International Journal of Physical Sciences

We have numerically shown that a high power Bessel-Gauss beam can be generated by a solid state thin disk laser using an axicon-based resonator. Ytterbium ions doped in the Yttrium Aluminum Garnet (YAG) crystal was utilized in this configuration as an active medium. We obtained the output power, the intensity as well as phase profiles on the output coupler and the active medium.


INTRODUCTION
Diffraction-free beam and optical invariant field are special cases of optical wave propagation that have caught general attention in the last decade.Temporal solitons were the first class of optical signals to be recognized as being invariant under propagation (Hasegawa and Tapert, 1973) and to be experimentally demonstrated (Mollenaure et al., 1980).But, they require a nonlinearity to preserve their shape.It was pointed out by Durnin that nonlinearity is not a necessity for invariant propagation; indeed, he proved that Bessel beams can propagate in a purely linear material without deformation (Durnin, 1987;Dallaire et al., 2009).On the other hand, Lopez and Helmerson have recently demonstrated that non-diffracting beams with arbitrary transverse shapes can be generated (Lopez-Mariscal and Helmerson, 2010).They have tailored a nondiffracting beam of an arbitrary pattern via an azimuthal modulation of the angular spectra of Helmholtz-Gauss wave fields.
On the other hand, power scalability and minimal thermal lensing are the particular features of the thin-disk laser (TDL).High output power, high efficiency and good beam quality are further advantageous qualities of this type of solid-state lasers (Giesen et al., 1994).The highest reported output power of a Yb:YAG single disk is 5.3 kW in multi-mode operation (Giesen and Speiser, 2007).
In our previous work, we considered a passive axiconbased configuration to generate Bessel-like beam with a flat output mirror (Aghbolaghi et al., 2010).In this work, an axicon based disk type laser with spherical output mirror is numerically studied.This numerical model is performed in two steps; at first, the transverse profile of beam is obtained employing the Collins integral and at the second step, the photon rate equation accounts for the amplification and saturation effect.These two steps are repeated successively approaching a steady state.We calculated the output power, intensity and phase distribution on rear surface of the axicon and output coupler.

CONFIGURATION
The configuration of a solid state thin disk laser with a Bessel-Gauss resonator is shown in Figure 1.It consists of a composite Yb:YAG/YAG or Yb:YAG/Diamond crystal and a concave spherical output mirror separated by distance L. The composite crystal includes a Yb-doped thin disk as the active medium and a conical-shaped undoped part (pure YAG) as an axicon with an apex angle γ.Such as the composite laser martials (Yb:YAG/YAG), this structure can be made through heat diffusion bonding technique (Dashkasan et al., 2012).The back surface of the active medium is coated with a high reflection layer which acts as a back mirror of resonator.r a , t, d ax , and R denote, respectively, the radius of the axicon, thickness of the active medium, thickness of the axicon, and curvature radius of the output coupler.We assume the refractive indices of the active medium and the axicon are the same.Also, we suppose d = t + d ax .
It is customary to replace each end mirror by two lenses with focal length ݂ = ܴ and locate the reference planes between them.The ABCD ray transverse matrix between two reference planes is obtained as follows: where where, r is the distance of the ray from the optical axis and n is the refractive index of the axicon and disk.Stability condition is given by: The stability parameter of this resonator versus the radius of the axicon for different curvature radius is shown in Figure 2. It should be noted that the following parameters are used in our calculations: The cavity length ‫ܮ‬ = 782.4݉݉ is calculated according to the resonance condition ).The point ‫ݎ‬ = ೌ ଶ = 1 ܿ݉ that locates in the intersection of curves corresponds to the eigenvalue driven by Gutie´rrez-Vega (2003).

Diffraction
The Huygens-Fresnel diffraction integral may be expressed in terms of the ABCD matrix elements (Collins, 1970).The incident amplitude distribution ‫ݑ‬ (ߩ , ߠ ), and the diffracted one ‫ݑ‬ (ߩ , ߠ )on two reference planes separated by a distance L are interconnected by the following integral: where ߩ and ߠ are the cylindrical coordinates of the incident plane; ‫ݑ‬ (ߩ , ߠ ) = ‫ݑ‬ (ߩ ) ൜ cos ݈ߠ sin ݈ߠ is the incident optical field; ‫ݑ‬ (ߩ , ߠ ) is the diffracted optical field, ߣ is the light wavelength, ݇ is the wavenumber given by ݇ = 2ߨ/ߣ, and A, B, and D are elements of the ray matrix.In addition, an angular dependency is introduced making it possible to simulate transverse-mode diffraction.
The upper limit ‫ݓ‬ corresponds to the radius of the aperture of the first reference plane.A transform factor ܶ (ߩ ) is called the transmittance function of the optical element or aperture and ‫ܬ‬ ‫)ݔ(‬ is the ݈th-order Bessel function.This makes it possible to reduce the twodimensional integral to only one dimension.Finally, time dependency can be expressed by the factor of exp ‫.)ݐ߱݅−(‬The constant phase shift of exp (−݅݇‫)ܮ‬can be ignored in the integral equation (Equation 5).

AMPLIFICATION
First, we assume that the amplifier is made from a thin disk within which we neglect diffraction.The level scheme in Yb: YAG can be considered as quasi 4 level scheme.But the level system can be treated as two level system under the assumption that there are fast relaxations between the levels within one manifold compared to the fluorescence lifetime of the laser transition (Ostermeyer and Straesser, 2007;Bordet and Bartniki, 2006).The rate equations for the pump process can be formulated for the population density of the upper and the lower manifolds with the emission and absorption cross sections for the pump at 941 nm and the laser at 1030 nm.Here, we define some parameters and recall the formula obtained in (Bordet, 2000;Bordet and Bartniki, 2006).The back reflection regenerative pumping is used in this work.The absorbed pump intensity can be written as: where ߛ is the transmission of the recycling apparatus taking into account the reflection-transmission of the optical components of the recycling loop and the mode matching of pump beam, R ୫ ୮ is back reflection of the active medium at pump wavelength, and the saturated single pass transmission of the amplifier medium, Γ, is given by where G and d are the gain saturated and thickness of the active medium, respectively.Also, ݂ , ݂ , ߙ and are ݃ given by the following equations: ߪ and ߪ are respectively the absorption and emission cross-sections, ܰ is the ytterbium concentration, ݂ and ݂ ௨ are the Boltzmann occupation factors of the Stark levels ݅ and ݆ of the lower and upper levels, respectively.Let us call I i impinging laser intensity with medium.The amplification of the intensity of the beam reads as ‫ܫ‬ ାଵ = ‫ܫ‬ ‫ܩ‬ ଶ .After the beam is amplified, the saturated gain can be computed by (Ostermeyer and Straesser, 2007): where R ୫ ୪ and d are the reflectivity of back side of active medium and thickness of it, respectively.Note that ‫ܫ‬ and ‫ܫ‬ are normalized to the corresponding saturation intensities.

NUMERICAL CALCULATIONS
The calculation process is carried out as follows: At the first reference plane, that is, on the boundary surface between the active medium and the axicon, random distributions for both the amplitude and the phase are chosen.This light is transmitted through the axicon and then is propagated in free space toward the output coupler then reflected and propagated backward to the axicon.Finally, we calculated the double-pass amplification through the active medium.This procedure is successively repeated until a steady state is obtained.In our analysis, we ignored deformation and thermal lensing of the active medium.

Phase and intensity distribution
The loss of fundamental mode on the geometrical parameters of the cavity was examined with the quantities given in Table 1.For the radius of the pump area, ‫ݎ‬ = 7 ݉݉, the resonance condition gives the cavity length as L= 49.19 cm.For our configuration, typically, several hundred round trips are required in order for iteration to converge starting from an arbitrary rotational symmetric field distribution on refractive axicon.The intensity distribution and phase of the field at the surface of the active medium (just before the axicon) are shown in Figure 3a and b for flat output mirror, and in Figure 4a and b, for a concave spherical mirror with R = 50 L .They cover a broad area of the active medium.The power density distributions are shown versus the radius of the pump area r in deferent curvature of the output coupler in Figure 5a and b for R > L and R < L, respectively.It can be seen that for curvature radii R smaller than resonator length L the power density distributions are strongly narrow.Specifically for R = L, it is distributed narrowly around r/2.On the other hand, for R > L, the intensity distributions spread over large areas.Thus, in order to use the maximum gain area, it is most reasonable to choose output couplers with the radius curvatures greater than the length of cavity.It can be very important for power scaling.As expected, the phase curve corresponds The population of the lowest sublevel of the fundamental manifold fl1 0.88 The population of the sublevel j of the excited state fuj 0.17   for both flat and curved output couplers; this fact confirms that the field at the axicon behaves as a conic wave.
The output field at flat output mirror (Figure 3c and d) corresponds to the calculated mode.It can be observed that the lowest mode in resonator falls off faster than the ideal zero-order Bessel beam J (k ୲ ρ).It becomes virtually zero outside the interval ൣ0, ‫ݎ‬ /2൧.In Figure 3d, the transverse phase distribution is represented.Similar to an ideal Bessel beam, the phase pattern jumps of π at the zeros of the Bessel function.The output field and its phase in this configuration with spherical mirror are shown in Figure 4b.The field has a Bessel shape, but now the Gaussian modulation is more evident.

Effects of varying the geometrical parameters on the loss
The loss per round trips with respect to the transverse radius of output mirror‫ݓ‬ , axicon radius ‫ݎ‬ or ‫ݓ‬ and cavity length is investigated.In Figure 6a, the results of wave optics show that the loss decreases drastically versus w m and approaches zero at w m =w p .We used it in this work.According to Figure 6b, it slowly varies versus ‫ݓ‬ at large amount and change fast at small ‫ݓ‬ .It can be useful for the disk laser configuration with a single active medium.The loss in terms of the cavity length and the wedge angle of the axicon are represented in Figure 6c and d condition (distance L 0 ).The rate of loss increases at L > L 0 is faster than it at L < L 0, as we expect, because the cavity length decreases.
Let us consider the effect of varying the curvature radius of the output mirror.Consider the loss behavior corresponding to the lowest mode resonating within the cavity.The loss as a function of the normalized R/L is depicted in Figure 7.The dashed line represents the loss in the limit by left and bottom axis when R is greater than L. The solid line represents it by right and above axis when R is smaller than L. In spite of Ref.13, the resonator behavior in range RϵሾL, 10Lሿ is investigated, where the loss of resonator fluctuated and has some extremums.It may be related to shift frequency in it because of varying the radius curvature of the output coupler.The loss is minimum about R = L.In range, Rϵ(0, Lሿ, based on our simulation the output beam profile is nearly far from the Bessel-Gauss beam.Our numerical results have good agreement with Ref. 13 in large R.

Output power
We employed the configuration of Figure 1 as a thin disk with axicon Bessel-Gauss resonator.It is assumed that 10% Yb doped in YAG crystal as an active medium.We calculated absorption (for the back reflection regenerative pumping), the diffraction and amplification into cavity by equations mentioned earlier.The required information for calculating is given in Table 1.The transverse intensity profile on the active medium (black curve) on left and bottom axis and the phase of field (red curve) on right  right and top axis in Figure 8b.

CONCLUSION
We have presented the thin disk solid state laser with Bessel and BG resonator.We have analytically stable and unstable area by geometrical optics viewpoint in term of different geometrical parameters.The traditional Gaussian beam resonator in which the spot size of the fundamental mode on the active media is very small.on the other hand, the spot size on the active medium in this configuration is very large.It can be very important for the disk laser configuration with a single active medium.The loss of resonator per round trips is calculated in terms of different resonator parameters by the viewpoint of wave optics.These results illustrated this configuration can be used for thin disk lasers that their gain per round trip is low.In this configuration, based on our computer model, a high power BG laser beam with small spot size can be obtained.

Figure 1 .
Figure 1.The configurations of solid state thin disk laser with an axicon-based resonator.

Figure 2 .
Figure 2. The stability of an axicon-based resonator versus position of ray on the axicon for different radii of the curvature of the output coupler.

Figure 3 .
Figure 3.The intensities transverse distribution (a) and optical phase distribution (b) on the active medium, the intensity transverse distribution (c) and optical phase distribution (d) on the output coupler.

Figure 4 .
Figure 4. intensities transverse distribution on the active medium and output coupler (a), the intensity transverse distribution and its phase on output coupler (b).

Figure 5 .
Figure 5. Schematic representation of the intensities transverse distribution on the active medium in term of radius.

Figure 6 .
Figure 6.The loss versus wm.It has been to the pumping spot size (a), the loss versus pumping spot size (b), the loss in terms of the cavity length (c), the loss in terms of the wedge angle of the axicon (d).

Figure 7 .
Figure 7.The dashed line represents the loss in the limit by left and bottom axis when is greater than one.The solid line represents it by right and above axis when R/L is smaller than one.

Figure 8 .
Figure 8.The transverse intensity distribution Ia and phase øα on the active medium, (a), The transverse intensity distribution Io and øo on the output coupler, (b), and output power in term of number of roun-trips, (c).

Table 1 .
Parameters of the disk laser with axicon based resonator.