ—pump energy could generate 96 ps pulses. This would be the shortest pulse duration. From this observation, the spectro-temporal evolution of the laser pulse emission at 360 μJ—pump energy is calculated and shown in Figure 13.

On the other hand, the suppression of the laser emission is evident where the Ce:LiCAF's gain coefficient is lower, particularly on either side of the 288.5-nm wavelength. Because of this, the maximum intensity of the broadband UV laser emission is expected at about 288.5 nm. Figure 14 confirms that the gain coefficient is still maximum at around 288.5 nm and hence, a shift in the position of the laser peak is not expected. Figure 10d shows the temporal evolution of the laser pulse from 284 to 293 nm. This wavelength range spans the whole bandwidth of the UV laser

<sup>τ</sup><sup>c</sup> <sup>¼</sup> <sup>L</sup> <sup>þ</sup> n lð Þ � <sup>1</sup> cð Þ 1 � ln ð Þ R1R<sup>2</sup>

The photon cavity lifetime (τc) decreases as the cavity length, L, or the mirror reflectivities, R<sup>1</sup> and R2, decrease. Short-pulse laser emission in solid-state gain media can be achieved through the combination of a photon cavity lifetime (τc) that is smaller compared to the duration of the pump laser pulse and moderate resonator transients. The latter is brought about by the interaction between the photons in the cavity and the excess population inversion. Previous works have reported using resonator transients in dye lasers to obtain laser pulse durations that are an order of magnitude shorter than the pulse duration of the pump laser [34–36].

In order to extend the technique of resonator transients to solid-state gain media, the rate equations were solved for a variety of output coupler reflectivities and cavity lengths. The optimum condition for setting up a transient cavity was first determined by using a constant value for the output coupler reflectivity while varying the cavity lengths from L = 2 mm to L = 10 mm. The following summarizes the values of the parameters that were kept constant: R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, l = 1 mm, r = 100 μm, pumping energy = 140 μJ, and Δt = 75 ps. The total number of Ce3+ ions in the crystal considered here is 6.3 � <sup>10</sup>13. By integrating Eq. (4), the absorption saturation of the 266 nm pump begins at pump energy of 142.8 μJ. Therefore, the maximum energy used in all calculations involving this 1-mm crystal, including Figures 15–18, is 140 μJ. Note that the 30% output coupler reflectivity is the same as what was used in the experiment of reference [71]. Calculations show that shorter pulse durations are obtained when the cavity length is shortened. These results are presented in Figure 15a–d. It should be noted

Figure 15. Temporal evolution of the Ce:LiCAF laser emission for different cavity lengths, L, of (a) 2 mm (b) 5 mm, (c) 8 mm, and (d) 10 mm. I(t) is represented by the solid plot while N1(t) is represented by the dashed plot. The values of the

n = 1.41, and pump energy = 140 μJ. The pump energy was chosen based on the effect of pump energy on pulse duration

, l = 1 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps,

following parameters that were kept constant are: <sup>N</sup> = 5 � <sup>10</sup><sup>17</sup> cm�<sup>3</sup>

shown in Figure 16.

(6)

149

Ultrashort Pulse Generation in Ce:LiCAF Ultraviolet Laser http://dx.doi.org/10.5772/intechopen.73501

Figure 13. Spectro-temporal evolution of the shortest possible laser pulse duration (96 ps) achievable using experimental resonator parameters <sup>N</sup> = 5 1017 cm<sup>3</sup> , l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, n = 1.41, and 360-μJ pump energy.

Figure 14. Spectral dynamics of the shortest achievable laser pulse duration (96 ps). The corresponding temporal dynamics is shown in Figure 10d.

emission from Ce:LiCAF. By measuring the full width at half maximum (FWHM), the duration of the laser pulse is estimated to be about 96 ps. These calculations indicate that the shortest pulse duration that can be achieved without damaging the crystal is around 96 ps with a slope efficiency of around 23% assuming that the same parameters for the laser cavity and gain medium from the experiment of reference [71] are used.

The design of the laser oscillator cavity is important for optimizing laser emission especially for solid-state gain media. The cavity transient method, which uses the relationship between the cavity round trip time and the fluorescence decay time of the laser gain medium offers a simplified means of generating short laser pulses directly from a cavity that is optically pumped [34, 35]. With this method, factors such as the cavity lifetime of the photon, energy of the pump, and duration of the laser pump pulse strongly influences the pulse duration of laser emission. Moreover, the cavity lifetime of the photon (τc) which is described by Eq. (6) is determined by the length of the cavity and the reflectivity of the mirrors.

Ultrashort Pulse Generation in Ce:LiCAF Ultraviolet Laser http://dx.doi.org/10.5772/intechopen.73501 149

$$\tau\_c = \frac{L + n(l - 1)}{c(1 - \ln\left(R\_1 R\_2\right))}\tag{6}$$

The photon cavity lifetime (τc) decreases as the cavity length, L, or the mirror reflectivities, R<sup>1</sup> and R2, decrease. Short-pulse laser emission in solid-state gain media can be achieved through the combination of a photon cavity lifetime (τc) that is smaller compared to the duration of the pump laser pulse and moderate resonator transients. The latter is brought about by the interaction between the photons in the cavity and the excess population inversion. Previous works have reported using resonator transients in dye lasers to obtain laser pulse durations that are an order of magnitude shorter than the pulse duration of the pump laser [34–36].

In order to extend the technique of resonator transients to solid-state gain media, the rate equations were solved for a variety of output coupler reflectivities and cavity lengths. The optimum condition for setting up a transient cavity was first determined by using a constant value for the output coupler reflectivity while varying the cavity lengths from L = 2 mm to L = 10 mm. The following summarizes the values of the parameters that were kept constant: R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, l = 1 mm, r = 100 μm, pumping energy = 140 μJ, and Δt = 75 ps. The total number of Ce3+ ions in the crystal considered here is 6.3 � <sup>10</sup>13. By integrating Eq. (4), the absorption saturation of the 266 nm pump begins at pump energy of 142.8 μJ. Therefore, the maximum energy used in all calculations involving this 1-mm crystal, including Figures 15–18, is 140 μJ. Note that the 30% output coupler reflectivity is the same as what was used in the experiment of reference [71]. Calculations show that shorter pulse durations are obtained when the cavity length is shortened. These results are presented in Figure 15a–d. It should be noted

emission from Ce:LiCAF. By measuring the full width at half maximum (FWHM), the duration of the laser pulse is estimated to be about 96 ps. These calculations indicate that the shortest pulse duration that can be achieved without damaging the crystal is around 96 ps with a slope efficiency of around 23% assuming that the same parameters for the laser cavity and gain

Figure 14. Spectral dynamics of the shortest achievable laser pulse duration (96 ps). The corresponding temporal dynam-

Figure 13. Spectro-temporal evolution of the shortest possible laser pulse duration (96 ps) achievable using experimental

, l = 10 mm, L = 15 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, n = 1.41, and

The design of the laser oscillator cavity is important for optimizing laser emission especially for solid-state gain media. The cavity transient method, which uses the relationship between the cavity round trip time and the fluorescence decay time of the laser gain medium offers a simplified means of generating short laser pulses directly from a cavity that is optically pumped [34, 35]. With this method, factors such as the cavity lifetime of the photon, energy of the pump, and duration of the laser pump pulse strongly influences the pulse duration of laser emission. Moreover, the cavity lifetime of the photon (τc) which is described by Eq. (6) is

medium from the experiment of reference [71] are used.

resonator parameters <sup>N</sup> = 5 1017 cm<sup>3</sup>

148 Numerical Simulations in Engineering and Science

360-μJ pump energy.

ics is shown in Figure 10d.

determined by the length of the cavity and the reflectivity of the mirrors.

Figure 15. Temporal evolution of the Ce:LiCAF laser emission for different cavity lengths, L, of (a) 2 mm (b) 5 mm, (c) 8 mm, and (d) 10 mm. I(t) is represented by the solid plot while N1(t) is represented by the dashed plot. The values of the following parameters that were kept constant are: <sup>N</sup> = 5 � <sup>10</sup><sup>17</sup> cm�<sup>3</sup> , l = 1 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, n = 1.41, and pump energy = 140 μJ. The pump energy was chosen based on the effect of pump energy on pulse duration shown in Figure 16.

that the calculation model does not consider energy loss due to cavity length and therefore, the numerical results could over estimate experimental results particularly when the cavity length is long. Regardless of the over estimation, the numerical results show that it is favorable to have a shorter cavity length in order to achieve a shorter pulse durations As Eq. (5) predicts, smaller photon cavity lifetimes are obtained from smaller cavity lengths and as a consequence, shorter laser pulse durations are also obtained. However, the length of the crystal, l, dictates the limit on the practical size of the cavity, although it would appear from Eq. (5) that ultrashort pulses could be obtained by using ultrashort cavity lengths. From the point of view of crystal growth, 1 mm is a practical crystal length for a LiCAF crystal doped with 1 mol% Ce3+, and for fluoride crystals with 1 mol% rare earth doping in general. A 140-μJ pump energy was used based on Figure 16, which shows how the pump energy affects the pulse duration. The output energy is about the

Figure 16. Effect of pump energy on the pulse duration of the Ce:LiCAF laser emission.

same for the cavity lengths considered (L = 2–10 mm) as shown in Figure 17. The slope efficiency

30 μJ output energy is obtained using L = 15 mm and l = 10 mm for the same 140 μJ pump energy.

Figure 18. Temporal profile of the laser pulse for different pump energies using a L = 2 mm laser oscillator (short cavity). The energies used are: (a) 30 μJ, (b) 50 μJ, (c) 70 μJ, (d) 100 μJ, (e) 120 μJ and (f) 140 μJ. I(t) is represented by the solid plot while N1(t) is represented by the dashed plot. The values of the following parameters that were kept constant are: N = 5

, l = 1 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, and n = 1.41. Figure 19 summarizes the laser output energy and pulse duration for the energies used in Figure 18. A duration of 31.5 ps is the shortest pulse duration that can be obtained for L = 2 mm and l = 1 mm (short resonator cavity). The damage threshold and the absorption saturation of the crystal limit this pulse duration. Therefore, optimizing growth conditions and doping levels can achieve shorter pulse durations. Nevertheless, this is much shorter than the pulse duration that can be obtained using the same values for the pump energy (140-μJ), and output coupler reflectivity (30%), but with a cavity length of L = 15 mm and a crystal length of l = 10 mm. As Figure 12 shows, this laser oscillator that is longer produces pulse duration of 123 ps. On the other hand, a lower output energy is obtained for the same pump energy when the length of the crystal and hence the length of the cavity are shortened. For instance, about 10 μJ output energy is obtained using L = 2 mm and l = 1 mm whereas about

Ultrashort Pulse Generation in Ce:LiCAF Ultraviolet Laser http://dx.doi.org/10.5772/intechopen.73501 151

Figure 18a–f shows the temporal evolution of the laser pulse from a short cavity (L = 2 mm) for various pump energies. The same crystal and laser cavity parameters used in simulating Figure 15a–d were used in Figure 18a–f. A single laser pulse with ps pulse duration is achieved when the energy of the pump laser is varied from 50 μJ to 140 μJ, with a 31.5 ps pulse duration obtained at 140 μJ. Even though it is not shown in the figure, it is worth noting that when the energy of the pump is greater than 3 mJ, lasing occurs almost immediately. This pump energy leads to excess population inversion, as it is much higher than the lasing threshold. As a result of the interaction between the excess population inversion and the photons in the cavity, resonator transients are formed and these are manifested as damped relaxation oscillation or spiking in the laser pulse profile. The laser pulse will eventually approach the shape of the pump pulse when the energy of the pump is increased further,

is about 10%.

1017 cm<sup>3</sup>

Figure 17. Output energy for the different cavity lengths considered.

that the calculation model does not consider energy loss due to cavity length and therefore, the numerical results could over estimate experimental results particularly when the cavity length is long. Regardless of the over estimation, the numerical results show that it is favorable to have a shorter cavity length in order to achieve a shorter pulse durations As Eq. (5) predicts, smaller photon cavity lifetimes are obtained from smaller cavity lengths and as a consequence, shorter laser pulse durations are also obtained. However, the length of the crystal, l, dictates the limit on the practical size of the cavity, although it would appear from Eq. (5) that ultrashort pulses could be obtained by using ultrashort cavity lengths. From the point of view of crystal growth, 1 mm is a practical crystal length for a LiCAF crystal doped with 1 mol% Ce3+, and for fluoride crystals with 1 mol% rare earth doping in general. A 140-μJ pump energy was used based on Figure 16, which shows how the pump energy affects the pulse duration. The output energy is about the

150 Numerical Simulations in Engineering and Science

Figure 16. Effect of pump energy on the pulse duration of the Ce:LiCAF laser emission.

Figure 17. Output energy for the different cavity lengths considered.

Figure 18. Temporal profile of the laser pulse for different pump energies using a L = 2 mm laser oscillator (short cavity). The energies used are: (a) 30 μJ, (b) 50 μJ, (c) 70 μJ, (d) 100 μJ, (e) 120 μJ and (f) 140 μJ. I(t) is represented by the solid plot while N1(t) is represented by the dashed plot. The values of the following parameters that were kept constant are: N = 5 1017 cm<sup>3</sup> , l = 1 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, and n = 1.41. Figure 19 summarizes the laser output energy and pulse duration for the energies used in Figure 18. A duration of 31.5 ps is the shortest pulse duration that can be obtained for L = 2 mm and l = 1 mm (short resonator cavity). The damage threshold and the absorption saturation of the crystal limit this pulse duration. Therefore, optimizing growth conditions and doping levels can achieve shorter pulse durations. Nevertheless, this is much shorter than the pulse duration that can be obtained using the same values for the pump energy (140-μJ), and output coupler reflectivity (30%), but with a cavity length of L = 15 mm and a crystal length of l = 10 mm. As Figure 12 shows, this laser oscillator that is longer produces pulse duration of 123 ps. On the other hand, a lower output energy is obtained for the same pump energy when the length of the crystal and hence the length of the cavity are shortened. For instance, about 10 μJ output energy is obtained using L = 2 mm and l = 1 mm whereas about 30 μJ output energy is obtained using L = 15 mm and l = 10 mm for the same 140 μJ pump energy.

same for the cavity lengths considered (L = 2–10 mm) as shown in Figure 17. The slope efficiency is about 10%.

Figure 18a–f shows the temporal evolution of the laser pulse from a short cavity (L = 2 mm) for various pump energies. The same crystal and laser cavity parameters used in simulating Figure 15a–d were used in Figure 18a–f. A single laser pulse with ps pulse duration is achieved when the energy of the pump laser is varied from 50 μJ to 140 μJ, with a 31.5 ps pulse duration obtained at 140 μJ. Even though it is not shown in the figure, it is worth noting that when the energy of the pump is greater than 3 mJ, lasing occurs almost immediately. This pump energy leads to excess population inversion, as it is much higher than the lasing threshold. As a result of the interaction between the excess population inversion and the photons in the cavity, resonator transients are formed and these are manifested as damped relaxation oscillation or spiking in the laser pulse profile. The laser pulse will eventually approach the shape of the pump pulse when the energy of the pump is increased further, although the resonator transients (spikes) will still be visible. These spikes have not been observed experimentally since the pump energy in experiments is not high enough.

reflectivity is 30%. Taking a closer look at the temporal dynamics of the laser pulse for various output coupler reflectivity (Figure 21a–d) shows that the threshold for laser emission is reached earlier when the reflectivity is increased. As a result, the laser pulse duration will be longer when the output coupler is highly reflecting. Consequently, a low-Q laser resonator that is established using an output coupler with low reflectivity is required in order to generate short laser pulses low-Q cavity is therefore desirable for generating short-pulse laser emission. Figure 21a–d was simulated using laser pump energy of 140 μJ. The choice of pump energy is

Figure 21. Temporal evolution of the laser emission for various output coupler reflectivity (R2). The length of the short Ce: LiCAF cavity oscillator is L = 2 mm. The various R<sup>2</sup> values considered are: (a) 10%, (b) 30%, (c) 50%, and (d) 70%. The

pumping energy is 140 μJ. The pump energy was chosen based on results in Figures 19 and 20. Solid graph represents I(t)

Figure 22. Output energy as a function of pump energy for different output coupler reflectivities ranging from 10 to 70%. A short cavity oscillator (<sup>L</sup> = 2 mm) was assumed and the other parameters were kept constant as follows: <sup>N</sup> = 5 1017 cm<sup>3</sup>

, l = 1 mm, R<sup>1</sup> = 100%, Δt = 75 ps, r = 100 μm, n = 1.41, and

Ultrashort Pulse Generation in Ce:LiCAF Ultraviolet Laser http://dx.doi.org/10.5772/intechopen.73501 153

,

following parameters were kept constant: <sup>N</sup> = 5 1017 cm<sup>3</sup>

l = 1 mm, R<sup>1</sup> = 100%, r = 100 μm, Δt = 75 ps, and n = 1.41.

while dashed graph represents N1(t).

As discussed earlier, the pump energy and pulse duration of the laser pump as well as the photon cavity lifetime which is determined by the length of the oscillator cavity and the reflectivity of the mirrors strongly influences the pulse duration of the resulting laser emission. Therefore, numerical simulations are performed to quantify the effect of the reflectivity of the output coupler on the laser pulse duration for various pump energies. The results are shown in Figure 20. Other parameters are kept constant as follows: <sup>N</sup> = 5 1017 cm<sup>3</sup> , L = 2 mm, l = 1 mm, R<sup>1</sup> = 100%, r = 100 μm, and Δt = 75 ps. It can be observed that a low-Q cavity results to a short laser pulse. The shortest pulse duration is about 31.5 ps when the output coupler

Figure 19. Output energy and pulse duration of the laser pulse from a short cavity oscillator (L = 2 mm) for different pump energies. The following parameters were kept constant: <sup>N</sup> = 5 <sup>10</sup><sup>17</sup> cm<sup>3</sup> , l = 1 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%, r = 100 μm, Δt = 75 ps, and n = 1.41. The slope efficiency is about 24%.

Figure 20. Dependence of pulse duration on pump energy for different output coupler reflectivities: 10, 30, 50, and 70%. A short cavity oscillator (<sup>L</sup> = 2 mm) was assumed and the other parameters were kept constant as follows: <sup>N</sup> = 5 1017 cm<sup>3</sup> , l = 1 mm, R<sup>1</sup> = 100%, r = 100 μm, Δt = 75 ps, and n = 1.41.

reflectivity is 30%. Taking a closer look at the temporal dynamics of the laser pulse for various output coupler reflectivity (Figure 21a–d) shows that the threshold for laser emission is reached earlier when the reflectivity is increased. As a result, the laser pulse duration will be longer when the output coupler is highly reflecting. Consequently, a low-Q laser resonator that is established using an output coupler with low reflectivity is required in order to generate short laser pulses low-Q cavity is therefore desirable for generating short-pulse laser emission. Figure 21a–d was simulated using laser pump energy of 140 μJ. The choice of pump energy is

although the resonator transients (spikes) will still be visible. These spikes have not been

As discussed earlier, the pump energy and pulse duration of the laser pump as well as the photon cavity lifetime which is determined by the length of the oscillator cavity and the reflectivity of the mirrors strongly influences the pulse duration of the resulting laser emission. Therefore, numerical simulations are performed to quantify the effect of the reflectivity of the output coupler on the laser pulse duration for various pump energies. The results are shown in

l = 1 mm, R<sup>1</sup> = 100%, r = 100 μm, and Δt = 75 ps. It can be observed that a low-Q cavity results to a short laser pulse. The shortest pulse duration is about 31.5 ps when the output coupler

Figure 19. Output energy and pulse duration of the laser pulse from a short cavity oscillator (L = 2 mm) for different

Figure 20. Dependence of pulse duration on pump energy for different output coupler reflectivities: 10, 30, 50, and 70%. A short cavity oscillator (<sup>L</sup> = 2 mm) was assumed and the other parameters were kept constant as follows: <sup>N</sup> = 5 1017 cm<sup>3</sup>

pump energies. The following parameters were kept constant: <sup>N</sup> = 5 <sup>10</sup><sup>17</sup> cm<sup>3</sup>

r = 100 μm, Δt = 75 ps, and n = 1.41. The slope efficiency is about 24%.

l = 1 mm, R<sup>1</sup> = 100%, r = 100 μm, Δt = 75 ps, and n = 1.41.

, L = 2 mm,

, l = 1 mm, R<sup>1</sup> = 100%, R<sup>2</sup> = 30%,

,

observed experimentally since the pump energy in experiments is not high enough.

152 Numerical Simulations in Engineering and Science

Figure 20. Other parameters are kept constant as follows: <sup>N</sup> = 5 1017 cm<sup>3</sup>

Figure 21. Temporal evolution of the laser emission for various output coupler reflectivity (R2). The length of the short Ce: LiCAF cavity oscillator is L = 2 mm. The various R<sup>2</sup> values considered are: (a) 10%, (b) 30%, (c) 50%, and (d) 70%. The following parameters were kept constant: <sup>N</sup> = 5 1017 cm<sup>3</sup> , l = 1 mm, R<sup>1</sup> = 100%, Δt = 75 ps, r = 100 μm, n = 1.41, and pumping energy is 140 μJ. The pump energy was chosen based on results in Figures 19 and 20. Solid graph represents I(t) while dashed graph represents N1(t).

Figure 22. Output energy as a function of pump energy for different output coupler reflectivities ranging from 10 to 70%. A short cavity oscillator (<sup>L</sup> = 2 mm) was assumed and the other parameters were kept constant as follows: <sup>N</sup> = 5 1017 cm<sup>3</sup> , l = 1 mm, R<sup>1</sup> = 100%, r = 100 μm, Δt = 75 ps, and n = 1.41.

based on the results shown in Figures 19 and 20, where pulse duration decreases with increasing energy within the limits of absorption saturation. Theoretically, shorter pulse durations can be achieved using an output coupler with less than 10% reflectivity. Practically, the threshold energy and the slope efficiency limit the choice of reflectivity. As Figure 22 shows, higher pump energies are needed to achieve lasing in a low-Q laser resonator. If the reflectivity of the output coupler is 10%, for instance, the lasing threshold for obtaining a 31.5-ps laser pulse is 80 μJ and the slope efficiency is 8%. However, increasing the output coupler reflectivity to 30% decreases the threshold energy to 40 μJ and increases the slope efficiency to 10%.

According to Figures 19–22, the optimal transient cavity laser resonator has a 2-mm long cavity and a 30% output coupler reflectivity. These figures also indicate that about 31.5-ps laser pulse duration and about 10% slope efficiency is possible when a 1 mol% Ce-doped LiCAF crystal that is 1 mm long is excited by a 266-nm wavelength pump laser with 75-ps pulse duration and 140 μJ pump energy. These conditions already take into account the crystal's damage threshold, which is about 600-μJ of pump energy for a 100-μm-beam radius as well as its absorption saturation, which is about 142.8 μJ. The 31.5-ps pulse duration is significantly shorter than the experimental pulse duration obtained by reference [71]. The spectro-temporal and the spectral profiles of the broadband 31.5-ps laser pulse are shown in Figures 23 and 24, respectively. The spectral profile is consistent with the trend observed in Figure 11, regardless of resonator cavity parameters. Maximum gain coefficient is also

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In summary, the transient cavity method is extended to a solid-state gain medium. Numerical simulations show that the same principles used to generate ultrashort laser pulses in dye lasers using this technique can be applied to solid-state gain media to generate ultrashort broadband pulses in the UV region. The laser gain medium was represented as a system of two homogeneous broadened singlet states and the numerical simulations solved the laser rate equations for broadband emission. The spectral and temporal evolution of the resulting laser emission was investigated in order to find the optimal cavity length and output coupler reflectivity that will give rise to the formation of resonator transients in the laser oscillator cavity. The calculations reveal that a laser oscillator with a short cavity and a low Q is ideal for the formation of resonator transients, which then lead to ultrashort (ps) laser emission. Specifically, a 2-mm cavity length and a 10% output coupler reflectivity can be used to generate a single 31.5 ps pulse using a 1-mm long Ce:LiCAF crystal with 1 mol% Ce3+ ion doping concentration. Although this work used Ce: LiCAF crystal as the laser gain medium, the transient cavity method can also be applied to

This research was supported by the Massey University Research Fund 2018 (MURF 2018 Project No. 1000020752), Institute of Laser Engineering, Osaka University Collaborative Research Grant (Grant no. 2017B1-RADUBAN), JSPS-VAST Joint Research Project (2011– 2014), the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Numbers 103.06.89.09 and 103.03-2015.29. M. Cadatal-Raduban and M.V. Luong are very grateful to K.G. Steenbergen and P. Schwerdtfeger for their input and valuable discussions on the numerical simulation of the electronic properties of the LiCAF and

generate ultrashort laser pulses using other rare earth-doped fluoride crystals.

achieved at around 188.5 nm.

6. Conclusion

Acknowledgements

LiSAF host.

Figure 23. Spectro-temporal evolution of the broadband, short-pulse Ce:LiCAF laser emission from an optimized low-Q (R<sup>2</sup> = 30%), short cavity (L = 2 mm) oscillator. A short laser pulse with about 31.5-ps pulse duration, broadband emission centered at 288.5-nm wavelength, and 10 μJ output energy can be obtained practically from a 1-mm long, 1 mol% Ce3+ doped LiCAF crystal when pumped by a 266-nm, 75-ps pump pulse with 140 μJ pump energy. A slope efficiency of about 10% is also feasible with pump energies that are far from the crystal's absorption saturation and damage threshold.

Figure 24. Spectral profile of the broadband, short-pulse Ce:LiCAF laser emission from an optimized low-Q (R<sup>2</sup> = 30%), short cavity (L = 2 mm) oscillator. Temporal dynamics is shown in Figure 21b.

According to Figures 19–22, the optimal transient cavity laser resonator has a 2-mm long cavity and a 30% output coupler reflectivity. These figures also indicate that about 31.5-ps laser pulse duration and about 10% slope efficiency is possible when a 1 mol% Ce-doped LiCAF crystal that is 1 mm long is excited by a 266-nm wavelength pump laser with 75-ps pulse duration and 140 μJ pump energy. These conditions already take into account the crystal's damage threshold, which is about 600-μJ of pump energy for a 100-μm-beam radius as well as its absorption saturation, which is about 142.8 μJ. The 31.5-ps pulse duration is significantly shorter than the experimental pulse duration obtained by reference [71]. The spectro-temporal and the spectral profiles of the broadband 31.5-ps laser pulse are shown in Figures 23 and 24, respectively. The spectral profile is consistent with the trend observed in Figure 11, regardless of resonator cavity parameters. Maximum gain coefficient is also achieved at around 188.5 nm.
