please solve the rate equation in Matlab and we need to reach the same results as shown in the paper.
Posted: Wed May 04, 2022 10:46 am
please solve the rate equation in Matlab and we need to reach
the same results as shown in the paper.
2. Rate-equation systems The following coupled differential equation system was used to describe the behaviour of the DFDL, containing a mixed solution of lasing and saturable absorber dyes [9]: dn(t) Tell C n(t) - Ip(t) op[N — n(t)] n(t) q(1) (1) dt n T dq(t) (ell - σm)c Qn(t) na,(t) q(t) dt n T + Oeal C n [N₁ (2) dn₁ (t) n(t) - Ip (1) pan₁ (1) + n₁(t)]q(t) + dt Ta 252 + - laa)[Na a)[N₁ n₁ (t)L} − 1) (3) - T₁ L(o eaa nL³ T. (1) max (4) 8сπ² - = q(t) Tc (1) n(t) q(t) [N₁ - n₁ (t)]g(t) cal C [Na 11 (exp {(Của caa - laa [n(t) (σell + Olal C 17 ₁)V², nL 10c Taal C n n. (t)]q(t) N₁ aal n C n₁ (1)q(t)
The meanings of the symbols and the values of the constants used in the calculations are as follows: n(t), n, (t) spatially averaged density of laser and absorber dye molecules in the upper (S₁) and lower (So) level, respectively (cm-³) q(1) N, N₂ density of DFDL photons having wavelength of 2₁ = 555 nm (cm³) density of laser and absorber dye molecules, respectively (9 x 10¹8 cm- (1.5 x 102 moll), 1.08 × 10¹7 cm3 (1.8 x 10-4 moll-')) I, (t) spatially averaged pump photon intensity (cm-² s-¹), Gaussian shaped in time, having 9 ns FWHM T, Ta decay time of laser and absorber dye S, state (4 ns, 4 ns) equivalent cavity decay time (s, see [4, 9]) Te op, pa Tejk, Jajkijk absorption cross-section of laser and absorber dye ground state at the pumping wavelength, respectively (1.15 × 10-¹7 cm², 3.8 × 10-¹7 cm²) emission, ground-state absorption and excited-state absorption cross- sections of the absorber dye (j a) or laser dye (j→ 1) molecules on the A lasing wavelength (k 1) or on the 2₁ = 580 nm wavelength (k→ > a), where the maximum of the amplified spontaneous emission (ASE) of the absorber dye is (ell = 1.1 x 10-¹6 cm², = 0.4 x 10-¹6 cm², o aal 4.1 x 10-16 cm², 3.05 x 10-16 cm², speed of light (3 x cal = 0.9 x 106 cm², al 1.3 x 10-16 cm², caa a = 2.0 x 10 cm²) 10⁰ cms ¹) с n refractive index of the dye solution (1.32) Ω, Ω factor determining the fraction of the spontaneously emitted photons by excited laser and absorber dye molecules that propagates into the angular and spectral ranges of the DFDL and the ASE of the absorber, respectively (5.2 x 10-8 and 2.5 x 10-6) L length of the DFDL (0.3 cm) V visibility of the interference pattern on the dye cell (0.38).
These parameters correspond to the experimental conditions used in [9], where an excimer laser was used to pump a DFDL, consisting of a methanol solution of Coumarin 153 as lasing dye and Rhodamine B as absorber. The last term in Equation 3 is unusual in rate-equation models. This term takes into account the decrease in the excited-state (and, consequently, the increase in the ground- state) density of the absorber molecules, caused by the ASE emitted by the excited absorber molecules. A detailed description of this term, as well as that of the whole system of Equations 1 to 4 can be found in [9]. For an SCDL the cavity decay time is constant, because it depends only on constant values, such as the reflectivity of the mirrors forming the cavity and the distance between the mirrors. Because of this, the behaviour of a passively Q-switched SCDL (using a laser-saturable absorber dye mixture) can be described by using only Equations I to 3 with constant , if the mirrors of the SCDL have no reflection at the wavelength of the ASE of the excited absorber molecules. However, the mirrors of the SCDL usually have reflection at both the laser and the absorber ASE wavelengths. In this case lasing can also occur with the absorber dye at the wavelength of the absorber ASE. To describe this case, the last term in Equation 3 (which describes the absorber ASE) must be replaced by a term that takes into account the effect of the stimulated emission caused by the absorber laser photons. (Actually, we should use both terms; however, in the presence of lasing the ASE is insignificant, so we can omit the corresponding term.) The modified equation is then acal C N₁ - n₁(1) dn, (1) dz = - [₂ (1)σpa ¹₁ (1) + [N₁ n. (1)]q(1) + n Taal C 11 n. (1)q(1) + [N₁ - n₁ (1)] q₁ (1) (3') n where q. (1) is the density of the absorber laser photons. To describe the change of q.(t), an additional equation has to be used: dq, (t) (0 can laa)C 9. (1) [N₁ - n₁(1)] - [N₁ - n₁ (1)]4₂ (1) + (5) dr n Tca where is the cavity decay time for the absorber laser photons. mirrors of the SCDL cavity depends on the wavelength, Tea If the reflectivity of the Te, otherwise Tea = Te For all three of the above equation systems the power of the laser light emitted by the DFDL or SCDL was calculated as Pout 1 hc q(t)abL 22₁ (1) (6) where (1 (Na₂ - N₁0p)-¹ (7) is the penetration depth of the pump beam into the dye solution, h is Planck's constant and b = 0.02 cm is the height of the excited volume. داد 0
3. Results for DFDL Fig. la shows the calculated time dependence of I,. n, and t, obtained by numerical integration of Equations 1 to 4, using the above parameters. The time of appearance of the DFDL pulses and their peak power are indicated by the arrows. The DFDL pulses are about 600 times shorter than the pump pulse, so it is not possible to show the time dependence of I, and P on the same timescale. Fig. Ib shows the time dependence of P n, and t, around the first DFDL peak. According to Fig. 1, initially for a long time there is a slow increase in t. This is caused by the pumping, which is the significant process in this 378 Pulse shortening and stabilization by Q-thing time range, where the population inversion is below the instantaneous threshold value. Over the same time range there is a slow decrease in ,, which is also caused directly by the pumping. A few hundred picoseconds before the appearance of the DFDL pulse the population () exceeds the threshold value and the decrease of a, becomes increasingly faster, and the increase of t, becomes less, then t, is also decreasing. During the DFDL pulse (mainly on its rising side) a very fast decrease is achieved in w, as well as in T,. This decrease, caused by the DFDL photons, leads to a DFDL pulse that is shorter than it would be without these changes. The decrease of 1,. caused by the DFDL photons, is the self-Q- switching [4], which takes place in every dynamic DFDL. The decrease in ,, caused by the DFDL photons, is the passive Q-switching (10). Thus, we can say that in a DFDL consisting of a mixture of a laser and an absorber dye, a double Q-switching takes place (9). Still, we should emphasize that the two Q-switching processes cause an opposite change in the Q-factor of the (equivalent) cavity and, consequently, in the q-photon density. One can easily recognize this by comparing the second and fourth terms on the right-hand side of Equation 2. These two terms describe the self-Q-switching and (neglecting the usually less significant fifth and sixth terms of Equation 2) the passive Q-switching. This opposite change of the Q-factor, caused by the decrease of w, and T,, means that the laser photon density tends to change more slowly in a passive Q-switched DFDL, than in a similar passive Q-switched SCDL, where t, is constant. Fortunately, in the DFDL the decrease of , appears slightly later than the decrease in a, (see Fig. 1b), so the two processes do not quench each other's effect. According to the following results of the calculations, the passive Q-switched DFDL can produce even shorter pulses than the passive Q-switched SCDL, if the value of Vis near unity. The reason why this is so is that P, depends on r, (see Equation 6). According to the calculations, the passive Q-switched DFDL emits a pulse series if the pump power exceeds the threshold considerably. However, just above the threshold there is a pump power range (the so-called dynamic range), where the DFDL emits a single pulse. The pulse duration decreases when the pump power is increased. These properties of the passive Q-switched DF are qualitatively similar to the properties of the common DFDL. However, there is some significant quantitative improvement, as was pointed out in [9). For a given parameter set, the rate-equation system was solved using different pump power maximum values. The A/ duration (FWHM) and the E, pulse energy (the product of Ar and the maximum of P) could be concluded from the solution. By plotting these values as a function of the pump power (normalized to the threshold value), we obtain the pulse duration and pulse energy characteristics of the DFDL. Fig. 2 shows these charac teristics by using the parameters listed above. Starting from these characteristics we can determine that Ar 15ps and E= 1.27 µl will be the predicted pulse duration and pulse energy of the possible shortest single pulse for the given parameter set. We also obtain the result that the dynamic range is D-9.7%. A further useful datum obtainable from the energy characteristics is the derivative of these characteristics. This quantity, denoted S shows how much greater the relative change in E, is than the relative change in the pump power. For a nitrogen-laser-pumped DFDL [11] an excellent agreement was found between the measured fluctuation of E, and the product of S and the fluctuation of the pump power. Before investigating the effect of the different parameters on the passive Q-switched DFDL, we should mention that in Equation 4 the contribution of the spatial absorption modulation caused by the spatial hole burning to the feedback is neglected. In order to
1.5 20 1.0 15 10 0.5 2nd 3rd 4th 5th 6th 5 TITT 1,0 1.1 1.2 1.3 Pump power (arbitrary units) Figure 2 Typical pulse duration and pulse energy characteristics of the first pulse of a passive Q-switched DFDL. The appearance of the other pulses is indicated by arrows. estimate the error caused by this, a series of calculations using T= max nL 8cm² z {n(1) (Gau − ¢m) + [N₂ − n₂(0)]9cm}²V², 1L) (4') instead of Equation 4 were carried out. Equation 4 gives a lower value and Equation 4' a higher value for t, than the correct value. According to these calculations, using Equation 4' instead of Equation 4 results in a 30% shorter pulse duration and a 50% larger pulse energy. So the results of the calculations somewhat overestimate the pulse duration and underestimate the pulse energy. In the first series of calculations the visibility of the interference pattern was changed. Fig. 3 shows the results of the calculations. All three parameters, first of all at low visibility 15 30 0.5 0 0.2 0.4 0.8 1.0 0.6 Visibility Figure 3 The (-) pulse duration, (-) pulse energy and (---) dynamic range of DFDL as a function. of the visibility of the interference pattern. (mm) 13 10 10 (۳۶) 13 (%) 0 10 At (ps) 20 At (ps)
1.0 10 EL (KJ) D (%) 0.5 5 60 9 At (ps) 20 0 0.6 1.2 1.8 ca (x10-4 mol 1-1¹) Figure 4 The (--) pulse duration, (-) pulse energy and (---) dynamic range of the DFDL as a function of the absorber dye concentration value, significantly depend on V. The dependence of Ar on Vis more pronounced for the passive Q-switched DFDL than for a common DFDL. Although a passive Q-switched DFDL produces a three-times shorter pulse duration than the common DFDL, even for V = 0.2, this factor becomes higher than 10 for V = 1. The other calculations on DFDL were carried out using a value of V = 0.38, which is typical for an excimer-laser pumped DFDL [9]. Fig. 4 shows the dependence of the DFDL pulse energy, pulse duration and dynamic range on the concentration of the saturable absorber dye (Rhodamine B). All three quantities change favourably (that is, E, and D increase, whereas Ar decreases) over the whole concen- tration range investigated if the degree of concentration is raised. However, the changes become significant only for a concentration larger than 6 x 10 moll-¹. For the wavelength of the DFDL the small-signal transmission through a 6 x 10³ moll-¹ Rhodamine B solution having a thickness equal to the length of the DFDL (L 0.3 cm) is 1.2%. For the largest concentration value (1.8 x 10 moll) used in the calculations, this small-signal transmission is equal to 1.7 x 10. According to Fig. 4 the advantageous changes should continue for even higher values of absorber concentration. However, our model is not correct in this high- concentration range, because the predicted pulse duration is shorter than the transit time of the light through the DFDL (in this case it is necessary to use a space-dependent model [12]), and the Förster energy transfer between the excited laser dye molecules and the absorber mol- model). ecules turns out to be significant (and this type of energy transfer is not included in our As was pointed out in the early theoretical work on passive Q-switching [10], for efficient Q-switching a larger absorber cross-section is needed than the emission cross-section of the lasing material. This is demonstrated in Fig. 5, where the pulse duration and pulse energy (a) and the dynamic range and the stability parameter (b) are shown plotted against the al(ell) cross-section ratio. ( - is the effective emission cross-section of the laser dye.) These curves were obtained from calculations in which and N, were the changing parameters, and their product was kept at = 44.3cm-1 In this way the
(1) 2.0 1.0 (%)0 C (4) 10 5 0 3 1 9 11 60 9 20 0 AC (ps) 10 55 3 5 7 9 11 (b) 6aal /(all-6111) Figure 5 (a) The (---) pulse duration and (-) pulse energy and (b) the (-) dynamic range and (---) stability parameter for a passive Q-switched DFDL plotted against the ratio of the effective cross-sections. loss before the Q-switching was the same for every da/(m) value. In Fig. 5 the corresponding at 1, E₁, D and S for N, = 0 (no passive Q-switching) are also indicated by horizontal lines. By using a saturable absorber, for which the cross-section ratio is equal to unity, a slight improvement is achieved in the dynamic range and in the stability of the DFDL. Neverthe- less, it results in a larger pulse duration and a smaller pulse energy. A significant improve- ment can be achieved in all four parameters of the DFDL pulse investigated, if the cross-section ratio is > 3. Over the 3 < (el) < 8 range the pulse parameters are very sensitive to the cross-section ratio. If this ratio is equal to 8, the passive Q-switching results an eight-fold shortening of the pulse duration and a ten-fold decrease in the pulse energy fluctuation. This latter indicates that the calculations predict a smaller fluctuation of the passive Q-switched DFDL, than the fluctuation of the pump power is. There are further improvements in the DFDL pulse parameters if the cross-section ratio is being increased; however, we should note that, here again, the predicted pulse duration turns out to be shorter than the transit time, therefore our model is not adequate for that range. We should also note that it is difficult to find a laser-abosrber dye pair for which the cross-section ratio is > 10.
the same results as shown in the paper.
- Please Solve The Rate Equation In Matlab And We Need To Reach The Same Results As Shown In The Paper 1 (299.92 KiB) Viewed 37 times
The meanings of the symbols and the values of the constants used in the calculations are as follows: n(t), n, (t) spatially averaged density of laser and absorber dye molecules in the upper (S₁) and lower (So) level, respectively (cm-³) q(1) N, N₂ density of DFDL photons having wavelength of 2₁ = 555 nm (cm³) density of laser and absorber dye molecules, respectively (9 x 10¹8 cm- (1.5 x 102 moll), 1.08 × 10¹7 cm3 (1.8 x 10-4 moll-')) I, (t) spatially averaged pump photon intensity (cm-² s-¹), Gaussian shaped in time, having 9 ns FWHM T, Ta decay time of laser and absorber dye S, state (4 ns, 4 ns) equivalent cavity decay time (s, see [4, 9]) Te op, pa Tejk, Jajkijk absorption cross-section of laser and absorber dye ground state at the pumping wavelength, respectively (1.15 × 10-¹7 cm², 3.8 × 10-¹7 cm²) emission, ground-state absorption and excited-state absorption cross- sections of the absorber dye (j a) or laser dye (j→ 1) molecules on the A lasing wavelength (k 1) or on the 2₁ = 580 nm wavelength (k→ > a), where the maximum of the amplified spontaneous emission (ASE) of the absorber dye is (ell = 1.1 x 10-¹6 cm², = 0.4 x 10-¹6 cm², o aal 4.1 x 10-16 cm², 3.05 x 10-16 cm², speed of light (3 x cal = 0.9 x 106 cm², al 1.3 x 10-16 cm², caa a = 2.0 x 10 cm²) 10⁰ cms ¹) с n refractive index of the dye solution (1.32) Ω, Ω factor determining the fraction of the spontaneously emitted photons by excited laser and absorber dye molecules that propagates into the angular and spectral ranges of the DFDL and the ASE of the absorber, respectively (5.2 x 10-8 and 2.5 x 10-6) L length of the DFDL (0.3 cm) V visibility of the interference pattern on the dye cell (0.38).
These parameters correspond to the experimental conditions used in [9], where an excimer laser was used to pump a DFDL, consisting of a methanol solution of Coumarin 153 as lasing dye and Rhodamine B as absorber. The last term in Equation 3 is unusual in rate-equation models. This term takes into account the decrease in the excited-state (and, consequently, the increase in the ground- state) density of the absorber molecules, caused by the ASE emitted by the excited absorber molecules. A detailed description of this term, as well as that of the whole system of Equations 1 to 4 can be found in [9]. For an SCDL the cavity decay time is constant, because it depends only on constant values, such as the reflectivity of the mirrors forming the cavity and the distance between the mirrors. Because of this, the behaviour of a passively Q-switched SCDL (using a laser-saturable absorber dye mixture) can be described by using only Equations I to 3 with constant , if the mirrors of the SCDL have no reflection at the wavelength of the ASE of the excited absorber molecules. However, the mirrors of the SCDL usually have reflection at both the laser and the absorber ASE wavelengths. In this case lasing can also occur with the absorber dye at the wavelength of the absorber ASE. To describe this case, the last term in Equation 3 (which describes the absorber ASE) must be replaced by a term that takes into account the effect of the stimulated emission caused by the absorber laser photons. (Actually, we should use both terms; however, in the presence of lasing the ASE is insignificant, so we can omit the corresponding term.) The modified equation is then acal C N₁ - n₁(1) dn, (1) dz = - [₂ (1)σpa ¹₁ (1) + [N₁ n. (1)]q(1) + n Taal C 11 n. (1)q(1) + [N₁ - n₁ (1)] q₁ (1) (3') n where q. (1) is the density of the absorber laser photons. To describe the change of q.(t), an additional equation has to be used: dq, (t) (0 can laa)C 9. (1) [N₁ - n₁(1)] - [N₁ - n₁ (1)]4₂ (1) + (5) dr n Tca where is the cavity decay time for the absorber laser photons. mirrors of the SCDL cavity depends on the wavelength, Tea If the reflectivity of the Te, otherwise Tea = Te For all three of the above equation systems the power of the laser light emitted by the DFDL or SCDL was calculated as Pout 1 hc q(t)abL 22₁ (1) (6) where (1 (Na₂ - N₁0p)-¹ (7) is the penetration depth of the pump beam into the dye solution, h is Planck's constant and b = 0.02 cm is the height of the excited volume. داد 0
3. Results for DFDL Fig. la shows the calculated time dependence of I,. n, and t, obtained by numerical integration of Equations 1 to 4, using the above parameters. The time of appearance of the DFDL pulses and their peak power are indicated by the arrows. The DFDL pulses are about 600 times shorter than the pump pulse, so it is not possible to show the time dependence of I, and P on the same timescale. Fig. Ib shows the time dependence of P n, and t, around the first DFDL peak. According to Fig. 1, initially for a long time there is a slow increase in t. This is caused by the pumping, which is the significant process in this 378 Pulse shortening and stabilization by Q-thing time range, where the population inversion is below the instantaneous threshold value. Over the same time range there is a slow decrease in ,, which is also caused directly by the pumping. A few hundred picoseconds before the appearance of the DFDL pulse the population () exceeds the threshold value and the decrease of a, becomes increasingly faster, and the increase of t, becomes less, then t, is also decreasing. During the DFDL pulse (mainly on its rising side) a very fast decrease is achieved in w, as well as in T,. This decrease, caused by the DFDL photons, leads to a DFDL pulse that is shorter than it would be without these changes. The decrease of 1,. caused by the DFDL photons, is the self-Q- switching [4], which takes place in every dynamic DFDL. The decrease in ,, caused by the DFDL photons, is the passive Q-switching (10). Thus, we can say that in a DFDL consisting of a mixture of a laser and an absorber dye, a double Q-switching takes place (9). Still, we should emphasize that the two Q-switching processes cause an opposite change in the Q-factor of the (equivalent) cavity and, consequently, in the q-photon density. One can easily recognize this by comparing the second and fourth terms on the right-hand side of Equation 2. These two terms describe the self-Q-switching and (neglecting the usually less significant fifth and sixth terms of Equation 2) the passive Q-switching. This opposite change of the Q-factor, caused by the decrease of w, and T,, means that the laser photon density tends to change more slowly in a passive Q-switched DFDL, than in a similar passive Q-switched SCDL, where t, is constant. Fortunately, in the DFDL the decrease of , appears slightly later than the decrease in a, (see Fig. 1b), so the two processes do not quench each other's effect. According to the following results of the calculations, the passive Q-switched DFDL can produce even shorter pulses than the passive Q-switched SCDL, if the value of Vis near unity. The reason why this is so is that P, depends on r, (see Equation 6). According to the calculations, the passive Q-switched DFDL emits a pulse series if the pump power exceeds the threshold considerably. However, just above the threshold there is a pump power range (the so-called dynamic range), where the DFDL emits a single pulse. The pulse duration decreases when the pump power is increased. These properties of the passive Q-switched DF are qualitatively similar to the properties of the common DFDL. However, there is some significant quantitative improvement, as was pointed out in [9). For a given parameter set, the rate-equation system was solved using different pump power maximum values. The A/ duration (FWHM) and the E, pulse energy (the product of Ar and the maximum of P) could be concluded from the solution. By plotting these values as a function of the pump power (normalized to the threshold value), we obtain the pulse duration and pulse energy characteristics of the DFDL. Fig. 2 shows these charac teristics by using the parameters listed above. Starting from these characteristics we can determine that Ar 15ps and E= 1.27 µl will be the predicted pulse duration and pulse energy of the possible shortest single pulse for the given parameter set. We also obtain the result that the dynamic range is D-9.7%. A further useful datum obtainable from the energy characteristics is the derivative of these characteristics. This quantity, denoted S shows how much greater the relative change in E, is than the relative change in the pump power. For a nitrogen-laser-pumped DFDL [11] an excellent agreement was found between the measured fluctuation of E, and the product of S and the fluctuation of the pump power. Before investigating the effect of the different parameters on the passive Q-switched DFDL, we should mention that in Equation 4 the contribution of the spatial absorption modulation caused by the spatial hole burning to the feedback is neglected. In order to
1.5 20 1.0 15 10 0.5 2nd 3rd 4th 5th 6th 5 TITT 1,0 1.1 1.2 1.3 Pump power (arbitrary units) Figure 2 Typical pulse duration and pulse energy characteristics of the first pulse of a passive Q-switched DFDL. The appearance of the other pulses is indicated by arrows. estimate the error caused by this, a series of calculations using T= max nL 8cm² z {n(1) (Gau − ¢m) + [N₂ − n₂(0)]9cm}²V², 1L) (4') instead of Equation 4 were carried out. Equation 4 gives a lower value and Equation 4' a higher value for t, than the correct value. According to these calculations, using Equation 4' instead of Equation 4 results in a 30% shorter pulse duration and a 50% larger pulse energy. So the results of the calculations somewhat overestimate the pulse duration and underestimate the pulse energy. In the first series of calculations the visibility of the interference pattern was changed. Fig. 3 shows the results of the calculations. All three parameters, first of all at low visibility 15 30 0.5 0 0.2 0.4 0.8 1.0 0.6 Visibility Figure 3 The (-) pulse duration, (-) pulse energy and (---) dynamic range of DFDL as a function. of the visibility of the interference pattern. (mm) 13 10 10 (۳۶) 13 (%) 0 10 At (ps) 20 At (ps)
1.0 10 EL (KJ) D (%) 0.5 5 60 9 At (ps) 20 0 0.6 1.2 1.8 ca (x10-4 mol 1-1¹) Figure 4 The (--) pulse duration, (-) pulse energy and (---) dynamic range of the DFDL as a function of the absorber dye concentration value, significantly depend on V. The dependence of Ar on Vis more pronounced for the passive Q-switched DFDL than for a common DFDL. Although a passive Q-switched DFDL produces a three-times shorter pulse duration than the common DFDL, even for V = 0.2, this factor becomes higher than 10 for V = 1. The other calculations on DFDL were carried out using a value of V = 0.38, which is typical for an excimer-laser pumped DFDL [9]. Fig. 4 shows the dependence of the DFDL pulse energy, pulse duration and dynamic range on the concentration of the saturable absorber dye (Rhodamine B). All three quantities change favourably (that is, E, and D increase, whereas Ar decreases) over the whole concen- tration range investigated if the degree of concentration is raised. However, the changes become significant only for a concentration larger than 6 x 10 moll-¹. For the wavelength of the DFDL the small-signal transmission through a 6 x 10³ moll-¹ Rhodamine B solution having a thickness equal to the length of the DFDL (L 0.3 cm) is 1.2%. For the largest concentration value (1.8 x 10 moll) used in the calculations, this small-signal transmission is equal to 1.7 x 10. According to Fig. 4 the advantageous changes should continue for even higher values of absorber concentration. However, our model is not correct in this high- concentration range, because the predicted pulse duration is shorter than the transit time of the light through the DFDL (in this case it is necessary to use a space-dependent model [12]), and the Förster energy transfer between the excited laser dye molecules and the absorber mol- model). ecules turns out to be significant (and this type of energy transfer is not included in our As was pointed out in the early theoretical work on passive Q-switching [10], for efficient Q-switching a larger absorber cross-section is needed than the emission cross-section of the lasing material. This is demonstrated in Fig. 5, where the pulse duration and pulse energy (a) and the dynamic range and the stability parameter (b) are shown plotted against the al(ell) cross-section ratio. ( - is the effective emission cross-section of the laser dye.) These curves were obtained from calculations in which and N, were the changing parameters, and their product was kept at = 44.3cm-1 In this way the
(1) 2.0 1.0 (%)0 C (4) 10 5 0 3 1 9 11 60 9 20 0 AC (ps) 10 55 3 5 7 9 11 (b) 6aal /(all-6111) Figure 5 (a) The (---) pulse duration and (-) pulse energy and (b) the (-) dynamic range and (---) stability parameter for a passive Q-switched DFDL plotted against the ratio of the effective cross-sections. loss before the Q-switching was the same for every da/(m) value. In Fig. 5 the corresponding at 1, E₁, D and S for N, = 0 (no passive Q-switching) are also indicated by horizontal lines. By using a saturable absorber, for which the cross-section ratio is equal to unity, a slight improvement is achieved in the dynamic range and in the stability of the DFDL. Neverthe- less, it results in a larger pulse duration and a smaller pulse energy. A significant improve- ment can be achieved in all four parameters of the DFDL pulse investigated, if the cross-section ratio is > 3. Over the 3 < (el) < 8 range the pulse parameters are very sensitive to the cross-section ratio. If this ratio is equal to 8, the passive Q-switching results an eight-fold shortening of the pulse duration and a ten-fold decrease in the pulse energy fluctuation. This latter indicates that the calculations predict a smaller fluctuation of the passive Q-switched DFDL, than the fluctuation of the pump power is. There are further improvements in the DFDL pulse parameters if the cross-section ratio is being increased; however, we should note that, here again, the predicted pulse duration turns out to be shorter than the transit time, therefore our model is not adequate for that range. We should also note that it is difficult to find a laser-abosrber dye pair for which the cross-section ratio is > 10.