Feasibility study of a conical-toroidal mirror resonator for solar-pumped thin-disk lasers

M. Endo

doi:10.1364/OE.15.005482

1. Introduction

Collecting the solar power in space and transmit it to the earth has been studied for long time to provide energy infrastructure of the mankind after the depletion of the fossil fuel[1]. Among the several proposals, direct generation of the coherent light by solar-pumped solid-state lasers and direct it to the earth is advantageous because of its simplicity, durability, and potential high-efficiency.

However, there are several obstacles to develop a practical solar-pumped, high-power solid-state laser operating in space. First, the sunlight is not a good nature as a pumping source in terms of its spectral purity and its spatial coherence. The former problem could be solved by using a Nd/Cr codoped medium, which has a broad-band absorbability to the sun’s visible spectrum. On the other hand, the later problem, especially, the optical resonator configuration suited to a large-scale space-based solar-pumped lasers, has not been discussed seriously to our knowledge. Our proposal is a new class of resonator that is simple, scalable, and suited to fully utilizing the moderately concentrated solar light as a pump source.

Fig. 1. Schematic drawing of a resonator with a conical-toroidal reflector element and a thin-disc active medium.

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Figure 1 shows a schematic drawing of the proposed optical resonator. It consists of a unit of conical mirror and toroidal mirror on which an active medium is set, and an output coupler. From the viewpoint of the active medium, it is a kind of thin-disk lasers[3]. However, unlike the ordinary thin-disk lasers or so-called active mirror lasers, the optical ray inside the active medium travels radially, parallel to the surface of the active medium.

For this purpose, the medium is shaped doughnut-like and its inner and outer circumferences are optically finished. A toroidal reflector at just outside the medium serves as a feedback mirror while a conical mirror is inserted at the central hole of the medium to serve as a mode converter, from radial to paraxial or vice versa. The oscillation mode is traveling active medium internally by total internal reflection at the surfaces.

The primary merit of the thin-disk media as a solar-pumped laser is its broad and circular surface exposed to the sunlight, so that no elaborated solar light concentrating optics is required. In addition, thin-disk configuration is beneficial for high-power lasers because of its efficient cooling capability by a direct contact to the heat sink, and low susceptivity to the index gradient of the medium to the beam quality since the laser beam travels parallel to the index gradient. For space applications, the former advantage is very important because the cooling of active medium is made to be simple, efficient, and maintenance-free. For the beam quality issue, our resonator uses the active media in a different way, and the latter merit of the thin-disk configuration is not directly connected to our case. Nevertheless, since the oscillation mode travels with total internal reflections, the index gradient is averaged out and not to be a serious problem.

The main idea of the proposed resonator is to say that it can be pumped from the broad area of the active medium with a moderately concentrated solar light, yet the small signal gain could be high enough to extract power from the medium efficiently because the optical path is comparable to that of the rod resonators. To ensure the rigidity and robustness of the resonator, the conical element and toroidal element are better to be fabricated from a single solid block with a diamond-turning technique. On the other hand, the lateral position of the active medium relative to the resonator is not strict to the wavelength accuracy, because the optical length of the medium does not change by the lateral shift of the medium in the first order approximation.

The output beam mode of this resonator is a confined, doughnut-like, whose diameter is basically defined by the thickness of the active medium and regardless its diameter. Therefore, a high-power system with good beam quality could be easily achieved. Because the wavenumber variation by the tilt of the mirror is not determined by the diameter of the toroidal mirror, but the diameter of the cone, its alignment sensitivity is comparable to the spherical mirror resonators whose aperture is the same as the size of the cone.

It should be pointed out that if the conventional thin-disk configuration is applied to the solar-pumped lasers, problems may arise because the round-trip gain per disk is very small due to the inefficient solar-pumping, and the scaling of the system with expanding the aperture leads to the difficulty of single transverse-mode oscillation.

In this paper, a numerical simulation of this resonator based on the Fresnel-Kirchhoff diffraction theory is presented, and the feasibility of the proposed resonator for various aspects, including the thermomechanical consideration of the medium and alignment sensitivity, are discussed.

2. Numerical model

A numerical simulation code has been developed specially for this study. The code is derived from the simulation code of the w-axicon optical resonator with a movable axicon[4].

Fig. 2. Schematic drawing of the resonator geometry and calculation model.

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Figure 2 shows the schematic drawing of the proposed resonator and its calculation model. The principle of the simulation is the well-known Fox-Li type iterative propagation based on the Fresnel-Kirchhoff diffraction integral[5]. Due to the complicated optical path of the proposed resonator, different propagator scheme should be used for different regions.

Let the electric field start from the output coupler (O. C.). A two-dimensional grid in (x_P,y_P) coordinates is defined here and the electric field at each point is defined by the vector components (E_x,E_y). The propagation of the electric field to the conical reflector is calculated by the ordinary two-dimensional Fresnel-Kirchhoff integral , for each component of the electric vector.

At the conical reflector, The electric field vectors are analytically converted and mapped to a rectangular region, whose abscissa is mapped from zero to 2π, and ordinate represents the actual length (y_R), as shown in the figure. Its conversion rule is described by the following formula,

y_{R} = - {(x_{P}^{2} + y_{P}^{2})}^{\frac{1}{2}} + H - \frac{t}{2}, zwφ = \tan^{- 1} (\frac{y_{P}}{x_{P}})

E_{p} (x_{P}, y_{P}) = E_{x} (x_{P}, y_{P}) \cos φ + E_{y} (x_{P}, y_{P}) \sin φ

E_{s} (x_{P}, y_{P}) = - E_{x} (x_{P}, y_{P}) \sin φ + E_{y} (x_{P}, y_{P}) \cos φ

E_{y} (φ, y_{R}) = r_{p} {(x_{P}^{2} + y_{P}^{2})}^{\frac{1}{2}} E_{p} \exp [- ikH]

E_{φ} (φ, y_{R}) = r_{s} {(x_{P}^{2} + y_{P}^{2})}^{\frac{1}{2}} E_{s} \exp [- ikH]

where φ and y_R are the coordinates in the radial region, r_p and r_s are the complex reflectivity of the mirror surface for p and s polarization respectively, and k is the wavenumber. Now the optical power of the propagating mode is converted from [W/m²] to [W/m∙rad] on the radial-region two-dimensional array.

From the axicon to the active medium, a Fresnel-Kirchhoff integral is operated to the array same as the paraxial region. However, a cyclic boundary condition is fulfilled in the φ coordinate, because the electric field in the φ direction is expanded throughout the definition area of the Fourier transformation.

At the entrance of the active medium, the height of the optical field is clipped at the thickness of the medium (±t/2), and its propagator is changed from the free-space mode to the waveguide mode.

Fig. 3. Description of the waveguide mode propagation model.

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The propagation inside the medium is modeled by the combination of the radial free-space propagation and total internal reflection. Figure 3 shows the propagation model inside the medium. The propagation is divided into several discrete steps. The electric field defined at the nth station is first amplified by the rule of the homogeneously broadened active medium, namely,

E_{0} = E_{00} \exp [\frac{δ l}{2} (\frac{g_{0}}{1 + \frac{I}{I_{s}}} - α)],

where E ₀ is the amplified electric field, E ₀₀ is the original electric field, g ₀ is the small-signal gain, α is the small signal loss of the medium, δl is the spacing between adjacent stations, I is the sum of the optical intensity in both directions, and I_s is the saturation intensity of the medium.

Then the optical field is freely propagated as a rule of radial Fresnel-Kirchhoff integral described above. The resultant electric field maybe laterally spread as shown at the (n + 1)th station. Then the electric field outside the waveguide is folded by the following rule,

E (y) = {E'}^{(0)} + {E'}^{(- 1)} + {E'}^{(+ 1)} + {E'}^{(- 2)} + {E'}^{(+ 2)} + \dots

As the regions with odd numbers are expected to experience odd number total internal reflections, the superposition of these sections are reversed.

It is concerned that the accuracy of the model is limited because the propagation internal the medium is approximated by the discrete steps. To see the error caused by the discretization, we calculated the output power and beam quality of a loaded resonator with number of the stations varied from 2 to 8. The result showed that the output power decreased by approximately 10% as the number of the station increased, but the transverse distribution of the converged electric field was perfectly identical. Thus, the accuracy of the model has been shown to be enough for this study, and we selected the number of steps to be six. The index gradient in the active medium could be incorporated, however, we have ignored this effect in the present study to avoid complexity.

3. Results of calculations

3.1. Bare cavity calculation

First, the size of the gain medium was determined, then main parameters of the resonator that fits the medium were determined. We defined the size of the disk as ϕ210 mm with ϕ10 mm hole at the center. To enjoy the benefits of the thin-disk geometry as much as possible yet practical absorption with double-pass was ensured, the thickness was determined to be 2.5 mm. Other parameters of the resonator were determined as Table 1.

Table 1. Dimension of the calculated resonator

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The reflection losses at the surfaces of the gain medium for pump light and laser mode was assumed to be zero, and reflection losses of toroidal mirror and output coupler has not been assumed. The conical element was assumed to have different reflectivities for s-polarization and p-polarization, since oblique incident reflective coatings normally have slightly different reflectivities for each polarization. In this work, the difference in electric field reflectivity was determined to be 1%. We have calculated both cases, namely, (r_p = 1.00 and r_s = 0.99) and (r_p = 0.99 and r_s = 1.00).

Upon the determined main parameters, we have several parameters to be optimized. These are the curvature radius of the toroidal mirror R and its vertex position relative to the centerline of the gain medium Δy. We have calculated the loss of the bare cavity as functions of these parameters. Instead of the active medium, a transparent medium whose index was same as GSGG (1.97) was assumed.

Fig. 4. Calculated resonator loss per round trip as a function of the vertex position Δy for different R. Left: (r_p > r_s) and right: (r_p < r_s).

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We took three deferent R’s and calculated bare-cavity round-trip losses of the resonator as a function of the vertex position Δy. The results are shown in Fig. 4. No difference is seen between (r_p > r_s) and (r_p < r_s) cases. It is seen that the resonator loss is sensitive to the vertex position, and its optimum value differs for each curvature radius of the toroidal mirror. Nevertheless, the minimum loss for each case was identical. Beam quality of the oscillating mode at the minimum loss was the same, too. Therefore, the selection of the R could be arbitrary made. As seen in Fig. 4, mode switching occurred as Δy was increased in R = 3 m case, while the oscillation mode was insusceptible with varying Δy in R = 5 and 10 m cases. From this result, we selected the curvature of toroidal mirror R = 10 m in favor of better mode discrimination.

As examining the loss of the optimized resonator, 75% of the loss was identified as the insertion loss at the inner surface of the medium. Nevertheless, the absolute value, less than 0.2%, was quite small and that may not cause any obstacles for efficient oscillation of a high-power laser. Reflection loss at the conical reflector was negligible since the oscillation mode was purely p-polarized.

In summary, the diffraction loss of the proposed resonator is quite small in spite of having free-space/waveguide mode conversions. Hereafter, we will discuss the results of loaded cavity calculations with R = 10 m resonator.

3.2. Loaded cavity calculation

We have simulated the laser oscillation of the proposed resonator assuming a Nd/Cr: GSGG as an active medium. It is known as a suitable medium for solar-pumped lasers because of its very wide absorption in the visible spectrum. A comparative study of the active media for solar-pumped lasers has been done by Hwang et al[6]. We referred the optical properties of Nd/Cr: GSGG from their work, that is summarized in Table 2.

To estimate the small signal gain as a function of the input solar irradiance, we have to calculate the absorption coefficient of the material over a wide range of the spectrum. To simplify the problem, we used the same approach as Ref [6]. First, the absorption cross section σ_ab(λ) [m²] of the Nd/Cr: GSGG was approximated by discrete, rectangular functions as in the Ref. [6]. We assumed a normal incident of the solar light to the active medium and double-pass absorption. Upon this assumption, the photon absorption rate per unit area of the active medium R_p[1/m²∙s] over a 1 AM0 (air mass zero) solar radiation is given by

R_{p} = \int I_{AM 0} (λ) \frac{λ}{hc} {1 - \exp [- 2 σ_{ab} (λ) Nt] d λ}

where I _AM0(λ) is the spectral intensity [W/m²/ nm] of the AM0 solar radiation at a given wavelength λ [nm]. In Eq. (7) and hereafter, the integration is conducted over the whole spectrum of the solar radiation. However, the ultraviolet (UV) and infrared (IR) absorption of the active medium are not taken into account. As in the lamp-pumped solid-state lasers, these bands are filtered out prior to reaching the active medium to avoid solarization or unwanted heating of the active medium. The effect of UV/IR cut-off filter is incorporated in σ_ab(λ).

Table 2. Optical properties of the Nd/Cr: GSGG[6]

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For simplicity, we assumed that all the absorbed photon is effectively used to pump Nd ion to the upper laser state. Disregarding the depth dependent pump intensity distribution, pumping probability of each Nd ion W_p[s^-1] is simply

W_{p} = \frac{R_{p}}{N_{t}} .

Small signal gain g ₀[m^-1] as a function of solar concentration S_c is obtained by

g_{0} = σ N W_{p} t_{sp} S_{c} .

A few useful figures are derived here from the above assumption. The “absorbable power” I_ab [W/m²] is defined by

I_{ab} = \int I_{AM 0} (λ) δdλ

where δ is zero when the absorption cross section of the medium is zero, otherwise 1. Calculated I_ab for our simplified GSGG absorption spectrum became 382 W/m², a 28% of the total solar radiation. In the same way, “absorbable photon flux” F_ab[1/m²∙s] is defined as

F_{ab} = \int I_{AM 0} (λ) \frac{λ}{hc} δdλ .

The upper limit of the optical-optical conversion efficiency (laser power/solar power) η _L(max) is described by

η_{L (max)} = \frac{F_{ab} h v_{L}}{\int I_{AM 0} (λ) d λ}

where hv_L denotes laser photon energy. The calculated η _L(max) for our case was 0.157. Next, the “averaged absorption coefficient” α_ab of the medium is defined by

α_{ab} = \frac{\int I_{AM 0} (λ) σ_{ab} Ndλ}{I_{ab}} .

It represents the average absorption coefficient of the effective band of the solar light, and is used to calculate the thermomechanical consideration of the active medium. The calculated value for our medium is α_ab = 2.56 × 10² m^-1. Portion of the absorbable power (I_ab) that is actually absorbed by the medium, η_ab is readily calculated from the above assumption as

η_{ab} = 1 - \exp [- 2 α_{ab} t],

and that was 0.722 for our case. Approximately 28% of the absorbable power is unused when the thickness of the medium is 2.5 mm. The detailed band-by-band absorption calculation was made, and the result was 0.695, showing the validity of above approximation.

Fig. 5. Laser output and beam quality of the loaded resonator calculation, as a function of the vertex position Δy. Input irradiance is S_c=1,000.

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Figure 5 shows the output power and beam quality of the loaded resonator when the medium was pumped with S_c=1,000, as a function of the vertex position. Unexpectedly, the highest power was obtained not at the lowest small-signal loss of the bare cavity. Presumably, the gain-induced intensity distribution caused the slightest beam deflection, and that caused the change in the optimum vertex position at the given condition. We changed the vertex position of the loaded cavity calculations from 1.2 mm to 0.6 mm accordingly. The near-field and far-field patterns of the output beam at the condition of Δy = 0.6 mm is shown in Fig. 6. The mode pattern looks a pure Laguerre-Gaussian LG¹ ₀, and its electric field is radially or azimuthally polarized in accordance with the choice of the preferred polarization. It is notable that only a small fraction of difference in reflectivity can select pure polarization state of oscillation modes.

The output power and conversion efficiency of the resonator was calculated as a function of the solar concentration. Figure 7 shows the result. The transmittance of the output coupler was optimized at each solar concentration. The oscillation threshold of the resonator is only S_c=100, and the conversion efficiency saturates at around S_c=2,000, showing the advantage of the proposed resonator. The theoretical limit of the conversion efficiency from the above discussion is η_L(max)η_ab, and it is 0.113 in our configuration. The actual conversion efficiency at S_c=3,000 is 0.056, that is only a half of the limit. The defect is mainly caused by the rather large absorption loss of the crystal assumed, α = 2 × 10^-3 cm^-1. We have conducted a calculation with a loss-free crystal, and obtained η_L=0.10.

Fig. 6. Near-field and far-field patterns of the resonator. Input irradiance is S_c=1,000, vertex position is Δy = 0.6 mm.

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Fig. 7. Output power and optical-optical conversion efficiency of the loaded resonator.

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3.3. Thermomechanical consideration

Temperature distribution inside the active medium and thermomechanical limitation of the pumping intensity was analyzed. The model considered was a infinitely wide, one-dimensional slab with a heat sink at fixed temperature T ₀ attached on one end. Normal incident of the pump light was assumed, and it double-passed the medium.

Using the defined α_ab, the absorbed power per unit volume P_ab [W/m³] as a function of the position z originated at the back surface is described by

P_{ab} (z) = α_{ab} I_{ab} S_{c} (1 - η_{L}) [e^{- α_{ab} (t - z)} + e^{- α_{ab} (t + z)}]

Temperature gradient over the medium was calculated by the one-dimensional finite-difference heat diffusion code. Thermal properties of the material was taken from Ref. [7] and summarized in Table 3. Values were approximated to be temperature invariant. In these calculations, reduction of the heat deposition by the stimulated emission was assumed to occur at the lasing efficiency shown in Fig.5.

Fig. 8. Model of the thermomechanical analysis of the active medium.

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Table 3. Thermal properties of the Nd/Cr: GSGG[7]

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Fig. 9. Absorbed power per unit volume and temperature of the active medium as a function of the position from the heat sink.

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Figure 7 shows the absorbed power and temperature distribution as a function of z when input solar concentration is 1,000. Pump power nonuniformity is seen to be ±10%, the assumption of the uniform pump distribution is justified. Temperature difference between two surfaces is 57 K.

A problem of uniform volumetric heat deposition on a infinitely-wide slab and its removal from one surface is analytically solved. The temperature difference between two surfaces is

T_{S} - T_{0} = \frac{I_{ab} S_{c} η_{ab} (1 - η_{L}) t}{2 κ} .

When solar concentration is 1,000, the analytical solution of the temperature difference is 55 K, that is close to the numerical simulation result.

The limitation of the pump power intensity is determined by the thermal stress caused by the thickness-wise temperature gradient. The detailed analysis of the pump power limit for thin-disk media is given by Vetrovec[8]. According to Ref. [8], the fracture limit of the uniformly-pumped thin-disk media which is cooled from one surface is given by

{(T_{S} - T_{0})}_{max} = \frac{3 R_{T}}{2 κ} .

Giving the values for Nd/Cr: GSGG yields (T_S - T ₀)_max=164 K, and this temperature difference is reached when pumping intensity of S_c =2,950. At the limit pumping intensity, small signal gain of the medium is 6.9×10^-2 cm^-1 and that is orders less than the commonly reachable values of the LD-pumped thin-disk lasers. One of the reasons of this low limitation of the pump intensity is that there is a large quantum defect between the absorption band of the Cr ion and lasing wavelength, that results in a larger amount of waste heat. Thus the proposed geometry is said to be attractive compared to the face-output thin-disk lasers.

3.4. Misalignment assessment

Fig. 10. Relative laser output power as a function of the tilt of the output coupler relative to the feedback mirror.

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Finally, the robustness of the proposed resonator to misalignment was assessed. The only degree-of-freedom of this resonator is the relative tilt of the conical-toroidal mirror to the output coupler, since the conical-toroidal mirror is rigidly fabricated by diamond-turning technique.

Figure 10 shows the laser output degradation as a function of the tilt of the output coupler. The proposed resonator was operated at S_c = 1,000. The 10% degradation of the output power was met at 12 μrad misalignment. For comparison, the alignment tolerance of a positive-branch confocal unstable resonator was calculated and the result is shown in the same figure. The resonator calculated was having the same aperture (cone diameter) and same length with the proposed resonator, magnification was 2.0, (NF_eq = 0.74) and round-trip small signal gain of the medium was e ^2g₀l=400%. It is seen that the proposed resonator has the same degree of alignment sensitivity as an ordinary unstable resonator.

4. Conclusions

A conical-toroidal mirror resonator combined with a thin-disk solid-state medium is proposed. Unlike the conventional thin-disk lasers, the optical ray path inside the media is radial direction, parallel to the face of the disk. This scheme is suited for solar pumping because the pump light could be irradiated to the wide surface of the disk, yet small-signal gain of the resonator could be reasonably high due to the novel beam propagation scheme. The output power and optical-optical conversion efficiency of the laser with Nd/Cr: GSGG as an active medium was calculated. When the medium was pumped at 3,000 solar concentration, the optical-optical conversion efficiency reached 0.056, a half of the theoretical limit. The small-signal loss of the medium was sorely responsible for this defect, because a theoretical limit efficiency was reached by a calculation with a loss-free media. Thermomechanical analysis of the active medium was conducted and it was shown that the fracture limit pumping intensity was 2,950 solar concentration. The small signal gain at the limiting pump intensity was 6.9×10^-2 cm^-1, considerably lower than the LD pumped thin-disk lasers, partly because a large quantum defect due to the white-light pumping generates larger temperature gradient. Therefore, the advantage of the proposed resonator over a conventional thin-disk geometry was apparent. Misalignment susceptivity of the proposed resonator was assessed, and it was shown that its susceptivity was same as the positive-branch confocal unstable resonator of the same aperture.

Acknowledgment

This research was supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 15860812, 2006.

References and links

1. P. E. Glaser, “Power from the Sun: Its Future,” Science 22, 857–861, (1968). [CrossRef]

2. R. L. Fork, “High-energy lasers may put power in space,” Laser Focus World, 113–114, September (2001).

3. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1-kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000). [CrossRef]

4. M. Endo, “Numerical simulation of an optical resonator for generation of a doughnut-like laser beam,” Opt. Express 12, 1959–1965 (2004). [CrossRef] [PubMed]

5. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

6. I. H. Hwang and J. H. Lee, “Efficiency and threshold pump intensity of CW solar-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2129–2134 (1991). [CrossRef]

7. W. F. Krupke, M. D. Shinn, J. E. Marion, J. A. Caird, and S. E. Stokowski, “Spectroscopic, optical, and thermo-mechanical properties of neodymium- and chromium-doped gadolinium scandium gallium garnet,” J. Opt. Soc. Am. B 3, 102–114 (1986). [CrossRef]

8. J. Vetrovec, “Active-mirror amplifier for high average power,” Proc. SPIE 4270, 45–55 (2001). [CrossRef]

Resonator length	L	[m]	1.0
Active medium diameter	D_a	[m]	0.21
Active length	l	[m]	0.10
Active medium thickness	t	[m]	2.50×10^-3
Toroidal mirror diameter	D_M	[m]	0.22
Cone height	H	[m]	2.55×10^-3

Doping density(Nd)	N	[1/cm³]	1.0×10²⁰
Doping density(Cr)	N	[1/cm³]	1.0×10²⁰
Stimulated emission cross section	σ	[cm²]	3.2×10^-19
Lifetime of upper laser level	t_sp	[μs]	240
Loss coefficient	α	[1/cm]	2×10^-3
Refractive index	n		1.97

Specific heat	C_p	[kJ/kgK]	0.4029
Density	ρ	[kg/m³]	6.495×10³
Thermal conductivity	κ	[W/mK]	6.02
Thermal stress resistance	R_T	[W/m]	660

Resonator length	L	[m]	1.0
Active medium diameter	D_a	[m]	0.21
Active length	l	[m]	0.10
Active medium thickness	t	[m]	2.50×10^-3
Toroidal mirror diameter	D_M	[m]	0.22
Cone height	H	[m]	2.55×10^-3

Doping density(Nd)	N	[1/cm³]	1.0×10²⁰
Doping density(Cr)	N	[1/cm³]	1.0×10²⁰
Stimulated emission cross section	σ	[cm²]	3.2×10^-19
Lifetime of upper laser level	t_sp	[μs]	240
Loss coefficient	α	[1/cm]	2×10^-3
Refractive index	n		1.97

Feasibility study of a conical-toroidal mirror resonator for solar-pumped thin-disk lasers

Abstract

1. Introduction

2. Numerical model

3. Results of calculations

3.1. Bare cavity calculation

3.2. Loaded cavity calculation

3.3. Thermomechanical consideration

3.4. Misalignment assessment

4. Conclusions

Acknowledgment

References and links

Cited By

Figures (10)

Tables (3)

Equations (20)

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