Miter bend polarizers in CW ECRH systems: groovedesign, ohmic loss, and cooling in numericalanalysesJawad Taheri ( javad.taheri.99@gmail.com )

Nuclear Science and Technology Research InstituteDavoud Iraji

Amirkabir University of Technology

Research Article

Keywords: grating polarizer, groove design, high-power ECRH system, ohmic loss, water cooling

Posted Date: August 19th, 2022

DOI: https://doi.org/10.21203/rs.3.rs-1952720/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

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Miter bend polarizers in CW ECRH systems: groove design, ohmic loss, and

cooling in numerical analyses

Jawad Taheri1,*, Davoud Iraji2,*

1 School of Plasma Physics and Nuclear Fusion, Nuclear Science and Technology Research Institute, Tehran,

Iran.

2 Department of Energy Engineering and Physics, Amirkabir University of Technology, Tehran, Iran.

*corresponding authors: javad.taheri.99@gmail.com, iraji@aut.ac.ir

Abstract

Polarizer miter bends are important components in the transmission lines of electron cyclotron

resonance heating systems, as they create a desired polarization for the electron cyclotron wave before

coupling in the fusion plasma. Normally, a pair of grooved mirrors with different groove depths is

enough to form a universal polarizer made of polarization twister and elliptical polarizer. The groove

shape, and how the grooved mirror tolerates in a modern high-power system with continuous wave

operation, can influence the entire operation of the system. This work contains various investigations

on the grooved mirror including the parametric groove design, low ohmic loss, and effective cooling

with an appropriate design of the cooling channel. Since the rounded trapezoidal grooves have some

advantages over rectangular and sinusoidal grooves, the study is mainly dedicated to the trapezoidal

grooves in both polarizer mirrors. The steady-state thermal analysis is carried out for 1 MW power at

110 GHz.

Keywords: grating polarizer, groove design, high-power ECRH system, ohmic loss, water cooling.

1. Introduction

To enhance the electron cyclotron resonance heating and current drive (ECRH & CD) of plasma at a

desired position, a particular elliptical polarization is required to be induced in the millimeter wave.

Normally this can be achieved by a pair of rotatable grooved mirrors as grating polarizers with different

groove depths installed into successive miter bends of the transmission line. One of them has the groove

depth that corresponds to λ/4 and the other has the grooved depth corresponding to λ/8 where λ is the

wavelength [1-3]. The former is referred to as polarization twister (PT) and the latter is referred to as

elliptical polarizer (EP) in this study.

The groove profile is also important when taking into account the arc discharge on the sharp edges of

the mirror corrugations, especially in high-power transmission lines. Therefore, it is recommended to

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round the edges of the corrugations (or grooves). In high-power systems, the operation can be also

limited by the high ohmic loss on the grooved mirrors as the grooves and their orientation with respect

to the incident polarization can lead to a significantly higher loss than a plane mirror. So with the

development of high-power, long-pulse gyrotron in modern systems such as ITER ECRH/ECCD with

power up to 1 MW, the loss in the transmission line components cannot be neglected [4, 5]. In such

systems, active water cooling is imperative for the compact polarizer miter bends.

This paper not only investigates the groove profile on the polarizer mirrors but also studies the ohmic

loss on the mirrors and effective active cooling for the ohmic heat dissipation. Extensive work is done

to design optimum profiles that are parametric based on the wavelength and can create suitable

polarization for each polarizer. The parametric feature makes the profiles to be adopted for any single

frequency transmission line. The proposed groove profiles are in trapezoidal shapes with rounded edges.

Since a rectangular groove is a special case of trapezoidal grooves where the sidewalls are parallel, the

analysis also includes the rounded rectangular shapes. We used numerical simulations integrated with

vector theory for diffraction gratings. The simulation parts are performed in COMSOL and the vector

theory is developed in MATLAB. Then the loss on the mirrors with the suitable grooves is calculated

by the numerical simulation. This analysis together with water cooling analysis is done for a continuous

wave (CW) operation with 1 MW power at 110 GHz.

2. Groove design

2-1. Calculation method for polarization parameters

The geometry of a rotatable polarizer grooved mirror in a miter bend is shown in Fig. 1Error!

Reference source not found.. The incident angle is θ =45° to the mirror normal axis, and the mirror

rotation angle is denoted by ϕ. When the electric field of a linearly polarized incidence is perpendicular

to the incidence plane it is called the H-plane incidence, and the electric field parallel to the incidence

plane defines the E-plane incidence. The polarization states of the reflected wave can be characterized

by two parameters: the rotation angle, α, and the ellipticity, β, where −45° ≤ β ≤ 45°. Circularly

polarization is defined by |β| = 45°, while linear polarization is defined by β = 0°. At the grooved

surface, the incident wave field is divided into two orthogonal components where the component

parallel to the grooves reflects at the top surface of corrugations while the component perpendicular to

the grooves penetrates the groove and reflects at the bottom and therefore the phase shift, τ, is made.

The elliptical form of waves comes from the relative phase shift of the reflected field components. For

a normal incidence on a rectangular grooved surface, τ = 2kd = 4πd/λ where d is the groove depth

and k is the wavenumber. In a such case, the groove depth d = λ/4 corresponds to a linear polarization

with τ = π while d = λ/8 corresponds to a circular polarization with τ = π/2.

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Fig. 1. Polarizer geometry in a miter bend (θ = 45°). When the orientation of the grooves

is perpendicular to the incidence plane, ϕ = 0°. kr is normal to the polarization ellipse.

In non-normal incidence on a non-rectangular grooved surface, obtaining τ is much more complicated.

By the numerical simulation (which solves Maxwell’s equations) the reflected field can be obtained and

then by the relation

[Er𝜃Erϕ] = [cos ξsin ξ −sin ξcos ξ ] [eiτ 2⁄0 0e−iτ 2⁄ ] [ cos ξ−sin ξ −sin ξ− cos ξ] [Ei𝜃Eiϕ] (1)

where ξ = atan (cosθ tanϕ) ≈ atan (0.7 × tan ϕ), and using the integral method in the vector theory

of diffraction gratings discussed in [1, 6-8] τ can be found. When τ is found then the polarization

parameters (α, β) of the reflected wave can be obtained by: 𝛼 = 12 atan(tan 2γ × cos δ) (2) β = 12 asin(sin 2γ × sin δ) (3)

where γ = atan(|Erϕ|/|Erθ|) , and δ = δϕ − δθ = arg (Erϕ) − arg (Erθ) is the phase difference

between the reflected field components, inducing the ellipticity.

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2-2. Parametric Grooves in the PT and EP mirrors

In profiles of trapezoidal grooves, both the groove width, w, and corrugation thickness, t, have mean

values as shown in Fig. 2, and by introducing the sidewall slant parameter, c, the deviation from the

rectangular-shaped groove is characterized, where c = 0 gives a rectangular groove. Therefore, the

sidewall slop can be obtained by 𝑎𝑡𝑎𝑛 ( 𝑑2𝑐). The fillet radii on the rounded edges are defined by r.

Fig. 2. Design parameters of a trapezoidal groove used in the simulation. a)

basic trapezoidal profile, b) trapezoidal profile with rounded edges.

For the polarization analysis, we can use the Floquet theory to take advantage of the periodic structure

of the grooved surface on the mirror. Therefore, the model includes a segment of the groove or

corrugation on the mirror as well as the air bar as shown in Fig. 3. The lengths of the air bar and the

perfect-matched layer (PML) of the model are 7λ and 2.5λ respectively, and the length and width of

the groove/corrugation are λ and p respectively. The maximum mesh element size is 0.2λ ( in quadratic

type mesh). We can also use a smaller bar to reduce the computation time, but the largest part of the

computation is contributed to the groove area rather than the bar. The model can rotate around its central

axis by an increment of 3° representing the mirror rotation (ϕ) and it assumed the H-plane incidence.

The incidence angle is fixed at 45° to the normal axis of the port. In the miter bend where the incident

angle is 45°, the periodicity of the grooves on the mirror must satisfy the condition p < 0.586λ to

suppress high-order modes of the reflected wave.

In Fig. 3 three different bars are provided for three different rotation angles. Every bar demonstrates

three regions each with a local contour plot to show the details of the field distribution in each region.

Region 1 contains incident and reflected beam and is dominated by the linearly polarized incidence,

while region 2 represents only the reflected waves in elliptical (or rotational) polarization. The Floquet

port in the middle of the model separates the incident beam from the reflected beam. Region 3 is the

PML added to absorb the reflected beam. All the values are normalized by the field amplitude of region

1 at ϕ = 90°. That means at ϕ = 90° the field reaches its maximum value somewhere around the

groove. This is shown in Fig. 3 by a magnified figure of the groove where it is obvious that the field is

concentrated at the corrugation edges.

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Fig. 3. Electric field distribution of the microwave beam after being reflected by the

corrugation/groove. The contour shown on each bar is not global and each region has its unique

legends.

For designing the groove in the PT mirror, first, the general condition p < 0.586λ = p0 has to be

satisfied so the grating reflects the wave as a plane mirror, and the groove depth d ≈ λ/4 (for normal

incidence) has to be modified for an incident angle of 45° as d0 = (λ/4)/cos (π 4⁄ ) ≈ (λ/4)/0.7

where d0 and p0 are used for the parametric design in this study. Since d0 and p0 are parametric based

on the wavelength, the results in this section (obtained for 110 GHz) can be applied to any millimeter

wave frequency.

The analysis aims to find the maximum value of the polarization ellipticity (β) created by each profile

and then find the most suitable profiles for each polarizer. Maximum β is usually obtained at two

different rotation angles of the mirror with positive and negative values corresponding to the direction

of the wave rotation. Since the periodicity does not considerably affect the maximum ellipticity (as long

as p < p0) [9, 10], a relatively large period is considered for the groove design. It is recommended to

use w = 0.6p or w = 0.7p for trapezoidal grooves because, in general, wider grooves are relatively

easier to fabricate, and the grooves with w = 0.8p and higher widths result in too thin corrugations,

making it hard to apply good fillets on its edges due to the overlapping issues, especially for smaller

periodicity and bigger c. Moreover, wider grooves cause lower loss compared to the narrow grooves in

the E-plane incidence [11] (for the H-plane incidence this will be discussed in this study).

In Table 1, the effect of the groove depth, width, and sidewall slant parameter on the maximum β is

investigated (values are rounded to the nearest integer). All the profiles have p = 0.9p0 and each profile

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has two values of the maximum β corresponding to the two different widths: The first value is for w =0.6p while the second is for w = 0.7p. The fillet radius on the edges is set as follows: For c = 0

(rectangular) we used r = 0.07λ (for either w = 0.06p and w = 0.07p). In other profiles with non-

zero c, again for w = 0.06p we used r = 0.07λ and for w=0.7p we used r = 0.05λ. This is because

when w and c increase, applying the bigger fillet encounters overlapping issues. We observed that

changing the fillet radius from 0.05λ to 0.07λ does not change the result by more than three degrees.

The rounding effect on the maximum β is more or less similar to the effect of the sidewall slant

parameter.

Table 1. Maximum ellipticity during 180° rotation of the PT mirror having rounded trapezoidal grooves with p =0.9p0 and w = (0.6p, 0.7p) c = 0.25t c = 0.2t c = 0.15t c = 0.1t c = 0.05t c = 0

27, 25 24, 22 20, 20 17, 18 14, 15(IV) 12, 16 d = 0.9d0

18, 16 14, 13 11(II), 11(I) 14, 11 17, 13 20, 13(III) d = d0 13, 12 15, 13 18, 15 22, 18 24, 20 28, 20 d = 1.1d0 20, 18 23, 20 26, 23 32, 25 33, 28 37, 28 d = 1.2d0

In this study the lowest value of the maximum β during 180° of mirror rotation appeared to be ~10°,

therefore, the profiles that give a maximum β within 10°-15° can be candidates for PT. It can be seen

that, in each row or column, the maximum β goes to the lower or upper limit, 10°and 45° respectively,

with a routine and then comes back. By using the logical patterns between the data in each row or

column of Table 1 one can predict the value in the next cell or use interpolation to reach the desired

result. These logical patterns may be more important than the values themselves as they can provide a

helpful insight into groove design based on the previous results.

For designing the groove in the EP mirror, we now use d0 ≈ (λ/8)/0.7 and look for the groove shapes

that give a maximum β within 40°-45°. The effect of each groove parameter on the maximum β is

investigated for the EP and reported in Table 2 by using the approaches applied in the PT groove design.

Table 2. Maximum ellipticity during 180° rotation of the EP mirror having rounded trapezoidal

grooves with p = 0.9p0 and w = (0.6p, 0.7p) c = 0.25t c = 0.2t c = 0.15t c = 0.1t c = 0.05t c = 0

27, 28 28, 30 30, 31 31, 32 32, 34 33, 32 d = d0 31, 33 33, 35 35, 36 36, 37 38, 38 39, 37 d = 1.1d0 36, 37 38, 39 39, 41 41, 42 43, 44 45, 42 d = 1.2d0

7

40, 42 41,43 44 (VI), 44(V) 43, 42 41, 41 39, 42 d = 1.3d0

45, 43 43, 41 41, 39 38, 37 36, 35 33, 37 d = 1.4d0

Here again, the first result in each cell is for w = 0.6p while the second is for w = 0.7p and the

rounding of edges is similar to that of the PT grooves. In Table 2, some of the profiles give the maximum

β within 40°-45° which can be considered for the EP. The polarization parameters of all the profiles in

column c = 0.15t in Table 1 and a profile with c = 0.15t and d = 1.3d0 in Table 2, all with w = 0.7p

(second values in each cell) are shown in Fig. 4. By comparing the β graphs, it can be seen how the

graph changes when the maximum β goes near its lower limit, and then two other extrema (positive and

negative) appear at two different angles as the varying parameter (d) increases.

All the results obtained for both polarizers are valid for the smaller periodicity of the grooves. However,

shorter periods may increase the loss as the number of grooves increases. This can draw our attention

to an important advantage of the trapezoidal grooves with a larger duty cycle (w/p) over sinusoidal

grooves.

Fig. 4. The polarization parameters made by the PT and EP trapezoidal grooves. All

the grooves have w=0.7p and c=0.15t but different depths.

2-3. The field in the grooves

As a preliminary investigation of the ohmic loss analysis, the electric field strength in the suggested

grooves for PT and EP is calculated here to get an opinion on the loss created by each groove. If we

consider the loss mainly as a result of the wall current induced by the waves in the grooves, similar to

the standing wave between walls, so we can assume the higher field in the grooves induces higher loss.

In Fig. 5 the normalized field strength in the middle of the grooves with suggested profiles from Table

1 and Table 2 is depicted. The graphs are obtained using the Floquet model shown in Fig. 3. The

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suggested groove profiles in Table 1 and Table 2 are denoted by the Latin numbers (Profile III is the

rounded rectangular groove).

Fig. 5. Electric field strength at the center of grooves (d/2, w/2) obtained by the simulation using the Floquet theory. Four

groove profiles are from Table 1 (PT) and two profiles are from Table 2 (EP)

By comparing the graphs shown in Fig. 5 some conclusions can be made:

At the H-plane incidence, the field in the grooves is maximum at ϕ = 90°, so in this position, the

ohmic loss is expected to be maximum.

For different grooves, the one with the strongest field in the groove at ϕ = 90° has the weakest

field at ϕ = 0°. This happens because when the electric field and the groove orientation are

perpendicular to the incidence plane, the field cannot penetrate much the grooves due to the cutoff

frequency condition, so the field in the groove is the weakest. Thus more penetration in the

situation where the grooves are parallel to the incidence plane means a higher cutoff when the

grooves are perpendicular to the incidence plane.

For identical grooves with different widths, the one with larger width causes a weaker field in the

groove (this is also examined for smaller widths to make sure).

For identical grooves with different sidewall slant parameters, the one with a higher slant (higher

c) causes a weaker field in the groove.

For identical grooves with different depths, the one with a smaller depth causes a weaker field in

the groove.

It is worth reminding that the above conclusions are made for the H-plane polarization and the field in

the groove is evaluated at the center of grooves (d/2, w/2). Different grooves also have different field

gradients in the grooves. Considering the first conclusion above, the following loss analysis has been

carried out just for the mirror rotation angle of 90°.

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3. Ohmic loss and cooling considerations

The ohmic attenuation of a reflected microwave by a plane mirror at normal incidence can be obtained

by Pa = 4(Rs Z0⁄ ) where Rs = ρ/δ = √πρZ0/λ is the surface resistance, Z0 is the characteristic

impedance of free space (377 Ω), ρ is the resistivity, and δ is the skin depth. The incident angle of θ to

the mirror normal axis affects the absorbed power by 𝑃𝑎𝑐𝑜𝑠𝜃 for the H-plane configuration and 𝑃𝑎/𝑐𝑜𝑠𝜃 for the E-plane configuration [12]. In the case of a grooved mirror, the grooves and their

orientation with respect to the incident polarization can significantly increase the loss, especially when

the skin depth is smaller than the surface roughness.

The simulation here assumed a sawtooth model of the surface roughness with a root mean square value

of 1 µm (bigger than the skin depth, which is ~0.2 µm at room temperature). In a such model, the surface

resistance is increased by the surface roughness as [13, 14]: Rs.r = Rs.s [1 + 2π atan (1.4 (Δδ)²)] (4)

where Rs.r and Rs.s are the rough and smooth surface resistance respectively, Δ is the root mean square

value of the surface roughness and δ is the skin depth.

For thermal analysis related to the ohmic loss, the whole miter bend should be modeled with the inlet

and outlet waveguides (miter bend arms) as shown in Fig. 6. The waveguides are corrugated with a

37.75 mm diameter designed for 110 GHz. The power incident on the grooved mirror is ~1 MW in the

gaussian HE11 mode with the H-plane polarization. The polarizer mirror is a CuCrZr shaft with a re-

entrant cooling channel as shown in Fig. 7. In COMSOL, the selected material for the mirror has

temperature-dependent resistivity which counts for the ohmic loss calculation.

Fig. 6. The polarizer miter bend with the mirror rotation angle of 90˚.

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Fig. 7. Simulation model of the grooved mirror made of C18150 copper alloy with a re-entrant cooling

channel. The mirror diameter and height are 41.75 mm and 26 mm respectively. The maximum mesh size on

the grooved surface is 0.1λ. The maximum mesh sizes of the channel walls are 0.5 mm on the bottom and 1

mm on the sides.

The grooved mirror for either PT or EP is in ϕ = 90° position where the grooves are parallel to the

incidence plane. The groove profiles I, II, III, and IV for the PT, and profiles V and VI for the EP have

been used and the resultant losses have been studied. Another crucial consideration for the polarizer in

the high-power CW operation that should be taken into account is the cooling channel dimensions

designed carefully and effectively. In this study, all the possible effective parameters of the channel

geometry are examined for heat dissipation. All the important dimensions are shown in Fig. 7. By

performing various simulations, we found that H is the most influencing dimension on the cooling since

it can force the flow towards the bottom wall of the channel and increase its velocity upon the wall. For

more efficiency, it is better to expand Do as large as possible. This was to be expected since increasing Do expands the cooling surface of the bottom wall which is the most area of the cooling. For the model

in this study, we found that the best size for Di and K is 10 mm and 5 mm respectively, and smaller or

bigger sizes decrease the heat dissipation. Finally, the best dimensions were obtained as follows: Do =30 mm, Di = 10 mm, G = 3 mm, H = 3 mm, and K = 5 mm. We used these values for the following

simulations. The size of G, however, can be slightly bigger (4 or 5 mm) without affecting the cooling.

In Table 4 the maximum and average temperatures as well as the surface ohmic loss on the grooved

surface of the mirror with different groove profiles are reported (temperatures are rounded to the nearest

five). The first four profiles are from Table 1 (PT) and the last two are from Table 2 (EP). The flow rate

is 20 lit/min with an inlet temperature of 20 . The water outlet temperature is obtained within 30-40 , and the pressure loss is 0.06 bar.

11

Table 3. Ohmic loss, maximum and average temperature (in ) on the grooved

surface of polarizer mirrors with different groove profiles from Table 1 & Table 2,

radiated by 1 MW CW gaussian beam at 110 GHz

T (ave.) T (max.) Loss (%)

Profile I 325 765 1.14

Profile II 390 855 1.4

Profile III 355 850 1.26

Profile IV 325 760 1.14

Profile V (EP) 205 420 0.61

Profile VI (EP) 215 460 0.7

Table 3 shows that in the H-plane configuration, the wider grooves (w = 0.7p) produce less ohmic loss

compared to the narrower grooves (w = 0.6p) with the same other groove parameters, and consequently

experience lower temperature. The shorter groove depths can reduce the loss, and as was also expected

from the previous section, higher values of c produce a lower loss. This is why profile I produces lower

loss than profile III. As a general explanation, we can say that when the sidewall slant parameter c, is

higher, the field in the grooves can be reduced due to the outward reflection, and consequently, the

induced tangential current on the walls is smaller. A higher sidewall slant factor can also offer better

heat conduction from the corrugation ridge to the mirror interior region. It means that besides the higher

mechanical stability, the trapezoidal grooves are likely to have higher thermal tolerance than the

rectangular grooves. So, groove shapes I and IV for the PT, and V for the EP suggest very good thermal

stability in the high-power CW systems (profile I is shown in Fig. 3).

In Fig. 8 the loss and temperature distribution on the PT mirror are shown. It is noticeable that the

maximum temperature always occurs on the ridges of corrugations where the conductive heat transfer

is lower than the bottom part and therefore the loss on the rounded edges becomes higher due to the

increased resistivity. Rounding the corrugations can moderate the very high temperature on the edges

where the loss is concentrated. in Fig. 9, two cross-section contours are provided to show the mirror

interior temperature and water flow velocity. The peak velocity of flow occurs on the bottom wall of

the channel where the channel dimension H (shown in Fig. 7) plays a key role in adjusting this velocity

which affects the cooling of the wall.

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Fig. 8. Temperature on the mirror (a), and the grooved surface ohmic loss (b).

Fig. 9. Cross-sectional view of the mirror interior temperature (a) and the flow velocity (b).

As a more investigation, the effects of dimension H and flow rate on the cooling of the mirror with

groove profile I are provided in Table 4. By increasing the flow rate from 20 lit/min to 30 lit/min, the

average cooling on the grooved surface enhances by about 17% while the cooling on the ridge of the

corrugations enhances by 10%. For H = 3 mm, the peak velocity of the 30 lit/min flow in the channel

(on the bottom wall) is near 10 m/s, and the pressure loss is still as low as 0.15 bar. For H = 4 mm, the

velocity on the channel bottom wall is lower compared to H = 3 mm, therefore, the cooling is less

efficient. The melting point of C18150 is within 1050-1100 . So the corrugations with profiles I and

IV while experiencing high temperature, are within the safe thermal limit.

Table 4. Temperature (in ) on the grooved surface of the PT with groove profile I from Table 1

20 lit/min 30 lit/min

H=3 mm 300 (ave.), 705 (max.) 250 (ave.), 630 (max.)

H=4 mm 340 (ave.), 790 (max.) 280 (ave.), 710 (max.)

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4. Conclusion

In this study, the rounded trapezoidal grooves in the grating polarizers were designed based on the

wavelength, so they can be adapted for any frequency of high-power ECRH systems. The grooves are

studied for their output polarization, suitable for the PT and EP, using numerical simulation and vector

theory for diffraction gratings. Then we considered different groove profiles with desired polarization

for the ohmic loss analysis. The analysis is carried out for 1 MW power of CW at 110 GHz while the

incidence polarization was in the H-plane configuration and the orientation of the grooves was parallel

to the incidence plane. The main purposes of the analyses were to see how the groove width and sidewall

slant factor as well as the groove depth affect the ohmic loss and to what level the resultant temperature

on the grooved surface increases. The surface roughness on the grooved surface was assumed to be 1

µm. According to the results, to increase the safeness of a high-power CW operation, it is recommended

to consider wider grooves, with slanted sidewalls, shorter groove depths, as well as larger allowed

periodicity to significantly decrease the loss of the reflected beam on the mirrors. The mirrors had an

integrated water channel with optimum dimensions to dissipate the heat in the steady state operation.

Declarations

Ethical Approval: Not applicable.

Competing interests: The authors have no competing interests.

Authors' contributions: The manuscript is written by Jawad Taheri and reviewed by Davoud

Iraji.

Funding: There was no specific funding for this study.

Availability of data and materials: The authors confirm that the data supporting the findings

of this study are available within the article.

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