---
title: "A new method for increasing the laser damage threshold of metal mirrors"
authors: ["V.N. Konopsky"]
affiliation: "Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region 142092, Russia"
journal: "Optics & Laser Technology"
year: 2000
volume: "32"
pages: "15-21"
doi: "10.1016/S0030-3992(00)00011-6"
type: journal-article
site_group: "Photonic surfaces and metal mirrors"
url_abstract: ""
url_pdf: "https://valery.konopsky.com/additional_pdf/A new method for increasing the laser damage threshold of metal mirrors.pdf"
language: en
source_tex: ""
source_pdf: ""
---
## Abstract

In order to increasing the damage threshold of metal mirrors we propose
to create a special structure on the surface of the mirrors ("photonic
surface"). This structure must have the period about $\lambda /2$ and
will suppress propagation of surface plasmons with the frequency
$\omega_0 = 2\pi
c/\lambda$ along the surface. This structure also will slightly
increase the heat removal from the mirror’s surface by the excitation of
the thermostimulated plasmon emission from the surface. The heat removal
from the surface is estimated and possible implementation of this
approach for the radiation of $\rm CO_2$–lasers
($\lambda=10.6~\mu$m) and Nd-YAG–lasers ($\lambda=1.06~\mu$m) is
analyzed.

*Keywords: Photonic bandgap materials, Plasmons, Laser mirrors.*

# Introduction

Laser damage of metal mirrors is a serious problem in high-power
infrared laser technology. The damage of metal mirrors often accompanied
by developing of spontaneous, highly periodic surface structure or
"ripples" . These structures are reversible when the laser energy is
below the damage threshold and became permanent at high laser energy .

The processes of ripple formation show the important role of surface
plasmons (SPs) in the laser damage of metal mirrors. The SP is a
fundamental electromagnetic excitation mode of a metal-dielectric
interface . According to present understanding the surface ripples
appear due to interference of incident laser beam and SPs caused by
random initial disturbances in the surface properties of the illuminated
surface. These transverse variations may be separated analytically into
individual spatial frequency components or sinusoidal surface grating
along the surface. Incident laser beams can excite the traveling SP’s
through appropriate spatial frequency components of surface
imperfections, and interference between the incident beam and the
surface wave lead to spatially varying the total light intensity with
the same period as its appropriate spatial frequency component of the
surface imperfections. This intensity variation produces a growth rate
for the surface imperfections, increasing the amplitude of its relevant
spatial component of the surface variation, which will lead to increased
the coupling from the laser beam into surface wave and so on.

The propagation direction of the SPs and therefore the orientation of
the ripples is determined by the polarization of the incident laser
beam.

In recent years the optical properties of materials that possess a
periodic modulation of their refraction index on the scale of the
wavelength of light have received much attention . Such materials can
exhibit photonic band gaps that are much like the electronic band gaps
for electron waves travelling in the periodic potential of the crystal.
In both cases frequency intervals exist where wave propagation is
forbidden. Materials where band gaps take place for propagation of bulk
light waves are called "photonic crystals". Surfaces that produce such a
band gap for propagation of SPs waves are called "photonic surfaces" .
For a single corrugation of the surface the plasmon band gap will extend
only over a limited propagation range, but for more than one simple
corrugation of the surface the suppression of SP propagation in all
possible directions is possible . Barnes et al. have used the photonic
surface to block a decay channel of fluorescent molecules into SP modes
near a metal surface .

In this paper we examine the possible uses of the photonic surface for
increasing the damage threshold of metal mirrors. A photonic surface on
a metal mirror will suppress the propagation of SPs caused by surface
imperfections and will slightly increase the heat removal from the
surface through thermostimulated plasmon emission.

# Photonic band gap for SPs caused by surface imperfections

At first we consider a single corrugation of the surface with the Bragg
vector $\vec g$ and a pitch $\Lambda$ such that
``` math
\begin{eqnarray}
 g&=&2k_{\mathrm SP}\nonumber\\
\Lambda&=&{\lambda_{\mathrm SP}\over 2} \; .

\end{eqnarray}
```
The SP wavevector $k_{\mathrm SP}$ is 
``` math
\begin{equation}
k_{\mathrm SP}= k_0\left({\varepsilon_{\mathrm M}
\over \varepsilon_{\mathrm M}+1}\right)^{1/2} \; ,

\end{equation}
```
where $k_0=\omega_0 /c$ is the wavevector of the incident radiation
and $\varepsilon_{\mathrm M}$ the dielectric constant of the metal at
the given light frequency.

For SPs with $\vec k_{\mathrm SP}\parallel\vec g$, Bragg scattering on
the grating takes place. The two counter-propagating SP modes set up a
standing wave and, owing to the different surface charges and field
distributions on the grating surface, associated with the two
standing-wave solution, a bandgap in the dispersion of the mode opens
up . The presence of a gap in the dispersion relation of SP propagating
along a periodic surface has been known for a long time . On Figure 1
one can see the SP dispersion curve for a flat surface (dashed line) and
for the corrugated surface with $\Lambda=\lambda_{\mathrm SP}/2$
(solid line).

The bandgap width $\Delta\omega$ for small corrugation amplitude $h$
($h<\Lambda$) is 
``` math
\begin{equation}
 {\Delta\omega\over\omega_0}={2\pi
 h\over\Lambda\sqrt{|\varepsilon_{\mathrm M}|}}= {4\pi
h\over\lambda\sqrt{|\varepsilon_{\mathrm M}|}} \; .

\end{equation}
```
A plasmon bandgap occurs not only for SPs propagating in the direction
normal to the corrugation grooves, but also for SPs propagating in the
direction range $-\psi<0<\psi$, where the angle $\psi$ is 
``` math
\begin{equation}
\psi\simeq\arccos{\left(
{
2-{\Delta\omega/\omega_0}
\over
2+{\Delta\omega/\omega_0}
}
                   \right)}\; .

\end{equation}
```
To achieve a total suppression for the SPs propagation across the
surface in all possible directions more than one set of corrugations is
required. For hexagonal arrangement of photonic surface, that is, for
three gratings at $60^0$, $\psi$ must be $\geq 30^0$, and
consequently $\Delta\omega/\omega_0$ must be $\geq 0.14$. Such
values may be achieved in visible and near-infrared regions and not be
practicable for $\lambda\sim 10.6~\mu$m.

For numeric estimations we will used the data for two wavelength
regions: $\lambda=10.6~\mu$m ($\rm CO_2$–lasers) and
$\lambda=1.06$ mkm (Nd-YAG–lasers). The reasons for the choice of
these two regions are as follows: the reflection of the metal mirror
with refraction index $\eta= n + i\kappa$ for normal incident is
``` math
\begin{equation}
R= {(n-1)^2+\kappa^2\over (n+1)^2+\kappa^2} \; .

\end{equation}
```
One can see that the employment of metals as very high reflection
mirrors are possible when $n\gg 1$; $\kappa\gg
1$, and when $n\ll 1$; $\kappa\gg 1$. The former case is realized
in the middle and far infrared regions ($n_{\mathrm
Au,Ag,Cu}\geq 12$; $\kappa_{\mathrm Au,Ag,Cu}\geq 70$, for
$\lambda\geq 10.6$ $\mu$m ), and the later case is realized in the
visible and near infrared regions for gold and silver
($n_{\mathrm Ag}\leq 0.05$; $\kappa_{\mathrm
Ag}\sim 5$, $n_{\mathrm Au}\leq 0.2$; $\kappa_{\mathrm
Au}\sim 4$, for $\lambda\simeq
0.63$ and $1.06$ $\mu$m ).

So for $\lambda= 1.06$ $\mu$m, where
$\varepsilon_{\mathrm Ag}\simeq -60+ 0.6 i$
($\varepsilon_{\mathrm Au}\simeq -51+ 3.9 i$) the bandgap
$\Delta\omega$ reaches the required value of $14\%$ of the central
frequency $\omega_0$ at $h/\lambda\simeq 0.09 (0.08)$ and full
photonic surface creation for this wavelength seems quite possible.

But for $\lambda= 10.6$ $\mu$m ($\varepsilon_{\mathrm
Au,Ag,Cu}\approx -5000+ 2000 i$) the bandgap $\Delta\omega$ reaches
the required value of $14\%$ at $h/\lambda\sim 1$, according to
equation (<a href="#4" data-reference-type="ref" data-reference="4">[4]</a>).
This means that the approximation of small corrugation amplitude is
broken-down and we cannot use the
equation (<a href="#4" data-reference-type="ref" data-reference="4">[4]</a>).

If, for $\lambda= 10.6$ $\mu$m, we use a single corrugation with a
reasonable amplitude $h\simeq 1$ $\mu$m ($h/\lambda\simeq 1/10$)
the bandgap $\Delta\omega/\omega_0$ will be about $0.017$ and the
maximum range of $\psi$ will be only $\simeq\pm 10.57^0$. But the
suppression of the SP propagation even in such a narrow range may be
very useful for *linear* polarized laser radiation at a colinear
corrugation and polarization orientation (the grating vector $\vec g$
must be parallel of the $E$ field direction in the surface plane).

It is well known that the propagation direction of SPs and the
orientation of the grating vectors $\vec g_{\mathrm
ripples}$ of the formed ripples are confined to a narrow range of
angles very accurately parallel to the $E$ field direction (and only
for a very high-intensity condition, for single-shot macroscopic damage
of the surface the $\vec
 g_{\mathrm ripples}$ directions distributed over $5^0$ or $10^0$
about the $E$ field direction) .

So even such a narrow bandgap ($\Delta\omega/\omega_0\simeq
 0.017$) in one direction of the SPs propagation may increase the
damage threshold of metal mirrors and therefore this approach may be
used even for $10.6$ $\mu$m radiation.

In the rest of this section we compare the band gap width,
$\Delta\omega/\omega_0$ with the width of the plasmon curve,
$\delta\omega/\omega_0$, which arises due to SP dissipative damping.
The ratio $\Delta\omega/\delta\omega$ may be used for a quantitative
estimation of the decreasing of the SP intensity on the "photonic
surface". Width $\delta\omega$ may be estimated as
``` math
\begin{equation}
\delta\omega/\omega_0={{\mathrm
Im} \left(\sqrt{{\varepsilon_{\mathrm M} \over
\varepsilon_{\mathrm M}+1}}\right) \over {\mathrm
Re}\left(\sqrt{{\varepsilon_{\mathrm M} \over
\varepsilon_{\mathrm M}+1}}\right) } \; ,

\end{equation}
```
where Re() and Im() are real and imaginary parts of the function
respectively.

Expanding $\varepsilon_{\mathrm M}= \varepsilon'+i\varepsilon''$, and
assuming that $|\varepsilon''|\ll
|\varepsilon'|$, one can obtain :
``` math
\begin{eqnarray}
{\mathrm Re}\left(\sqrt{{\varepsilon_{\mathrm M}
\over \varepsilon_{\mathrm M}+1}}\right)&=&
\left({\varepsilon'
\over \varepsilon'+1}\right)^{1/2}\nonumber\\
{\mathrm Im}\left(\sqrt{{\varepsilon_{\mathrm M}
\over \varepsilon_{\mathrm M}+1}}\right)&=&
{\varepsilon''\over 2 \varepsilon'^2
}\left({\varepsilon' \over \varepsilon'+1}\right)^{3/2}\; ,

\end{eqnarray}
```
and
``` math
\begin{equation}
\delta\omega/\omega_0=
{
{\varepsilon''
\over
2 \varepsilon'
(\varepsilon'+1)}
}\; .

\end{equation}
```
Therefore (for pointed above values of $\varepsilon'_{\mathrm Ag}$ and
$\varepsilon''_{\mathrm
Ag}$) $\delta\omega/\omega_0\simeq 4\cdot 10^{-5}$ and
$\Delta\omega/\delta\omega\simeq 430$ for $\lambda= 10.6$ $\mu$m;
and $\delta\omega/\omega_0\simeq
8.5\cdot 10^{-5}$ and $\Delta\omega/\delta\omega\simeq 1900$ for
$\lambda= 1.06$ $\mu$m (we have taken $\Delta\omega$ at
$h/\lambda = 1/10$).

# Heat removal from the mirror through emission of thermostimulated plasmons

The incident laser radiation will not "see" the existence of the grating
on the photonic surface, because the grating pitch $\Lambda<\lambda/2$
(for normal incidence an even weaker condition $\Lambda<\lambda$ is
sufficient). But this surface structure will act as a common diffraction
grating for thermostimulated surface plasmons (TSPs) with
$\lambda_{\mathrm TSP}>2\Lambda$.

TSP is the SP excited by the heat energy of the body (see  and cites
therein). The TSP is also (like a SP) nonradiative electromagnetic mode
on the *flat* surface. On the *corrugated* surface part of the energy of
the TSP will be extracted into vacuum above the surface.

When the temperature of the grating is equal to the temperature of the
environment, the TSPs emit and absorb equal amount of the energy. But
when the temperature of the grating is increased (due to absorbtion of a
part of the laser radiation) heat removal from the grating through TSP
emission takes place.

In the Appendix we derive that the radiative heat removal from the flat
mirror may be estimated as:
``` math
\begin{eqnarray}
P_{\mathrm M}^{\mathrm flat}(T) & \simeq
\sigma T^4
{ \gamma\over \omega_{\mathrm p}}
S
\Biggl[                        &
{8\over 3}
-360{ \zeta(5)\over \pi^3 }
\frac
{kT}
{\hbar\omega_{\mathrm p}}
+{64\over 21}{ \pi^2 }
\left(\frac
{kT}
{\hbar\omega_{\mathrm p}}\right)^2
\nonumber \\
& &
+{\gamma\over\omega_{\mathrm p}}
\left(
{25\over 6}-2C
-2\ln \left( \frac
{\hbar\omega_{\mathrm p}}
{kT}
\right)
+180\frac {\zeta'(4)}{\pi^4}
\right)
\Biggr] \;,

\end{eqnarray}
```
where $\sigma$ is the Stefan-Boltzman constant, $\omega_{\mathrm p}$
is the plasma frequency of electrons,
$\gamma=\gamma_{293}(1+\alpha (T-293))$, $\gamma_{293}$ is the
collision frequency of electrons at 293 K, $\alpha$ is the temperature
coefficient of the electrical impedance of the material, $S$ is the
area of the mirror, $C\simeq 0.577$ is the Euler’s constant and
$\zeta(Z)$ is the Riemann Zeta function ($\zeta(5)\simeq
1.037$ and $\zeta'(4)\simeq -0.069$).

The heat removal from the flat mirror placed inside an environment with
the temperature 300 K $\Delta P^{}(T)=
P_{\mathrm M}^{\mathrm flat}(T)-
P_{\mathrm M}^{\mathrm flat}(300~{\mathrm K})$ is presented in
Figure 2.

For a mirror with one-dimensional corrugation (common diffraction
grating) we have not derived the analytical approximation for heat
removal, and will use numerical computation to estimate the heat removal
from the grating (see Appendix for details). The results of the
computation for radiation with $\lambda\sim 10.6$ $\mu$m is shown on
Figure 3. We take the silver, gold and copper gratings with the pitch
$\Lambda=5.3$ $\mu$m, and corrugation amplitude
$h=\lambda/10\simeq 1$ $\mu$m.

One can see that the extra radiative heat removal from the grating
respect to the radiative heat removal from the flat surface is not very
large (several percents). The physical reason is that an angular
dependence of the TSP emission at the given light frequency has a very
narrow width when corrugation amplitude of the grating is small
($\delta\theta \; \mbox{\rm [in
radians]}\sim\delta\omega/\omega_0$). If corrugation amplitude of the
grating increases, then starting with some point the angular width of
the SP resonance increases as well, but an amplitude of the SP resonance
decreases so that appropriate integral is taken by angle $\theta$
remains the same. The dependence of the heat removal respective to
corrugation amplitude of the gold grating at $700$ K is shown on
Figure 4.

We carry out our calculations for one-dimensional grating. For a
hexagonal arrangement of the corrugations (three gratings at $60^0$)
one may expect an increase of the heat removal by a factor of
approximately three.

# Conclusions

In this paper we have proposed the use of a "photonic surface" to
increase the damage threshold of metal mirrors. The SPs propagation is
forbidden on such a surface. Inasmuch as the SP wave is strongly
localized at the interface and its intensity is very large just above
the surface, the elimination of the SPs caused by the surface
imperfections must be very useful for increasing the damage threshold.

Moreover the photonic surface will slightly increase the heat removal
immediately from the interface.

The calculations of the frequency bandgaps, forbidden direction ranges
and radiative heat removal from the flat and corrugated surfaces are
performed for Ag, Au and Cu.

# Acknowledgments

Author thanks E.A. Vinogradov, who has pointed out at increasing of the
heat removal from the corrugated surface, and E.V. Alieva for pointing
at Ref. , containing optical constants of metals.

# Appendix

Let us start with estimation of the heat removal from the *flat* metal
mirror with permittivity given by the Drude’s model:
``` math
\begin{eqnarray}
\varepsilon_{\mathrm M}&=& \varepsilon'_{ }+i\varepsilon''_{
}=1-{\omega_{\mathrm p}^2 \over\omega^2+i\gamma \omega
}\nonumber \\ \varepsilon'_{ }&=& 1-{\omega_{\mathrm p}^2
\over\omega^2+\gamma^2} \simeq -{\omega_{\mathrm p}^2
\over\omega^2+\gamma^2} \\ \varepsilon''_{ }&=&
{\gamma\omega_{\mathrm p}^2 \over\omega(\omega^2+\gamma^2)}
\; , \nonumber

\end{eqnarray}
```
here $\omega_{\mathrm p}$ and $\gamma$ are plasma frequency and
collision frequency of electrons.

We will use the Leontovich approximation (see, for example ) for our
computations. In this approximation the Fresnel’s equations have the
form:
``` math
\begin{eqnarray}
r_{\mathrm
s}={E_{\mathrm s}\over E^0_{\mathrm s}}=-{1-z\cos(\theta)\over
1+z\cos(\theta)}\nonumber \\
r_{\mathrm
p}={E_{\mathrm p}\over E^0_{\mathrm p}}=-{\cos(\theta)-z\over
\cos(\theta)+z} \; ,

\end{eqnarray}
```
where $z=z'+iz''=1/\sqrt\varepsilon_{\mathrm M}$ is the surface
impedance of the metal.

Therefore
``` math
\begin{eqnarray}
R_{\mathrm s}&=&
r_{\mathrm s}r^*_{\mathrm s}=
{
(z'\cos(\theta)-1)^2
+z''^2\cos^2(\theta)
\over
(z'\cos(\theta)+1)^2
+z''^2\cos^2(\theta)
}\nonumber \\
R_{\mathrm p}&=&
r_{\mathrm p}r^*_{\mathrm p}
={
(\cos(\theta)-z')^2 +
z''^2
\over
(\cos(\theta)+z')^2 +
z''^2
} \; ,

\end{eqnarray}
```
where real and imaginary parts of the $z$ are:
``` math
\begin{eqnarray}
z'&=&{1\over\sqrt{2}}{\sqrt{
\sqrt{\varepsilon'^2_{ }+ \varepsilon''^2_{ }}
+\varepsilon'}\over
\sqrt{\varepsilon'^2_{ }+ \varepsilon''^2_{ }}} \nonumber \\
z''&=&-{1\over\sqrt{2}}{\sqrt{
\sqrt{\varepsilon'^2_{ }+ \varepsilon''^2_{ }}
-\varepsilon'}\over
\sqrt{\varepsilon'^2_{ }+ \varepsilon''^2_{ }}}\; .

\end{eqnarray}
```

To estimate the amount of the heat energy emitted from the surface we
will use the Kirchhoff’s law:
``` math
\begin{equation}
P_{\mathrm M}(\theta,\varphi,\omega,T)=\left(
1-R(\theta,\varphi,\omega) \right) P_{\mathrm
black}(\omega,T)  \; ,
\end{equation}
```
where $R$ is the reflectivity of the mirror, $\mathrm
 \theta$ is the incident angle of the plane wave with the frequency
$\mathrm\omega$, $\mathrm\varphi$ is the angle between plane of
incidence and the Bragg vector $\vec g$ ($\vec g$ is absent in the
case of the flat surface), $P_{\mathrm
black}$ is the spectral radiation intensity of the black body with the
temperature $T$, given by Plank’s law:
``` math
\begin{equation}
P_{\mathrm black}(\omega,T)=
{{\hbar}{\omega}^{3}\over 4{\pi }^{3}{c}^{2}\left
({\exp{{(\hbar \omega/kT ) }}}-1\right )}\; .

\end{equation}
```

The total heat removal from the surface is:
``` math
\begin{equation}
P_{\mathrm M}(T)=
\int_{0}^{\infty}
\left[
\int_{0}^{\pi/2}
\left[
\int_{0}^{2\pi}
 A(\theta,\varphi,\omega)
d\varphi
\right]
\sin(\theta)\cos(\theta)d\theta
\right]
P_{\mathrm black}(\omega,T)
d\omega
\; ,

\end{equation}
```
where
``` math
\begin{eqnarray}
A(\theta,\varphi,\omega)
&=&{1\over 2}(
A_{\mathrm s}(\theta,\varphi,\omega)+
A_{\mathrm p}(\theta,\varphi,\omega) )
\nonumber\\
&=&
1-{1\over 2}\left(
R_{\mathrm p}
(\theta,\varphi,\omega)
+R_{\mathrm s}
(\theta,\varphi,\omega)
\right)

\end{eqnarray}
```
is the isotropic absorbtion coefficient at given direction and
frequency.

The integrated absorbtions for s- and p- polarizations are:
``` math
\begin{eqnarray}
{A_{\mathrm s}(\omega)\over 2\pi}&=&
{1\over 2}-\int_0^{\pi/2}{
R_{\mathrm s}
}\sin(\theta)\cos(\theta)d\theta =
{1\over 2}-{       1\over 2
  \left ({z'}^{2}+{z''}^{2}\right )^{2}{z''}   } \times
\nonumber\\&&
\times\Biggl[
-8z'\left ({z''}^{3}-{z''}^{2}\arctan\left({\frac {z''}{1+
z'}}\right)\right )
\nonumber\\&&
\phantom{\times\Biggl[}
+2{z'}^{2}
\left ({z''}^{3}+4 z''\ln \left(\left (1+{
z'}\right )^{2}+{z''}^{2}\right) \right )
\nonumber\\&&
\phantom{\times\Biggl[}
-8{z'}^{3}
\left (
z''+\arctan\left({ \frac {z''}{1+z'}}\right)\right)
+{z'}^{4} z''
+{z''}^{5}
\Biggr]
\nonumber\\
{A_{\mathrm p}(\omega)\over 2\pi}&=&
{1\over 2}-\int_0^{\pi/2}{
R_{\mathrm p}
}\sin(\theta)\cos(\theta)d\theta =
\nonumber\\&&
-4 z' \left ( {  z''} \arctan\left({\frac {{
  z''}}{{{  z''}}^{2}+{ z'}+{{ z'}}^{2}}}\right)-1\right )
\nonumber\\&&
-4{{ z'}}^{2} \ln \left({\left (1+{
z'}\right ) ^{2}+{{  z''}}^{2}\over {{  z'}}^{2}+{{
z''}}^{2}}\right)
+4 {{  z'}^{3}\over {  z''}}
\arctan\left({\frac {{  z''}}{{{  z''}}^{2}+{  z'}+{{
z'}}^{2}}}\right)
\; .

\end{eqnarray}
```

The frequency dependence of the $z'$ and $z''$ may be found
from (<a href="#a22" data-reference-type="ref" data-reference="a22">[a22]</a>)
and (<a href="#b11" data-reference-type="ref" data-reference="b11">[b11]</a>):
``` math
\begin{eqnarray}
z'&= &
{1\over\sqrt {2}}
\sqrt {\sqrt {{\frac {{\omega}^{4}}{{{  \omega_{\mathrm
p}}}^{4}}}+{\frac {{\omega}^{2 }{{  \gamma}}^{2}}{{{
\omega_{\mathrm p}}}^{4}}}}-  {\frac {{\omega}^{2}}{{{
\omega_{\mathrm p}}}^{2}} }}
\nonumber \\
z''&=&-
{1\over\sqrt {2}}
\sqrt {\sqrt {{\frac
{{\omega}^{4}}{{{ \omega_{\mathrm p}}}^{4}}}+{\frac
{{\omega}^{2}{{ \gamma}}^{2}}{{{  \omega_{\mathrm
p}}}^{4}}}}+{\frac {{\omega}^{2}}{{{  \omega_{\mathrm p}}}^
{2}}}}\; .

\end{eqnarray}
```
Expanding
equations (<a href="#a5" data-reference-type="ref" data-reference="a5">[a5]</a>)
in a Taylor series respect to small parameter
$\gamma/\omega_{\mathrm p}$ one can find the approximation:
``` math
\begin{eqnarray}
z'&\simeq&
{1\over 2}
{\frac {\gamma}
{\omega_{\mathrm p}}} -
{1\over 16}
{\frac {\gamma^3}
{\omega^2\omega_{\mathrm p}}}
\nonumber \\
z''&\simeq&-{\frac {\omega}
{\omega_{\mathrm p}}} -
{1\over 8}
{\frac {\gamma^2}
{\omega\omega_{\mathrm p}}}\; .

\end{eqnarray}
```

To find the frequency dependence of the absorbtion, we substitute
equations (<a href="#a5" data-reference-type="ref" data-reference="a5">[a5]</a>)
in
equations (<a href="#a4" data-reference-type="ref" data-reference="a4">[a4]</a>)
and expand the result in Taylor series of the small parameter
$\gamma/\omega_{\mathrm p}$. Retaining the first and the second terms
we have:
``` math
\begin{equation}
A(\omega)\simeq
{8\over 3}
\pi{\frac {\gamma}
{\omega_{\mathrm p}}}
+{1\over 2}
\pi\left({\frac {\gamma}
{\omega_{\mathrm p}}}\right)^2
-2\pi
\left({\frac {\gamma}
{\omega_{\mathrm p}}}
\right)^2\ln\left(
{\omega_{\mathrm p}\over
\omega}
\right)-
\pi^2{\frac {\gamma\omega}
{\omega^2_{\mathrm p}}}
+{8\over 5}\pi{\frac {\gamma\omega^2}
{\omega^3_{\mathrm p}}}
\; .

\end{equation}
```

Performing the last integration in
equation (<a href="#a04" data-reference-type="ref" data-reference="a04">[a04]</a>)
we obtain the result:
``` math
\begin{eqnarray}
P_{\mathrm M}^{\mathrm flat}(T) & \simeq
\sigma T^4
{ \gamma\over \omega_{\mathrm p}}
S
\Biggl[                        &
{8\over 3}
-360{ \zeta(5)\over \pi^3 }
\frac
{kT}
{\hbar\omega_{\mathrm p}}
+{64\over 21}{ \pi^2 }
\left(\frac
{kT}
{\hbar\omega_{\mathrm p}}\right)^2
\nonumber \\
& &
+{\gamma\over\omega_{\mathrm p}}
\left(
{25\over 6}-2C
-2\ln \left( \frac
{\hbar\omega_{\mathrm p}}
{kT}
\right)
+180\frac {\zeta'(4)}{\pi^4}
\right)
\Biggr] \;,

\end{eqnarray}
```
here $\sigma=\pi^2 k^4/(60 c^2 \hbar^3 )$ is the Stefan-Boltzman
constant, $\gamma=\gamma_{293}(1+\alpha (T-293))$, $\gamma_{293}$ is
the collision frequency of electrons at 293 K, $\alpha$ is the
temperature coefficient of the electrical impedance of the material,
$S$ is the area of the mirror, $C\simeq 0.577$ is the Euler’s
constant and $\zeta(Z)$ is the Riemann Zeta function
($\zeta(5)\simeq 1.037$ and $\zeta'(4)\simeq -0.069$). We have taken
into account the temperature dependence of the $\gamma$, using the
fact that $\gamma\sim\varrho$ , where $\varrho$ is the electrical
impedance of the metal.
Approximation (<a href="#aF" data-reference-type="ref" data-reference="aF">[aF]</a>)
is true for $\hbar\gamma/(2.82 k)\ll T\ll\hbar\omega_{\mathrm p}/(2.82
k)$ (e.c. for Ag: $189~{\mathrm K}\ll T\ll 3.8\cdot
10^4$ K). For rough
estimation (<a href="#aF" data-reference-type="ref" data-reference="aF">[aF]</a>)
may be taken in the form:
``` math
\begin{equation}
P_{\mathrm M}^{\mathrm flat}(T)  \approx
{8\over 3}
\sigma T^4 S
{ \gamma_{293}(1+\alpha (T-293))\over \omega_{\mathrm p}}
\;.

\end{equation}
```

To calculate the heat removal from the *corrugated* surface (from the
diffraction grating) we repeat our computation performing the numerical
integration of the
equation (<a href="#a04" data-reference-type="ref" data-reference="a04">[a04]</a>).
The reflection from the grating may be found from the next equations :
``` math
\begin{eqnarray}
E_{\mathrm s}&=&
r_{\mathrm s}\left[
E^0_{\mathrm s}-2i\omega{|a_{\mathrm s}|^2
E^0_{\mathrm s}+
a_{\mathrm s}^*
a_{\mathrm p}
E^0_{\mathrm p}
\over F}      \right]\nonumber \\
E_{\mathrm p}&=&
r_{\mathrm p}\left[
E^0_{\mathrm p}-2i\omega{|a_{\mathrm p}|^2
E^0_{\mathrm p}+
a_{\mathrm s}
a_{\mathrm p}^*
E^0_{\mathrm s}
\over F}      \right]\; ,

\end{eqnarray}
```
where
``` math
\begin{eqnarray}
a_{\mathrm s}=
\left({|z|\omega\cos(\theta) \over 2}\right)^{1\over 2}
hg{\sin(\varphi)\over 1+z\cos(\theta)}\; ,\nonumber\\
a_{\mathrm p}=
\left({|z|\omega\cos(\theta) \over 2}\right)^{1\over 2}
hg{\cos(\varphi)\over \cos(\theta)+z}\; ,\nonumber\\
F=\omega^2
-\omega^2_{\mathrm sp_{\pm}}
+i\omega(
\gamma_{\mathrm d}
+\gamma_{\mathrm r})
\; ,\nonumber \\
\omega_{\mathrm sp_{\pm}}=
({\omega^2\sin^2(\theta)+g^2 c^2\mp 2\omega
gc\sin(\theta)\cos{\varphi}})^{1\over 2}(1-|z|^2/2)
\; ,\nonumber \\
\gamma_{\mathrm d}=-\omega{\mathrm Im} (z^2)
\; ,\nonumber \\
\gamma_{\mathrm r}=
|a_{\mathrm s}|^2+
|a_{\mathrm p}|^2\; .

\end{eqnarray}
```
Here $\gamma_{\mathrm r}$ is the constant of the radiative damping of
the SP, and $\gamma_{\mathrm d}$ is the constant of the dissipative
damping of the SP ( $\gamma_{\mathrm d}\simeq 2\delta\omega$ from the
equation (<a href="#7" data-reference-type="ref" data-reference="7">[7]</a>)).

The first term in the square bracket of
equations (<a href="#aR" data-reference-type="ref" data-reference="aR">[aR]</a>)
describes the reflection from the flat surface (nonresonant term).
$r_{\mathrm s}$ and $r_{\mathrm p}$ are given by the
equations (<a href="#a2" data-reference-type="ref" data-reference="a2">[a2]</a>).
The second term in the square bracket describes the role of the SPs
excitation in the reflection from the grating with the Bragg vector
$\vec g$ and the corrugation amplitude $h$. The results of the
calculations for silver, gold and copper gratings are illustrated in
Figure 3. The following parameters are used:\
Au: $\gamma_{293}=330$ cm$^{-1}$, $\alpha=0.00402$ K$^{-1}$
$\omega_{\mathrm p}=70200$ cm$^{-1}$;\
Ag: $\gamma_{293}=370$ cm$^{-1}$, $\alpha=0.0043$ K$^{-1}$
$\omega_{\mathrm p}=74500$ cm$^{-1}$;\
Cu: $\gamma_{293}=370$ cm$^{-1}$, $\alpha=0.00433$ K$^{-1}$
$\omega_{\mathrm p}=67000$ cm$^{-1}$; $\Lambda=5.3$ $\mu$m,
$h=\lambda/10\simeq 1$ $\mu$m.

Exceeding of the heat removal from the grating respect to the heat
removal from the flat surface $\delta  P(T)$ we present in the form
``` math
\begin{equation}
\delta P
(T)
={
\Delta P^{\mathrm \lambda/10}(T)
-\Delta P^{0}(T)
\over
\Delta P^{0}(T)
}100\%
\; ,

\end{equation}
```
here $\Delta P^{\mathrm \lambda/10}(T)=
P_{\mathrm M}^{\mathrm \lambda/10}(T)-
P_{\mathrm M}^{\mathrm \lambda/10}(300~{\mathrm K})$ and
$\Delta P^{0}(T)=
P_{\mathrm M}^{0}(T)-
P_{\mathrm M}^{0}(300~{\mathrm K})$ are the heat removal from the
grating (with $h=\lambda/10$) and from the flat surface ($h=0$)
placed inside of an environment at $300$ K. The heat removal from the
flat surface given by
equation (<a href="#10" data-reference-type="ref" data-reference="10">[10]</a>)(<a href="#aF" data-reference-type="ref" data-reference="aF">[aF]</a>)
agrees well with the heat removal from the grating with $h=0$
($P_{\mathrm M}^{0}(T)$), calculated by the numerical integration of
the
equation (<a href="#a04" data-reference-type="ref" data-reference="a04">[a04]</a>).

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[15]