Beam intensity reshaping by pump modification in a laser amplifier

Igor A. Litvin; Oliver J. P. Collet; Gary King; Hencharl Strauss

doi:10.1364/OE.23.030165

1. Introduction

Lasers beams with a specific intensity profile such as super–Gaussian, Airy or Doughnut-like are desirable in many applications such as laser materials processing, medicine and so on. Methods of producing such beams can be divided into two classes, namely extra and intra-cavity beam shaping. Extra-cavity (external) beam shaping can be achieved by manipulating the output beam from a laser with suitably chosen amplitude and/or phase elements, and has been extensively reviewed [1]. Unfortunately extra-cavity beam shaping results in unavoidable losses, while reshaping the beam by phase only elements suffers from sensitivity to environmental perturbations. Alternatively, intra–cavity beam shaping can achieve high efficiencies in a cavity by maximizing the pump and output mode overlap while producing the required beam [2,3]. However, the method of obtaining such laser modes is quite complicated due to complexity of the intra-cavity optics involve in the process [3].

High power end-pumped amplifiers can produce very high output powers with good efficiency and good beam quality [4]. Analysis of amplifier stages has shown that pump beam size and power can improve the beam quality of the seed through gain guiding effects [5].

In this paper we propose and investigate a theory of reshaping laser beams into a desirable beam profiles with an end-pumped laser amplifier that is pumped with a beam which has a modified intensity profile. Advantages of using this method are that beam shaping and amplification is combined, the elimination of costly diffractive or refractive optics and the ability to reshape multimode beams. Additionally this method can be extended to the cases were the seed beam is shaped to produce the desired output. Furthermore, a manipulation of the power of the pump beam results in a variation of the final intensity profile of the shaped beam. Such a unique feature can be useful in the processes where adaptive control of resulting beam is required. A disadvantage of the method is that the required profile of the pump beam is different from that produced by high brightness pump sources like laser diodes or standard Gaussian pump lasers. However one can design such lasers to produce outputs that close to a specific required pump profile, and still get decent beam shaping. Alternatively one can use cheaper (few level) extra-cavity or intra–cavity pump beam shaping to roughly obtain the required pump profile. The reshaping efficiency of conventional methods is comparable (or even slightly higher) than the proposed method and the final efficiency will depend on the particular reshaping scheme. However, the unique feature of adaptive beam shaping of non-coherent beams with simultaneous amplification is unique to this method.

We first derived an analytical equation, which describes the intensity transformation of a seed beam as a function of the pump intensity distribution in an unsaturated four level amplifier. We then developed a numerical technique to calculate the required pump intensity profile to reshape the seed beam into the desired profile in a general amplifier (pump power saturated/unsaturated, four level/three level). To illustrate both the analytical and numerical methods, we used the classical example of reshaping a Gaussian beam into a Flat –Top beam [1–3].

Lastly we experimentally verified the theory by pumping a Ho:YLF amplifier with a modified Tm:YLF pump laser which lased with a HG₀₁(Hermite–Gaussian) intensity profile. A decent Gaussian to Flat–Top transformation was achieved using this profile, even though it only roughly approximated the calculated required pump profile.

2. Analytical method

Before finding the pump intensity profile, which allows shaping of the seed beam into a desired intensity profile, we obtain an analytical equation, which describes the intensity transformation of a seed beam in a four level unsaturated amplifier as the function of a spatial intensity distribution of an arbitrary circular symmetric pump.

We consider a four level amplifier system end-pumped by a beam with circular symmetry (see Figs. 1 and 2(a)). Similarly the amplifier is seeded by a circular symmetric seed beam. The absorption of the pump and amplification of the seed at any point along the four level gain medium are governed by the following coupled equations [5,6],

\frac{\partial I_{p} (r,z)}{\partial z} = - σ_{a} n_{1} (r,z) I_{p} (r,z),

\frac{\partial I_{s} (r,z)}{\partial z} = σ_{e} n_{2} (r,z) I_{S} (r,z),

\frac{\partial n_{2} (r,z)}{\partial t} = σ_{a} n_{1} (r,z) \frac{I_{p} (r,z)}{h ν_{p}} - σ_{e} n_{2} (r,z) \frac{I_{S} (r,z)}{h ν_{p}} - \frac{n_{2} (r,z)}{τ_{2}},

were I_p (I_S) is the intensity of the pump (seed) beam, hν_p (hν_s) is the energy per photon at the pump (seed) wavelength, σ_a is the absorption cross section of the active ions in the ground energy level n₁, σ_e is the emission cross section of the upper laser energy level n₂, and τ₂ is the lifetime of level n₂ (see Fig. 1 as an example of such a four level system).

Fig. 1 Energy levels diagram of Nd:YVO₄ (four level system) [5].

Download Full Size | PDF

Fig. 2 (a) A schematic of beam shaping by a laser amplifier where 1 is the seed beam profile, 2 is the output beam profile and 3 is the pump beam profile. (b) The simulated upper laser level population density (n₂) along the z-axis of an 0.2% Nd:YVO₄ amplifier for different pump powers and pump and seed beam radii of 1.5 mm. For 20 W incident pump, the zero order solution (red dashed) as well as the solution of Eq. (2) without assumption on the population density (solid lines) are shown. For 40 W incident pump, both the zero (black dashed) and the first order (green dashed) solutions are shown.

Download Full Size | PDF

For the case where the pump and seed beams have no temporal change in intensity, the population of the upper laser level is assumed to be static and Eq. (1c), that governs the time evolution of the population density, is set to zero. In this way the system is simplified to a two boundary value problem that is coupled together by a balance equation. This may be further simplified by only considering the small signal gain operation regime of the amplifier, were σ_aI_p>>σ_eI_s, so that the balance equation, Eq. (1c) simplifies to,

σ_{a} \frac{I_{p} (r,z)}{h ν_{p}} (n_{tot} - n_{2} (r,z)) = \frac{n_{2} (r,z)}{τ_{2}},

where conservation of ions (n₁ = n_tot - n₂) has been assumed.

In order to obtain the population density of the ions in the upper level we assume that the gain medium is not saturated by the pump, namely that n_tot>> n₂ and $z \partial n_{2} / \partial z$ is small compared to n_tot - n₂ [7]. This assumption yields a solution for the zero order approximation of the upper level population (n⁰₂):

n_{2}^{0} (z) = \frac{I_{p} (z_{0}) n_{t o t} σ_{a} τ_{2}}{h ν_{p} e^{- σ_{a} n_{t o t} z} + σ_{a} τ_{2} I_{p} (z_{0})} .

Based on the zero order equation for

n_{2}

(see Eqs. (1b) and (3)) we can find the analytical connection between the intensity profiles of the seed and pump beams by integrating Eq. (1b),

I_{s} (r,L) = I_{s} ({r,z}_{0}) + I_{s} ({r,z}_{0}) \frac{σ_{e}}{σ_{a}} (L n_{t o t} σ_{a} + \log [\frac{h ν_{p} + τ_{2} σ_{a} I_{p} (r, z_{0})}{h ν_{p} e^{L n_{t o t} σ_{a}} + τ_{2} σ_{a} I_{p} (r, z_{0})}]),

where and I_s(r,z₀) and I_s(r,L) are the intensity profiles of the seed beam at the incident and output amplifier crystal planes respectively and I_p(r,z₀) is the intensity profile of the pump beam.

Equation (2) is a transcendental equation obtained without assuming n_tot>>n₂. In order to improve the obtained solution for n₂ (see Eq. (3)) we can implement an iterative approach. This involved substitution of the obtained population density into the left hand side of the balance equation Eq. (2) to find the next order of the solution.

As an example we consider an 0.2% (atm.) doped Nd:YVO₄ amplifier system (Fig. 1) end-pumped by a 808 nm pump beam and seeded by a 1064 nm seed beam, both which is circular symmetric (shown in Fig. 2(a)). The parameters for this crystal are given in Table 1. Typical pump maximum intensity values ( $I_{p}^{\max} = δ h ν_{p} / (σ_{a} τ_{2} (1 - δ))$ where δ = n₂^max/n_tot [7]) where Eq. (3) is valid (unsaturated regime) for this material is around 8-10 W/mm² (while for an equally doped Nd:YAG crystal it is around 25-28 W/mm²). Figure 2(b) shows the simulated upper laser level population density (n₂) along the z-axis of a 0.2% Nd:YVO₄ amplifier for different pump powers and pump and seed beam radii of 1.5 mm. The first order solution shows a slightly better fit than the zero order solution when compared to the solution of Eq. (2) without assumption on the population density. It can also be seen that with increasing pump powers, the accuracy of the analytical solutions decreases. This is because the assumption that $n_{2}$ is much smaller that n_tot becomes less valid for higher pump powers. However, we found that by comparing similar z dependences of n₂, that Eq. (3) is still valid for 0.2% (atm.) doped Nd:YVO₄ if the density of ions in the upper level is 14-16% of total ion density (see Figs. 2(b) and 3). For this example ~8% of the Neodymium ions are excited on the face of the crystal for 40 W of incident pump power.

Table 1. Some of the optical properties of Nd:YVO₄, Ho:YLF and Yb:YAG [8–10].

View Table

Fig. 3 The analytical and numerically simulated outputs of a 0.2% Nd:YVO₄ amplifier pumped with a super-Gaussian pump profile and seeded with a Gaussian seed profile.

Download Full Size | PDF

The analytical solution of Eq. (4) can be verified by comparing it with a numerical simulation. We use a method adapted from a standard algorithm [11] for the solution of Eqs. (1a)–(1c) at a transverse co-ordinate. The numerical simulation uses the standard laser rate equations to calculate the seed intensity at discreet transverse positions after the amplifier crystal. As an example we consider the amplification of a Gaussian seed beam by Flat–Top intensity profile.

Comparing the analytical results to the numerical results shown in Fig. 3 it is found that the two methods fit with good precision for maximum pump intensities as high as 9 W/mm², which corresponds to a 14% excitation of ions to the upper laser level.

3. Beam shaping

We can extract an analytical equation for the pump beams intensity profile that transforms a given seed beam into a desired output from the analytical relationship between the seed, pump and shaped-output beams from Eq. (4).

I_{p} (r, z_{0}) = \frac{- e^{L n_{t o t} σ_{a} + \frac{σ_{a}}{σ_{e}}} + e^{L n_{t o t} σ_{a} + \frac{σ_{a} I_{s} (r,L)}{σ_{e} I_{s} ({r,z}_{0})}}}{e^{L n_{t o t} σ_{a} + \frac{σ_{a}}{σ_{e}}} - e^{\frac{σ_{a} I_{s} (r,L)}{σ_{e} I_{s} ({r,z}_{0})}}} \frac{h ν_{p}}{τ_{2} σ_{a}} .

This solution may produce negative intensity values, which indicates that attenuation of the input beam is required at certain positions. This places restrictions on what beams can be converted using four-level gain media. In addition, there is a limit to the maximum amplification due to invalidity of the solution of Eq. (5) in the saturated regime discussed above (the higher pump power) [7]. With a ratio of upper laser level to ground densities of 14-16%, the maximum ratio between I_s(r,L)) and I_s(r,z₀) is around 1.6-1.8 [7].

Figure 4 shows an example of the how the required pump profile was calculated in order to convert a Gaussian seed beam (red dashed) to a super-Gaussian beam of order 6 (black dashed) with the same width and peak intensity. The pump profile was first calculated using Eq. (5) and is shown by the green dashed line. Subsequently the numerical simulation described earlier was used to calculate exactly the resultant seed intensity (red solid). There is relatively good agreement between the generated and the required seed beams at lower radial coordinates. However at larger radial coordinates the Gaussian seed intensity needs to be suppressed. It is therefore possible to reshape the beam in regions where positive gain is required, but the method is limited as no reduction of the existing beam is possible.

Fig. 4 Conversion of a Gaussian seed beam (red dashed) to a super-Gaussian beam of order 6 (black dashed). The required pump beam (green dashed) was calculated using Eq. (5). The actual shaped-output beam (solid red trace) was calculated with a standard algorithm [11] for the solution of Eqs. (1a)–(1c).

Download Full Size | PDF

In order to mitigate this limitation one can increase the amplification of seed beam and so obtain better contrast between the desired output beam and the intensity of the outer regions. However, this can lead to the invalidation of Eq. (5) due to the higher pump power.

For such high amplifications the implementation of Eq. (5) is invalid and we propose to use an iterative numerical method to find the required pump beam profile. The method uses the same numerical simulation described earlier in Fig. 3, which numerically solves Eqs. (1a)–(1c). The pump power is iteratively varied at discreet transverse (in this case radial) positions until the output intensity (calculated using the numerical simulation) match the target output profile. This method placed no limitations on intensity of seed and pump beams. Figure 5 presents the results of such a numerical calculation of the reshaping of a Gaussian beam into a super-Gaussian beam of order 6 (Flat-Top) as well as the required pump intensity profile for the high pump power (the saturated regime [7]). The contrast between the inner and outer parts (in the region where attenuation is required the pump beams is set to zero and the seed beam is transmitted) of the beam is significantly more due to the higher amplification and illustrates that the reshaping of the seed beam into desirable intensity profile is possible. The main limitation of the method is therefore suppressing unwanted radial components. Figure 5(b) shows the calculated efficiency of the amplifier (percentage of pump photons being transformed into seed photons) as a function of seed power. The efficiency of the amplifier increases with the seed power because the probability increases that a photon will encounter an excited ion, scale with the number of photons and tends towards the quantum efficiency. The higher power amplifier of Fig. 5(a) had a calculated efficiency of ~60% at seed powers of 250 W (indicated with the red dot).

Fig. 5 (a) Improvement of the output beam contrast by increased amplification of the seed (calculated using the numerical technique). (b) The efficiency of the amplification and transformation of the Gaussian beam into a Super-Gaussian beam of order 6 for different powers of the seed beam. The red point corresponds to the amplification presented in the Fig. 5(a). (c) Beam shaping of a Gaussian beam using an LG₀₃ pump beam in an amplifier.

Download Full Size | PDF

To implement this beam shaping technique we require shaped pump beams. The most obvious way to produce such beams use conventional extra-cavity beam shaping of laser beams [1–3] using amplitude and phase elements. However, the pump beam only has to approximate the required profile. Such beams can be easily generated by higher mode selection in multimode pump lasers or the superposition of different higher order Laguerre–Gaussian pump beams [12,13]. For instance in Fig. 5(c) a single higher order mode (LG₀₃) was used as pump beam. The resultant shaped beam was remarkably close to the required Super Gaussian beam sought in the previous examples.

It must also be noted that this research has significant implications for amplification of custom beams. For instance, a carefully crafted flat-top beam will not remain flat top if it is amplified in an amplifier with, for instance, a Gaussian profile.

4. Three level gain media

One of the disadvantages of four level gain media is their inability to produce negative amplification (attenuation of the seed beam). Three level gain media [9,10] are much better suited for this technique because of their (normally undesirable) property that they absorb the seed beam until they are pumped to transparency. They can therefore exhibit negative attenuation in areas where it is needed, which increases the contrast of the shaped amplified beam. However, the rate equations are more complicated for three level gain media, which makes analytical solutions similar to Eq. (5) difficult to find. A numerical method similar to the one described earlier was used to find approximate solutions to the three level coupled rate equations. Figure 6(a) plots the amplification as a function of both the pump and seed intensity of a 5 cm, 0.2% doped Ho:YLF amplifier. Notice that for low pump and seed intensities the three level medium absorbs the seed and the amplification is less than one (attenuates). The required pump intensity profile is found by taking the ratio of the input seed intensity and that of the desired output profile at each transverse position. This ratio is the amplification that is required to shape the beam at each transverse position to obtain desired overall profile. By matching this amplification to the seed intensity, the required pump intensity can be found on the horizontal axis. This was done by first emulating the calculated values with a 3rd order polynomial expansion in both the pump and seed intensity (shown in Fig. 6(b)). The required pump intensity for a given seed intensity and amplification can be determined because the emulator is invertible. However, a disadvantage of this approach is that the emulators are valid only for specific gain materials. The type of material, doping percentage and length are specific to each emulator.

Fig. 6 The numerical result (a) and 3rd order polynomial fit (b) of the amplification with a 5 cm, 0.2% doped Ho:YLF gain medium.

Download Full Size | PDF

The numerical method was applied to reshape a Gaussian seed beam into a 6th order super Gaussian beam. In Fig. 7(a) a 1.039 µm beam was reshaped using a 5 cm long, 0.5% doped Yb:YAG crystal and in Fig. 7(b) a 2.05 μm beam was reshaped using with a 5 cm, 0.2% doped Ho:YLF crystal. Both examples show that the method works well and that the negative amplification in the wings leads to acceptable contrast of the output beam.

Fig. 7 (a) The numerical result of the amplification and reshaping of a Gaussian seed beam into a 6th order Super–Gaussian by a (a) 5 cm (0.5%) Yb:YAG rod and (b) a 5 cm (0.2%) Ho:YLF rod by implementing a 3rd order polynomial emulator.

Download Full Size | PDF

5. Experimental verification

The theory was experimentally verified using a Thulium doped Yttrium Lithium Fluoride (Tm:YLF) slab laser to end pumped a Holmium doped Yttrium Lithium Fluorite (Ho:YLF) three level rod amplifier (5 mm × 8 mm × 60 mm (a × c × a), a-cut with the c-axis horizontal).

A modified version of the Tm:YLF slab laser described in [14] was used to pump both Ho:LiLuF seed laser and Ho:YLF amplifier at the same time. One of the problems associated with [4] Tm:YLF slab lasers in general is low beam quality. The measured output beam of the Tm:YLF laser in the horizontal axis was M2 = 200 at full pump power.

In order to maximize the performance of the Tm:YLF laser used to pump the 60 mm long, 3 level Ho:YLF crystal rod we used a cylindrical lens (CO) to manipulate the shape of the intra-cavity beam. The idea was to the increase the quality of output beam by decreasing the number of modes present in the resulting output beam. The slab was placed in the cavity horizontally with the long side of the slab paralleled to the laser base. This presented a rectangular face to the optical axis of the laser where the horizontal axis is much wider than the vertical axis. Therefore the increase of the beam width in the horizontal axis lead to a decrease of the number of oscillating modes due to the increased spatial size for higher number transverse modes. In order to maximize the beam width at the curved output coupler (OC) side of the laser cavity we placed the output coupler as close to its stability boundary as possible. Because the laser becomes unstable at about 300mm and longer, we place the OC at about 290 mm from the back reflector (see Fig. 8 (horizontal)). We then used cylindrical lens (CL) to decrease the beam width in the vertical direction to avoid truncating the obtaining oscillation modes by narrow side of the slab (see Fig. 8 (vertical)).

Fig. 8 The schematic of beam quality improving in the slab laser where OC is the output coupler; CLV is the cylindrical lens which effect on vertical direction; HR is the high reflection mirror.

Download Full Size | PDF

Based on this technique the M2 parameter in the horizontal direction of the laser beam was dramatically improved form having an M2 = 200 (no CL laser cavity) to an M2 = 15 (with CL) at full pump power for the Hermit – Gaussian output beam (HG₀₁) (see Fig. 9). This improvement of the M2 parameter of the output beam would allow for more efficient pumping of the long 3 levels amplifier rod. The choice of such an output beam (HG₀₁) was based on the requirement of the pump beam intensity profile in the laser amplifier to retransform successfully the Gaussian beam into a beam with a super Gaussian transversal intensity profile. One of the disadvantages of the implementation of the high order HG_nm beam is the corresponding undesirable increase of the M2 parameter.

Fig. 9 (a-c) The experimentally obtained intensity profiles of the beam emitted by corresponding laser crystals (see a1-c1 (photos of the mounted corresponding laser crystals)). (d) Experimental setup of a Tm:YLF(c1) beam shaping laser (1.885 μm) pumping a Ho:LiLuF seed laser (b1) with 45W and an Ho:YLF (a1) amplifier with 35W and the seed co-aligned to the 1.885 μm beam into the amplifier ; where (1) laser diode 0.808 μm, (2) polarizing beam splitter, (3) beam dump, (4) beam shaping lens for pump, (5) beam splitter HT for pump (0.808 μm) and HR for 1.885 μm, (6) lens, (7) HR mirror for 1.885 μm, (8) 85% output coupler, (9) HR mirror for 2.073 μm, (10) fold mirror HT for 1.885 μm and HR for 2.073 μm, (11) 95% output coupler for 2.073μm, (12) HR mirror for 2.073 μm, (13) f = 200 mm lens and (14) f = 105 mm lens.

Download Full Size | PDF

The Tm:YLF slab laser delivered 45 W of pump power at a wavelength 1885 nm. To pump the seed laser the waist of pump beam was imaged into a Ho:LiLuF crystal by a 200mm lens (Fig. 9(13)). After passing through Ho:LiLuF laser the pump beam has lost ~10 W of 1.885µm pump power. The remaining transmitted pump beam was re-imaged into the Ho:YLF amplifier with a f = 150mm lens (Fig. 9(14)).

The Ho:LiLuF seed laser output energy was ~0.33 W with an M2 = 1.5 (see Figs. 9(b, b1 and d(9-11)).The 2073 nm seed beam from the Ho:LiLuF laser was co-aligned back with the 1885 nm pump beam and together passed only once through the amplifier crystal. The seed laser (2073 nm) was circular symmetric and had a Gaussian spatial distribution. A Gaussian intensity distribution was obtained from the Ho:LiLuF laser even though it was being pumped by Hermit – Gaussian like pump beam (see Fig. 9(c)) by slightly misaligning output coupler mirror 11. This was not the most efficient output of this laser, but it did serve to experimentally verify our theory, because only one pump source was available. The pump beam and seed beams matched each other and had radii of ~0.6 mm near the front face of the crystal. The pump beam was horizontally polarized and the seed beam was vertically polarized.

Fig. 10 (a) Measured 2D beam profiles of the transformation of a Gaussian seed beam (red) into a Flat-Top beam (blue) in an amplifier being pumped by a lobe shaped pump beam (black). (b) The 1D numerical simulation of such a transformation and (c) the centroid profile of the experimental results as shown in Fig. 10(a).

Download Full Size | PDF

The seed beam was amplified from 0.33 W to 0.50 W at full 1.885 µm pump power of 35 W. Figure 9(b) shows the two dimensional 2.073 µm seed and 1.885 µm pump profiles in the amplifier crystal as well as the amplified beam after the crystal. Even this marginal amplification (due to the non-optimal seed wavelength) was sufficient to illustrate significant beam shaping in the horizontal direction from Gaussian to a roughly Flat-Top in one dimension. In principle, very high amplification factors are possible. However, the resulting pump shape will need to be adjusted for different amplifications. For Gaussian to Flat-Top conversion the required pump beam profile will have a central node for low amplification factors, which will “fill-up” for higher amplification factors. A practical limit will be when the required pump beam is also almost Flat-Top

Figure 10(b) shows the numerical simulation results of a Gaussian beam being shaped to Flat-Top beam in an amplifier being pumped by a Hermit – Gaussian pump beam (HG₀₁). The horizontal centroid profile of the experimental results is shown in Fig. 10(c) and they are in good agreement with the numerical prediction. The significant difference is the intensity distribution in the amplified beams wings. The loss of contrast of the obtained super Gaussian beam was the result of the pump beam profiles not being confined close to the lobes, but rather tapering off (see Figs. 10(a) and 10(c)). There was therefore no significant negative amplification in the wings the amplified beam. This will often be the case when using higher order mode beams (HG_nm) from stable cavities as a pump source in shaping amplifiers. However, the flat top profile that was achieved will be sufficient for a large number of applications.

6. Conclusion

In this paper we have outlined and investigated in detail, the theory of reshaping laser beams into a desirable beam profiles with end-pumped laser amplifiers. We used the example of reshaping a beam from Gaussian to Flat –Top throughout this paper, using different gain media. Analytical equations were developed for four-level laser materials which describes the intensity transformation of a seed beam in an amplifier crystal, pumped by an arbitrary pump profile. Equations were also developed to find the required profile of a pump beam that result in the reshaping of a seed beam to a specific target profile. In the case where high gain was required; we have shown that the analytical equations are no longer valid and we have subsequently proposed an alternative numerical method to calculate the required pump profile.

The theory was experimentally verified using a Tm:YLF pumped Ho:YLF amplifier by reshaping a Gaussian seed beam from Ho:LiLuF laser into a Flat-Top output beam by using HG₀₁ pump intensity in horizontal direction. The experimental result was in good agreement with the numerical prediction.

References and links

1. F. M. Dickey and S. C. Holswade, Laser Beam Shaping, Theory and Techniques (Marcel Dekker, Inc., 2000).

2. I. A. Litvin, “Implementation of intra-cavity beam shaping technique to enhance pump efficiency,” J. Mod. Opt. 59(3), 241–244 (2012). [CrossRef]

3. I. A. Litvin and A. Forbes, “Gaussian mode selection with intracavity diffractive optics,” Opt. Lett. 34(19), 2991–2993 (2009). [CrossRef] [PubMed]

4. H. J. Strauss, D. Preussler, M. J. D. Esser, W. Koen, C. Jacobs, O. J. P. Collett, and C. Bollig, “330 mJ single-frequency Ho:YLF slab amplifier,” Opt. Lett. 38(7), 1022–1024 (2013). [CrossRef] [PubMed]

5. Z. Xiang, D. Wang, S. Pan, Y. Dong, Z. Zhao, T. Li, J. Ge, C. Liu, and J. Chen, “Beam quality improvement by gain guiding effect in end-pumped Nd:YVO₄ laser amplifiers,” Opt. Express 19(21), 21060–21073 (2011). [CrossRef] [PubMed]

6. A. E. Siegman, Lasers (University Science Books, 1986), pp. 292–293.

7. I. A. Litvin and O. J. P. Collett, “Beam shaping with a laser amplifier,” Proc. SPIE 8960, 89601N (2014). [CrossRef]

8. C. Czeranowsky, “Resonatorinterne Frequenzverdopplung von diodengepumpten Neodym-Lasern mit hohen Ausgangsleistungen im blauen Spektralbereich,” PhD thesis, University of Hamburg (2002).

9. B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections, and radiative lifetimes of rare earth ions in solids: Application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83(5), 2772 (1998). [CrossRef]

10. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

11. W. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5(7), 1412–1423 (1988). [CrossRef]

12. S. Ngcobo, I. A. Litvin, L. Burger, and A. Forbes, “The digital laser,” arXiv:1301.4760.

13. D. Naidoo, T. Godin, M. Fromager, E. Cagniot, N. Passilly, A. Forbes, and K. Aït-Ameur, “Transverse mode selection in a monolithic microchip laser,” Opt. Commun. 284(23), 5475–5479 (2011). [CrossRef]

14. H. J. Strauss, M. J. D. Esser, G. King, and L. Maweza, “Tm:YLF slab wavelength-selected laser,” Opt. Mater. Express 2(8), 1165–1170 (2012). [CrossRef]

Parameter	Nd:YVO₄ [8]	Ho:YLF [9]	Yb:YAG [10]
Total active ion density, n_tot, [m⁻³]	0.2% of 1.25 × 10²⁸	0.2% of 1.25 × 10²⁸	0.5% of 1.39 × 10²⁸
Upper laser level lifetime,τ₂, [μs]	97	13 000	950
Ground state absorption cross section, σ_a, [m⁻³]	51 × 10⁻²⁴	1.07 × 10⁻²⁴	5 0.76 × 10⁻²⁴
Upper laser level emission cross section, σ_em, [m⁻³] Upper laser level emission cross section for pump, σ_eg (3 levels), [m⁻³] Ground state absorption cross section for seed, σ_au (3 levels), [m⁻³]	125 × 10⁻²⁴	1.49 × 10⁻²⁴ 7.8 × 10⁻²⁵ 5.43 × 10⁻²⁵	2.31 × 10⁻²⁴ 0.76 × 10⁻²⁴ (2.31/15) × 10⁻²⁴

Beam intensity reshaping by pump modification in a laser amplifier

Abstract

1. Introduction

2. Analytical method

3. Beam shaping

4. Three level gain media

5. Experimental verification

6. Conclusion

References and links

Cited By

Figures (10)

Tables (1)

Equations (7)

Optics Express