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241

Accurate determination of saturation parameters for Cr4+-doped solid-state saturable absorbers Alphan Sennaroglu, Umit Demirbas, Sarper Ozharar,* and Fatih Yaman† Departments of Physics and Electrical–Electronics Engineering, Laser Research Laboratory, Koç University, Rumelifeneri, Sariyer, Istanbul 34450 Turkey Received March 4, 2005; revised July 31, 2005; accepted September 7, 2005 We describe a systematic, rigorous procedure for the determination of the optical absorption saturation parameters for Cr4+:YAG and Cr4+:forsterite crystals at 1064 nm. A rate-equation approach was used to analyze the cw and pulsed transmission data of several crystals by accounting for the transverse as well as longitudinal variation of the beam intensity, saturation effects, and excited-state absorption. Use of an iterative procedure whereby the cw and pulsed data were simultaneously analyzed led to a considerable reduction in the error for the determination of cross sections. The average value of the absorption cross section ␴a and the normalized excited-state absorption cross section fp = ␴esa / ␴a were determined to be 6.13⫻ 10−19 cm2 and 0.45, respectively, for Cr4+:forsterite and 19.6⫻ 10−19 cm2 and 0.06, respectively, for Cr4+:YAG. Detailed comparison was also made with previous saturation measurements in the literature. Our results further show that lumped models based on the thin-length approximation should be used with caution in the determination of cross sections, especially when the pump beam is tightly focused inside the absorber. © 2006 Optical Society of America OCIS codes: 140.3580, 160.6990.

1. INTRODUCTION In addition to their use as tunable laser media in the near-infrared region,1 Cr4+-doped crystals such as Cr4+:YAG and Cr4+:forsterite have also been employed as saturable absorbers in passively Q-switched and–or mode-locked solid-state lasers near 1 ␮m,2–7 owing to the presence of strong absorption bands in that region. Two important factors that influence the performance of these materials in passive Q switching are the strength of ground-state absorption and the existence of unwanted excited-state absorption, characterized by the cross sections ␴a and ␴esa, respectively. Because accurate determination of the cross sections is very important in the evaluation of passive Q switches, investigation of the saturation mechanism in these materials has attracted a great deal of attention over the past decade.2,3,5,7–21 However, a large discrepancy still remains among the reported cross-section values of the Cr4+-doped media, calling for a systematic procedure for the determination of the saturation parameters. In this work, we present the results of a detailed investigation aimed at accurately determining the saturation parameters of Cr4+:YAG and Cr4+:forsterite saturable absorbers. A rate-equation analysis was employed to analyze the cw and pulsed transmission data of several samples. A four-level energy scheme was assumed, and effects due to saturation, excited-state absorption, and transverse intensity variations were taken into account. In the experiments, both cw and pulsed transmission data were collected at 1064 nm. Our results show that when cw and pulsed transmission data are analyzed separately, we see a large spread in the best-fit cross-section values. On the other hand, if cw and pulsed data are analyzed iteratively so that the small-signal absorption coefficient is determined from the cw data and the excited-state ab0740-3224/06/020241-9/$15.00

sorption cross section is determined from the pulsed data, a significant reduction in the spread of the cross-section values is observed. The paper is organized as follows. Section 2 describes the rate-equation models that we used for the analysis of cw and pulsed transmission data. The experimental setup is described in Section 3. The analysis of the experimental data and the details of the iterative fitting procedure are further discussed in Section 4.A. The average best-fit values of ␴a and fp 共fp = ␴esa / ␴a兲 were determined to be 6.13 ⫻ 10−19 cm2 and 0.45, respectively, for Cr4+:forsterite; in the case of Cr4+:YAG, ␴a and fp were 19.6⫻ 10−19 cm2 and 0.06, respectively. In Section 4.B, we assess the validity of lumped saturation models and show that use of the thinlength approximation in the determination of cross sections leads to a large discrepancy in best-fit values. This was because the pump beam was tightly focused inside most of the absorbers considered in this study.

2. THEORY A. Continuous-Wave Case In this section, we derive the differential equations that govern the spatial evolution of the beam power or energy through the saturable absorber. Let us consider the energy-level diagram in Fig. 1 to model a saturable absorber subject to excited-state absorption. In the presence of pump photons at the wavelength of ␭p, absorber ions are first excited from the ground state 兩g典 to the first excited-state 兩3典 as a result of stimulated absorption. Fast nonradiative decay then occurs to the upper level 兩2典. In the case of Cr4+:forsterite and Cr4+:YAG, level 兩2典 serves also as the upper laser level from which ions may decay via stimulated or spontaneous emission to the lower level 兩1典. In our analysis, we have neglected the stimulated © 2006 Optical Society of America

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Sennaroglu et al.

⳵Ip ⳵z

+

1 ⳵Ip vg ⳵t

= − ␴aIp共Ng + fpN2兲.

共3兲

In Eq. (3), fp is the normalized strength of the excitedstate absorption at ␭p, defined according to fp =

␴esa共␭p兲 ␴a共␭p兲

共4兲

.

For the cw case, we concentrate on the steady-state solution of Eqs. (1) and (2). Noting that N2 + Ng = NT, where NT is the total ion density, the steady-state expression for N2 becomes Fig. 1. Energy-level diagram of a saturable absorber exhibiting excited-state absorption. Zig-zagged lines indicate nonradiative decay processes.

emission processes. Fast nonradiative decay finally brings the ions from state 兩1典 back to the ground state 兩g典. Because nonradiative decay rates are very fast, in comparison with decay rates of level, populations of levels 兩1典 and 兩3典 are assumed to be negligible. Finally, we assume that excited-state absorption may also take place from level 兩2典 to a higher-lying level 兩4典 at the pump wavelength of ␭p and that ions excited to level 兩4典 undergo a rapid nonradiative decay back to 兩2典. The population densities Ng共r , z兲 (ground state) and N2共r , z兲 (upper level) obey the following rate equations:

⳵N2共r,z兲 ⳵t

=−

⳵Ng共r,z兲 ⳵t

=

␴a␭pIp共r,z兲 hc

Ng共r,z兲 −

N2共r,z兲

␶f

,

N2 = NT

Adz

⳵up ⳵t

⳵Ip ⳵z

共2兲 where up is the electromagnetic energy density at the wavelength ␭p and 共Ng␴aIp + N2␴esaIp兲 corresponds to the pump energy lost per unit volume per unit time owing to ground-state and excited-state absorption. Here, ␴esa is the excited-state absorption cross section. Simplifying and by using the relation up = vgIp (vg = group velocity), we obtain

共5兲

.

冢 冣 1 + fp

= − ␣p0Ip

1+

Ip

Isa

Ip

共6兲

.

Isa

In Eq. (6), ␣p0 共␣p0 = NT␴a兲 is the small-signal differential pump-absorption coefficient. To derive the differential equation satisfied by the beam power, we assume that the pump intensity 共Ip兲 has a transverse distribution given by Ip共r,z兲 = Pp共z兲⌽p共r,z兲,

共7兲

where Pp共z兲 is the pump power at the location z and r is the radial coordinate. The normalized distribution function ⌽P is assumed to be Gaussian: ⌽P =

冉 冊 2r2

2

␲␻p2

exp −

␻p2

共8兲

.

Above, ␻p is the position-dependent pump spot-size function. By integrating Eq. (6) over the beam cross section and by using Eqs. (7) and (8), we obtain

dPp dz

= 关Ip共z兲 − Ip共z + dz兲兴A − 共Ng␴aIp + N2␴esaIp兲Adz,

1 + Ip/Isa

Here, Isa 共Isa = hc / ␴a␭p␶f兲 is the absorption saturation intensity. By using the steady-state expressions for N2 and Ng, the differential equation describing the spatial evolution of the pump intensity can be written as

共1兲 where h is Planck’s constant, c is the speed of light, Ip共r , z兲 is the intensity of the pump radiation at ␭p, ␴a is the ground-state absorption cross section at ␭p, and ␶f is the fluorescence lifetime. The quantity 共␴a␭pIpNg / hc兲 gives the number of atoms per unit volume per unit time undergoing stimulated absorption. In the following analysis, we will drop the spatial dependence of the intensities and population densities. We next consider a thin slab of the saturable absorber to derive the differential equation satisfied by the pump intensity. The thickness and the cross-sectional area of the slab are taken to be dz and A, respectively. Conservation of energy gives

Ip/Isa

= − ␣p0Pp



0

冢 冣 1 + fp



dr2␲r⌽p

1+

P p⌽ p Isa

P p⌽ p

.

共9兲

Isa

Note from Eq. (9) that in the limit as Isa goes to infinity, the pump power Pp decays exponentially inside the saturable absorber as expected. B. Pulsed Case We next consider the case in which a periodic pulse train is incident on the saturable absorber. The pulse intensity ¯ are related Ip共t兲 and the integrated energy density E p through

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Vol. 23, No. 2 / February 2006 / J. Opt. Soc. Am. B





¯ , dtIp共t兲 = E p

Table 1. Length and Small-Signal Absorption Coefficient ␣p0 of Each Sample Used in the Saturation Measurementsa

共10兲

−⬁

where ␶ is much greater than the pulsewidth so that Ip共␶兲 is essentially 0. By using Eq. (1), Eq. (3) can be recast into the form

⳵Ip ⳵z

+

1 ⳵Ip vg ⳵t

= 共1 − fp兲

hc ⳵Ng ␭p ⳵t

− fp␣p0Ip .

共11兲

Integrating Eq. (1) from t = −⬁ to t = ␶ with the initial condition Ng共−⬁兲 = NT = ␣p0 / ␴a, and with the assumption that ␶ is much shorter than the fluorescence lifetime ␶f leads to

冉 冊

Ng = NT exp −

¯ E p

Esa

243

Sample

Sample Number

Length (mm)

␣p0 共cm−1兲

Beam Waist 共␮m兲

Cr4+:forsterite Cr4+:forsterite Cr4+:forsterite Cr4+:YAG Cr4+:YAG Cr4+:YAG Cr4+:YAG

1 2 3 1 2 3 4

20 20 20 20 20 1.56 0.72

1.63 0.60 1.51 1.11 1.09 1.45 2.00

25 25 25 25 25 100 100

a

The measured beam waist is also shown for each sample.

共12兲

,

where Esa 共Esa = hc / ␴a␭p兲 is the absorption saturation fluence. Next, we integrate Eq. (11) from t = −⬁ to t = ␶ and use Eq. (12) to obtain the differential equation satisfied by the pump energy density: ¯ ⳵E p

⳵z

冋 冉 冊册

= − 共1 − fp兲␣p0Esa 1 − exp −

¯ E p

Esa

¯ . − fp␣p0E p 共13兲

Similar to the continuous-wave case, we assume that the pulse energy density and the total energy Ep per pulse are related according to ¯ =E ⌽ , E p p p

共14兲

where ⌽p is given by Eq. (8). Integrating over the beam cross-section, we obtain

⳵Ep ⳵z

= − 共1 − fp兲␣p0Esa





0

− fp␣p0Ep .

冋 冉 冊册

dr2␲r 1 − exp −

E p⌽ p Esa

共15兲

Note that if the pulse repetition rate of the pulsed laser is f0, then the average incident power Pi and the incident energy Epi per pulse are related through f0Epi = Pi. Equations (9) and (15) were used to analyze the cw and pulsed transmission data of the Cr4+:YAG and Cr4+:forsterite samples.

3. EXPERIMENT The experimental setup is shown in Fig. 2. The cw and pulsed transmission data of the Cr4+:forsterite and Cr4+:YAG samples were measured by using a 1064 nm flashlamp-pumped Nd:YAG laser (Quantronix, Model 116) that could be operated in cw or Q-switched mode. In

Fig. 2. Sketch of the experimental setup (␭ / 2, half-wave plate at 1064 nm).

Fig. 3. Measured and fit variations of the spot-size function ␻p共z兲 as a function of position z. The best-fit values of the beam waist and the M2 factor were determined to be 25 ␮m and 1.04, respectively.

Q-switched mode, the laser produced 140 ns pulses at a 1 kHz repetition rate. Table 1 lists the dimensions and the small-signal absorption coefficient of the samples used in the experiments. Except for the two thin antireflection-coated Cr4+:YAG samples (Cr4+:YAG Samples 3 and 4 in Table 1), all the other crystals were Brewster cut. With the half-wave plate (␭ / 2 in Fig. 2), the pump-beam polarization was adjusted to minimize the Fresnel losses, which could be kept below 0.2%. All measurements were performed at room temperature. The pump beam was focused with a lens having a 10 cm focal length, and the transmitted power was measured with a power meter (Molectron, Model 5100). In the case of pulsed pumping, the average transmitted power was measured with the same power meter, and the corresponding transmitted energy was determined by diving the average power by the pulse-repetition rate (1 kHz in this case). The measured variation of the pump spot-size function ␻p共z兲 with relative position is shown in Fig. 3. Spot-size measurements were performed by using the knife-edge technique. In particular, a sharp, vertical razor blade was scanned across the beam cross section to determine the transverse locations x1 and x2 which gave 84% and 16% power transmission, respectively. The spot-size at that location is then given by 兩x2 − x1兩.To determine the beam parameters, ␻p共z兲 was assumed to have the functional form

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J. Opt. Soc. Am. B / Vol. 23, No. 2 / February 2006

冋 冉 冊册

␻p共z兲 = ␻p0 1 +

z − zfp

2

zRp

Sennaroglu et al.

1/2

,

共16兲

where ␻p0 is the pump beam waist, zfp is the beam-waist 2 / M2␭p, n0, refractive index location, and zRp (zRp = n0␲␻p0 2 of the crystal; and M , beam-quality factor) is the Rayleigh range. For this beam geometry, the best-fit values of M2 and ␻p0 were determined to be 1.04 and 25 ␮m, respectively. In the case of the thin Cr4+:YAG samples, we used a less tightly focused beam with a measured waist of 100 ␮m (Cr4+:YAG samples 3 and 4 in Table 1).

4. RESULTS AND DISCUSSION A. Cross-Section Values for Cr4+:Forsterite and Cr4+:YAG Equations (9) and (15) can be used to investigate the dependence of the transmission on the key parameters of the saturable absorber for the cw and pulsed cases. As an example, consider Figs. 4(a) and 4(b) which show how the transmission varies as a function of the incident pump power and the incident pulse energy for the cw and pulsed cases, respectively, for a hypothetical saturable absorber ( ␴a = 25⫻ 10−19 cm2, ␶f = 5 ␮s, ␣p0 = 1.5 cm−1, ␭p = 1064 nm, L0 = 20 mm, n0 = 1.635, zfp = L0 / 2, and␻p0 = 25 ␮m). Here, L0 is the crystal length. Calculations were done for different values of fp. Note that in each case, the maximum

Fig. 5. Calculated variation of the transmission for a hypothetical absorber as a function of the beam-waist location for the (a) cw and (b) pulsed cases for different levels of excited-state absorption. (Absorber parameters: ␴a = 25⫻ 10−19 cm2, ␶f = 5 ␮s, ␣p0 = 1.5 cm−1, ␭p = 1064 nm, L0 = 20 mm, n0 = 1.635, and ␻.p0 = 25 ␮m.)

transmission Tmax obtained at very high intensities is limited by the presence of excited-state absorption and is given asymptotically by Tmax = exp共− fp␣p0L0兲.

Fig. 4. Calculated variation of the transmission for a hypothetical absorber as a function of (a) incident power (cw case) and (b) incident pulse energy (pulsed case) for different amounts of excited-state absorption. (Absorber parameters: ␴a = 25 ⫻ 10−19 cm2, ␶f = 5 ␮s, ␣p0 = 1.5 cm−1, ␭p = 1064 nm, L0 = 20 mm, n0 = 1.635, zfp = L0 / 2, ␻p0 = 25 ␮m).

共17兲

Figures 5(a) and 5(b) show the calculated variation of the transmission as a function of the beam-waist location 共zfp兲 for the cw and pulsed cases. Here, the fixed parameters of the saturable absorber were ␴a = 25⫻ 10−19 cm2, ␶f = 5 ␮s, ␣p0 = 1.5 cm−1, ␭p = 1064 nm, L0 = 20 mm, n0 = 1.635, and ␻p0 = 25 ␮m. In Fig. 5(a), the incident pump power was kept at 5 W, whereas the incident pump energy was 50 ␮J in Fig. 5(b). Note that the transmission has a strong dependence on the beam-waist location and peaks when the beam is focused near the center of the crystal. This suggests that z-scan methods can also be employed to determine the value of the absorption cross section, as was done in previous studies.13,18 In our experiments, the samples were translated at a fixed incident pump power until the transmission was maximized, and then the variation of the transmission was measured as a function of the incident pump power or energy at this location. Two different approaches were used to determine the best-fit values of the cross sections. In the first case, the cw and pulsed transmission data of each sample were analyzed separately. Specifically, for a given set of 共␣p0 , ␴a , fp兲 trial values, the particular zfp that maximizes the transmission was first determined, and the corresponding deviation between the measured and the calculated transmissions was calculated. The param-

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Vol. 23, No. 2 / February 2006 / J. Opt. Soc. Am. B

eters ␣p0, ␴a, and fp were then varied to find the best-fit values that minimize the deviation. In principle, ␣p0 can also be measured accurately with a spectrophotometer to eliminate one of the unknown parameters. In this study, we chose to determine it by using the scheme described above, for two reasons. First, the procedure proposed in this paper requires no additional instrumentation other than the transmission measurement setup discussed in Section 3. Second, all of the 20 mm long samples used in our measurements (see Table 1) were Brewster cut, and this makes spectrum measurements with a spectrophotometer cumbersome. The three Cr4+:forsterite samples shown in Table 1 were first analyzed by use of the algorithm described above. The fluorescence lifetime ␶f and the crystal refractive index n0 were taken to be 2.9 ␮s and 1.635, respectively.21 In the measurements, the electric field was kept parallel to the b axis of the crystals (E储b). The results, which are displayed in Table 2, show a large spread in the best-fit values. In particular, the fractional spread, defined as the ratio of the standard deviation to the average value, came to 1.16 and 0.28 for ␴a and fp, respectively. These numbers were determined by considering the pulsed and cw transmission results of the three samples. The reason for the large variation in the best-fit values is that cw data alone do not reveal the behavior of the saturable absorbers in the high-intensity limit, hence making the accurate determination of fp very difficult. Similarly, owing to small signal-to-noise ratio at low power levels, determination of ␣p0 is very sensitive to fluctuations in the case of pulsed transmission data. We resolved this problem by using a second, iterative approach to determine the parameters of the saturable absorber. The steps of the analysis may be summarized as follows. The cw transmission data of each sample were first analyzed to determine the best-fit values of ␣p0, ␴a, and fp. The ␣p0 value determined from the cw data was

Fig. 6. Measured and fit variations of the transmission as a function of the (a) incident power (cw case) and (b) incident pulse energy (pulsed case) for the three Cr4+:forsterite crystals.

Table 4. Previously Reported ␴a and fp Values for Cr4+:Forsterite in the Literaturea

Table 2. Best-Fit Values of ␣p0, fp, and ␴a Obtained by Analyzing the cw and Pulsed Transmission Data of the Cr4+:Forsterite Samples Separately ␣p0 共cm−1兲

␴a 共⫻10−19 cm2兲

fp

Sample No

cw

Pulsed

cw

Pulsed

cw

Pulsed

1 2 3

1.63 0.60 1.55

2.09 0.71 1.66

0.63 0.56 0.74

0.38 0.40 0.42

11.4 9.5 67.0

13.7 9.7 8.5

Table 3. Best-Fit Values of ␣p0, fp, and ␴a for Cr4+:Forsterite Samples Obtained by Using the Iterative Analysis Schemea ␴a 共⫻10−19 cm2兲

Sample Number

␣p0 共cm−1兲

fp

cw

Pulsed

1 2 3

1.63 0.60 1.51

0.46 0.45 0.45

6.3 6.5 8.8

5.1 4.3 5.8

a

The average values of f p and ␴a are 0.45 and 6.13⫻ 10−19 cm2, respectively.

245

a

␴a 共⫻10−19 cm2兲

fp

Reference

1.36 6 23 1.9 1.9 2.5 6.11 6.13a

0.13 0.48 — — — 0.3 0.31 0.45

23 16 2 17 9 8 Average This paper

If more than one sample was used, only the average value is given.

kept fixed in the analysis of the pulsed data, and only ␴a and fp were varied to determine the best-fit values. Next, the cw data were analyzed once again to determine ␴a. Here, the best-fit fp value obtained from the analysis of the pulsed data and the best-fit ␣p0 value from the original cw analysis were kept fixed. The best-fit values of the saturation parameters for Cr4+:forsterite obtained by using this scheme are shown in Table 3. Note that the use of the iterative procedure leads to a large reduction in the spread of the best-fit values. For example, the fractional spread was reduced from 1.16 to 0.25 and from 0.28 to 0.01 for the best-fit values of ␴a and fp, respectively. Figures 6(a) and 6(b) show the resulting measured and fit variation of the transmission as a function of the incident

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Table 5. Best-Fit Values of ␣p0, fp, and ␴a Obtained by Analyzing the cw and Pulsed Transmission Data of the Cr4+:YAG Samples Separatelya ␣p0 共cm−1兲

Sample Number Orientation 1 2 3 4

E 储 具1 , 0 , 0典 E 储 具1 , 0 , 0典 P1a P2a P1a P2a

␴a 共⫻10−19 cm2兲

fp

cw

Pulsed

cw

Pulsed

cw

Pulsed

1.11 1.06 1.59 1.58 1.99 1.99

0.63 0.87 1.11 0.96 1.37 1.83

0.07 0.01 0.37 0.58 0.04 0.00

0.10 0.08 0.00 0.00 0.00 0.00

26.2 14.2 103.3 152.2 23.5 23.1

8.1 14.9 8.3 5.8 5.1 8.9

the cross-section values determined in this study with previous reports. As can be seen from Table 7, the average ␴a value of 19.6⫻ 10−19 cm2 determined in this study is close to the average of the previously reported values 共23.3⫻ 10−19 cm2兲. The average fp value of the present study (0.06) is somewhat lower than the previous results (the average value in Table 7 is 0.16). Finally, since samples 3 and 4 were normal cut, we also compared the best ␣p0 values with those obtained from absorption spectrum measurements with a spectrophotometer. The aver-

a P1 and P2 refer to the two orthogonal electric field polarizations used in the transmission measurements for samples 3 and 4 共See the text兲.

Table 6. Best-Fit Values of ␣p0, fp, and ␴a for Cr4+:YAG Samples Obtained by Using the Iterative Analysis Schemea Sample Number 1 2 3 4

a

␴a 共⫻10−19 cm2兲

Orientation

␣p0 共cm−1兲

fp

cw

Pulsed

E 储 具1 , 0 , 0典 E 储 具1 , 0 , 0典 P1 P2 P1 P2

1.11 1.09 1.42 1.47 1.99 2.00

0.06 0.10 0.02 0.10 0.05 0.03

25.7 20.4 24.5 17.4 24.0 25.0

20.4 11.0 15.0 24.0 15.1 12.8

The average values of f p and ␴a are 0.06 and 19.6⫻ 10−19 cm2, respectively.

power or pulse energy for the Cr4+:forsterite samples. Here, the best-fit values obtained by using the iterative procedure were used. Table 4 lists some of the reported cross-section values in the literature for Cr4+:forsterite. The average ␴a and fp values of 6.13⫻ 10−19 cm2 and 0.45 determined for Cr4+:forsterite (E储b) agree well with the average of the previously measured values and come closest to those reported by Kuleshov et al.16 A similar analysis was carried out with Cr4+:YAG samples. In the calculations, the fluorescence lifetime ␶f and the crystal refractive index n0 were taken to be 4.5 ␮s and 1.82, respectively.21 Samples 1 and 2 shown in Table 1 were Brewster cut, and transmission measurements were performed by aligning the pump electric field along the ⬍1,0,0⬎ direction. Samples 3 and 4 were normal cut along the ⬍0,0,1⬎ direction. Since the complete orientation of each of these samples was not known, two sets of data were taken for each sample by aligning the electric field along two orthogonal directions. The results obtained for these polarizations are indicated as P1 and P2 in Tables 5 and 6. All of the measurements were averaged to determine the saturation parameters. Similar results were obtained as in the case of Cr4+:forsterite. Whereas separate analysis of the cw and pulsed data gave larger deviations in the cross-section values, use of the iterative procedure again led to a reduction in the fractional spread of the best-fit values from 1.41 to 0.26 and from 1.76 to 0.56 for ␴a and fp, respectively. Measured and fit values of transmission as a function of input power or pulse energy are displayed in Figs. 7(a) and 7(b). Finally, we compare

Fig. 7. Measured and fit variation of the transmission as a function of the (a) incident power (cw case) and (b) incident pulse energy (pulsed case) for the Cr4+:YAG crystals.

Table 7. Previously Reported ␴a and fp Values for Cr4+:YAG in the Literaturea ␴a 共⫻10−19 cm2兲 a

23.2 5.8 70 13 16.4a 8.7 57 11.2 30 14 14.1a 3.6 50 — 32 8.7 15 23.29 19.6a a

fp

Reference

⬍0.06 — 0.29 — 0.29 0.25 0.14 0.01 0.07 0.09 0.12 — 0.09 0.4 0.14 0.25 0.07 0.16 0.06

20 13 15 12 18 5 3 21 10 19 24 25 26 27 7 28 29 Average This paper

If more than one sample was used, only the average value is given.

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Vol. 23, No. 2 / February 2006 / J. Opt. Soc. Am. B

age best-fit values for ␣p0 were 1.45 cm−1 and 2.00 cm−1 for samples 3 and 4, respectively, in good agreement with 1.54 and 1.95 cm−1 determined from spectrophotometer measurements. Because the absorbed pump power could lead to lensing and modify the spot-size distribution inside the saturable absorber, we also investigated the effect of thermal lensing on cross-section measurements. The variation of the pump beam parameter q(z) due to thermal lensing effects can be calculated by solving the differential equation 1

d

q共z兲2

+

冉 冊 1

dz q共z兲

+ ␤共z兲2 = 0,

␤共z兲 =



nTh0共z兲 2␬n0



1/2

.

共19兲

Here, nT = d n / d T is the thermal index coefficient, ␬ is the heat conductivity, n0 is the refractive index, and h0共z兲 is the axial value of the pump-induced heat deposition rate per unit volume given by h0共z兲 =

␣p0 1 + 共Ip/Isa兲

␩ hI p .

to investigate the saturation behavior of the samples. Best-fit saturation parameters of the lumped model are determined and compared with those of the distributed models (Sections 2.A and 2.B) to assess the validity of the thin-length approximation. When the transverse and longitudinal variation of the beam inside the sample is neglected, the cw transmission Tcw of the saturable absorber subject to excited-state absorption can be calculated from Tcw = 1 − q0

共18兲

where ␤共z兲 is the quadratic variation in the refractive index along the radial direction and is given by22

共20兲

The heating fraction ␩h appearing in Eq. (20) is close to unity for both Cr4+:forsterite and Cr4+:YAG.21 In the absence of thermal lensing, ␤共z兲 = 0 in Eq. (18), and one recovers the well-known solutions of Gaussian beam propagation in a homogeneous medium. First, we investigated the effect of cw thermal lensing in Cr4+:forsterite (nT = 2.2⫻ 10−6 / K, ␬ = 5 W / m/K, ␩h = 0.91). For each sample, the best-fit values of ␣p0 and fp were kept fixed, and the change in the best-fit value of ␴a was calculated in the presence of thermal lensing. The average fractional change in the best-fit ␴a values came to 6.2%. In the case of Cr4+:YAG (nT = 9.8⫻ 10−6 / K, ␬ = 12 W / m/K, ␩h = 0.93), the corresponding variation in the best-fit value of ␴a was less than 1%, owing to the stronger influence of saturation at high pumping levels [see Eq. (20) above for the axial heat source h]. Because the estimated fractional changes in ␴a were small, the influence of thermal lensing was neglected in our analysis. As discussed below, neglect of thermal lensing was also taken into account in order to determine the overall measurement errors. The error in absorption cross-section values was determined for both Cr4+:forsterite and Cr4+:YAG. In the experiments, errors in transmission and spot-size measurements were estimated to be ±1% and ±6%, respectively. For each sample, the best-fit fp and ␣p0 values were kept fixed, and the resulting error in the best-fit ␴a values was calculated. For Cr4+:forsterite, the overall average error due to measurement uncertainties and the neglect of thermal lensing was estimated to be ±6.4%. In the case of Cr4+:YAG, the corresponding error was ±12%. B. Validity of Lumped Models An alternative analysis of saturable absorbers involves the use of lumped models in which the absorber is treated as a thin medium. In this section, we follow this approach

247

1 + fp共Pp/Psa兲 1 + 共Pp/Psa兲

共21兲

,

where q0 is the small-signal absorption (q0 = ␣p0L0; L0 is the crystal length), Pp is the incident pump power, and Psa is the saturation power (Psa = IsaAeff, Aeff is the effective cross-sectional area of the beam). Equation (21) can be readily derived by using the rate-equation approach of Section 2 along with the “thin-length” and plane-wave approximations. For the pulsed case, the transmission Tpulsed becomes Tpulsed = 1 − q0

共1 − fp兲 Ep/EA

关1 − exp共− Ep/EA兲兴 − q0fp .

共22兲

In Eq. (22), Ep is the incident energy per pulse and EA is the saturation energy 共EA = EsaAeff兲. Note that as the pulse energy Ep in Eq. (22) tends to zero, the small-signal transmission becomes 1-q0, as expected. All of the Table 8. Best-Fit Values of fp, q0, Psa, and ␴ ᠪ aᠪ for the cw Transmission Data of Cr4+:YAG and Cr4+:Forsterite Samples Obtained with the Lumped Models Material

Sample

fp

q0

Psa (W)

␴a 共⫻10−19 cm2兲

Cr4+:YAG

1 2 3, P1 3, P2 4, P1 4, P2 1 2 3

0.15 0.07 0.46 0.64 0.18 0.06 0.85 0.79 0.96

0.89 0.88 0.22 0.22 0.13 0.13 0.96 0.70 0.95

5.8 9.6 1.6 1.1 5.6 6.6 39.1 5.9 2.5

4.7 2.8 68.5 99.7 19.6 16.6 1.1 7.5 17.7

Cr4+:forsterite

Table 9. Best-Fit Values of fp, q0, Ps, and ␴a for the Pulsed Transmission Data of Cr4+:YAG and Cr4+:Forsterite Samples Obtained with the Lumped Models Material

Sample

fp

q0

EA 共␮J兲

␴a 共⫻10−19 cm2兲

Cr4+:YAG

1 2 3, P1 3, P2 4, P1 4, P2 1 2 3

0.22 0.27 0.00 0.07 0.00 0.00 0.84 0.63 0.81

0.65 0.86 0.15 0.13 0.10 0.07 0.96 0.71 0.96

58 53 126 135 145 160 141 60 107

2.5 2.7 4.7 4.4 4.1 3.7 1.2 2.9 1.6

Cr4+:forsterite

248

J. Opt. Soc. Am. B / Vol. 23, No. 2 / February 2006

Cr4+:forsterite and Cr4+:YAG samples were also analyzed by using Eqs. (21) and (22) to determine the best-fit values of the parameters q0, Psa, EA, and fp. The results are shown in Tables 8 and 9 for the cw and pulsed cases, respectively. Note that the ␴a values shown in the last column of Tables 8 and 9 were determined by taking Aeff 2 = ␲␻rms where ␻rms is the minimum rms radius of the beam inside the sample. As can be seen from Tables 8 and 9, there is a large discrepancy in the results of the lumped models. In particular, the lumped models predict a larger amount of excited-state absorption in relation to the distributed models of Sections 2.A and 2.B. For example, in the case of Cr4+:YAG, the average fp comes 0.18, approximately 3 times larger than that determined by using the distributed models (See Tables 6 and 9). In the case of ␴a, both models predict approximately the same average best-fit value (19.6⫻ 10−19 cm2 in the distributed model, as opposed to 19.5⫻ 10−19 cm2 in the lumped model for Cr4+:YAG). However, the fractional spread increased considerably in the case of the lumped model (0.26 for the distributed model in comparison with 1.61 for the lumped model in the case of Cr4+:YAG). Similar results were obtained for Cr4+:forsterite. Hence, we conclude from this comparison that lumped models cannot be used to make accurate predictions of the saturation parameters in our case. This is attributed to the large variation of the pumpbeam spot size inside the samples due to tight focusing.

5. CONCLUSIONS In this paper, we have provided a detailed investigation of the saturation behavior of Cr4+:YAG and Cr4+:forsterite crystals. A rate-equation approach was used to analyze several samples with a distributed model that accounts for the transverse as well as the longitudinal variation of the beam intensity inside the saturable absorber. An iterative procedure was used to analyze the cw and the pulsed transmission data simultaneously. This led to a dramatic reduction in the spread of the best-fit crosssection values for both absorbers. The obtained results are close to the average of the previously reported values for both Cr4+:YAG and Cr4+:forsterite. Comparing the saturation parameters for the two media, we find that the ground-state absorption cross section is approximately three times larger in Cr4+:YAG than in Cr4+:forsterite. Also, the strength of excited-state absorption was found to be significantly lower in the case of Cr4+:YAG in comparison with Cr4+:forsterite. Both of these characteristics make Cr4+:YAG a more suitable passive Q switch for 1 ␮m lasers. Finally, the results of Section 4.B showed that in the analysis of saturable absorbers, lumped models based on the thin-length approximation should be used with caution, especially when the pump beam is tightly focused inside the samples.

Sennaroglu et al.

thor and can be reached by telephone at (90) 212 338 1429, by fax at (90) 212 338-1559, or by e-mail at [email protected].

*Present address, Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Florida 32816-2700. † Present address, Institute of Optics, University of Rochester, Rochester, New York 14627.

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Accurate determination of saturation parameters for ...

Received March 4, 2005; revised July 31, 2005; accepted September 7, 2005 ... excited-state absorption cross section fp = esa/ a were determined to be 6.1310−19 cm2 and 0.45, respectively, .... Note that if the pulse repetition rate of the pulsed laser is .... cal absorber as a function of (a) incident power (cw case) and (b).

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