Monolithic mirror and method for designing same
The present invention refers to a mirror comprising a plurality of one-dimensional photonic crystals, the mirror having very high reflectance in a very broad range of wavelengths, a broad range of directions, even hemispheric, and all the polarizations of the incident photons. The invention also refers to a method for designing said mirror and a photovoltaic cell comprising such a mirror.
1 . A method of manufacturing a mirror having total reflectance in a predefined vacuum wavelength range ([λ A , λ B ]) for incident unpolarized radiation with an angle of incidence (θ) lower than or equal to a predefined maximum angle of incidence (θ max ), comprising:
forming m stacked one-dimensional photonic crystals, wherein for each i-th photonic crystal a first dielectric material and a second dielectric material are selected, wherein each i-th photonic crystal is formed by stacking a plurality of alternate layers of the first dielectric material having a first thickness, and the second dielectric material having a second thickness, the first dielectric material having an index of refraction different than the index of refraction of the second dielectric material;
wherein each photonic crystal comprises a plurality of unit cells repeated identically a prescribed number of times, each unit cell comprising a layer of the first dielectric material and a layer of the second dielectric material,
wherein the reflectance of each photonic crystal as a function of vacuum wavelength (λ 0 ) shows the shape of a rectangular pulse of unity height with rounded corners in an interval
(
λ
0
L
,
λ
0
T
)
between a leading edge wavelength value
(
λ
0
L
)
and a trailing edge wavelength value
(
λ
0
T
)
,
said pulse in said interval being identified as total reflection band, the leading edge wavelength value and the trailing edge wavelength value being dependent on the angle of incidence (θ) and on the polarization of the incident radiation,
wherein for each i-th photonic crystal, a leading edge wavelength value
(
λ
0
l
,
i
L
)
of the total reflection band of the i-th photonic crystal is set for θ=0,
the first thickness (h al,i ) for the layer of first dielectric material of the i-th photonic crystal and the second thickness (h bl,i ) for the layer of second dielectric material of the i-th photonic crystal satisfy the following relationships:
h
al
,
i
=
λ
0
l
,
i
L
4
π
n
al
,
i
{
2
π
-
arccos
[
(
1
-
n
al
,
i
n
bl
,
i
)
2
-
4
(
n
al
,
i
/
n
bl
,
i
)
(
1
+
(
n
al
,
i
/
n
bl
,
i
)
)
2
]
}
h
bl
,
i
=
h
al
,
i
(
n
al
,
i
/
n
bl
,
i
)
wherein n ai,l and n bl,i are, respectively, the indices of refraction of the first dielectric material and the second dielectric material of the i-th photonic crystal, and
wherein the trailing edge wavelength value
(
λ
0
l
,
i
T
)
of the total reflection band of the i-th photonic crystal is given by:
λ
0
l
,
i
T
=
8
n
al
,
i
h
al
,
i
cos
(
θ
al
,
i
)
X
(
1
+
r
)
wherein parameter X is obtained by solving in X the equation α TM +1=0 for the predefined maximum angle of incidence (θ max ) and for transversal magnetic (TM) polarization, wherein said equation is solved by an iteration method whose initial value is X=1,
wherein
α
TM
=
cos
(
π
X
/
2
)
(
1
+
Z
)
2
4
Z
-
cos
(
π
rX
/
2
)
(
1
-
Z
)
2
4
Z
Z
=
n
bl
,
i
cos
(
θ
ai
,
i
)
/
(
n
ai
,
i
cos
(
θ
bl
,
i
)
)
r
=
cos
2
(
θ
al
,
i
)
-
(
Zh
bl
,
i
h
al
,
i
)
cos
2
(
θ
bl
,
i
)
cos
2
(
θ
al
,
i
)
+
(
(
Zh
bl
,
i
/
h
al
,
i
)
cos
2
(
θ
bl
,
i
)
)
θ
al
,
i
=
arcsin
(
sin
(
θ
max
)
n
al
,
i
)
θ
bl
,
i
=
arcsin
(
sin
(
θ
max
)
n
bl
,
i
)
wherein the leading edge wavelength value
(
λ
0
l
,
i
L
)
is set to:
a value equal to λ A , for i=1, and
a value equal to the trailing edge wavelength value
(
λ
0
l
,
i
-
1
T
)
of the total reflection band of the (i−1)-th photonic crystal for θ=θ max and TM polarization, for i>1,
wherein m is the number of the photonic crystal which fulfils that the trailing edge wavelength value
(
λ
0
l
,
m
T
)
of the total reflection band of said m-th photonic crystal for θ=θ max and TM polarization is equal to or greater than λ B .
2 . The method according to claim 1 , wherein λ A is comprised in the visible or near infrared range and/or λ B is comprised in the medium infrared range.
3 . The method according to claim 1 , wherein the layers of the photonic crystals are deposited on a substrate.
4 . The method according to claim 3 , wherein the substrate is covered with a layer of a reflective metal and the photonic crystals are deposited on said layer.
5 . The method according to claim 4 , wherein the reflective metal is silver or gold.
6 . The method according to claim 3 , wherein the photonic crystals are arranged in the mirror in an order defined by the transparency of the first and second dielectric materials of the photonic crystals, such that photonic crystals made of a material not transparent to radiation in a wavelength range comprised in the total reflection band of another photonic crystal are placed downstream of said another photonic crystal in the direction intended for incoming radiation.
7 . The method according to claim 1 , wherein the layers of the photonic crystals are covered with a protective thick transparent layer.
8 . The method according to claim 7 , wherein the protective thick transparent layer is transparent in the range [λ A , λ B ].
9 . The method according to claim 1 , wherein:
(a) the photonic crystals are arranged in the mirror in the order defined by the position of their total reflection bands from λ A to λ B , or
(b) the photonic crystals are arranged in the mirror in an order different to the order defined by the position of their total reflection bands from λ A to λ B .
10 . The method according to claim 1 , wherein the number of unit cells in each photonic crystal is greater than or equal to 5.
11 . The mirror manufactured according to the method of claim 1 , wherein m>1 and the predefined maximum angle of incidence (θ max ) is 0.99×π/2.
12 . A photovoltaic cell comprising the mirror manufactured according to claim 1 , wherein m>1 and the predefined maximum angle of incidence (θ max ) is 0.99×π/2, wherein the mirror is deposited on a transparent substrate and coated with a metal layer, the photovoltaic cell being a thermo-photovoltaic cell.
13 . A photovoltaic cell comprising the mirror manufactured according to claim 1 , wherein m>1 and the predefined maximum angle of incidence (θ max ) is 0.99×π/2; and a semiconductor substrate, wherein the mirror is deposited on a back face of the semiconductor substrate and coated with a metal layer, and the photovoltaic cell is a thermo-photovoltaic cell.
14 . A thermal insulation for an incandescent body, wherein the thermal insulation comprises at least one mirror manufactured according to claim 1 , wherein m>1 and the predefined maximum angle of incidence (θ max ) is 0.99×π/2.
15 . The method according to claim 1 , wherein the number of unit cells in each photonic crystal is greater than or equal to 7.
16 . The method according to claim 1 , wherein the number of unit cells in each photonic crystal is greater than or equal to 10.
17 . The method according to claim 1 , wherein the predefined maximum angle of incidence (θ max ) is <π/2.
18 . The method according to claim 17 , wherein the predefined maximum angle of incidence (θ max ) is 0.99×π/2.
19 . A method of manufacturing a mirror having maximum reflectance in a predefined vacuum wavelength range ([λ A ,λ B ]) for incident unpolarized radiation with an angle of incidence (θ) lower than or equal to a predefined maximum angle of incidence (θ max ), comprising:
forming m stacked one-dimensional photonic crystals, wherein for each i-th photonic crystal a first dielectric material and a second dielectric material are selected, wherein each i-th photonic crystal is formed by stacking a plurality of alternate layers of the first dielectric material having a first thickness, and the second dielectric material having a second thickness, the first dielectric material having an index of refraction different than the index of refraction of the second dielectric material;
wherein each photonic crystal comprises a plurality of unit cells repeated identically a prescribed number of times, each unit cell comprising a layer of the first dielectric material and a layer of the second dielectric material,
wherein the reflectance of each photonic crystal as a function of vacuum wavelength (λ 0 ) shows the shape of a rectangular pulse of unity height with rounded corners in an interval
(
λ
0
L
,
λ
0
T
)
between a leading edge wavelength value
(
λ
0
L
)
and a trailing edge wavelength value
(
λ
0
T
)
,
said pulse in said interval being identified as total reflection band, the leading edge wavelength value and the trailing edge wavelength value being dependent on the angle of incidence (θ) and on the polarization of the incident radiation,
wherein for each i-th photonic crystal, a trailing edge wavelength value
(
λ
0
t
,
i
T
)
of the total reflection band of the i-th photonic crystal is set for θ=θ max and TM polarization,
the first thickness (h at,i ) for the layer of first dielectric material of the i-th photonic crystal and the second thickness (h bt,i ) for the layer of second dielectric material of the i-th photonic crystal satisfy the following relationships:
h
at
,
i
=
λ
0
t
,
i
T
4
π
n
at
,
i
×
{
arccos
[
(
1
-
n
bt
,
i
cos
(
θ
at
,
i
)
/
(
n
at
,
i
cos
(
θ
bt
,
i
)
)
)
2
-
4
n
bt
,
i
cos
(
θ
at
,
i
)
/
(
n
at
,
i
cos
(
θ
bt
,
i
)
)
(
1
+
n
bt
,
i
cos
(
θ
at
,
i
)
/
(
n
at
,
i
cos
(
θ
bt
,
i
)
)
)
2
]
}
h
bt
,
i
=
h
at
,
i
n
at
,
i
cos
(
θ
at
,
i
)
/
(
n
bt
,
i
cos
(
θ
bt
,
i
)
)
wherein n at,i and n bt,i are, respectively, the indices of refraction of the first dielectric material and the second dielectric material of the i-th photonic crystal,
wherein
θ
at
,
i
=
arcsin
(
sin
(
θ
max
)
n
at
,
i
)
θ
bt
,
i
=
arcsin
(
sin
(
θ
max
)
n
bt
,
i
)
and
wherein the leading edge wavelength value
(
λ
0
t
,
i
L
)
of the total reflection band of the i-th photonic crystal is given by:
λ
0
t
,
i
L
=
8
n
at
,
i
h
at
,
i
X
(
1
+
r
)
wherein parameter X is obtained by solving in X the equation α TM +1=0 for θ=0, wherein said equation is solved by an iteration method whose initial value is X=3,
wherein
α
TM
=
cos
(
π
X
/
2
)
(
1
+
Z
)
2
4
Z
-
cos
(
π
rX
/
2
)
(
1
-
Z
)
2
4
Z
Z
=
n
bt
,
i
/
n
at
,
i
r
=
1
-
(
Z
h
bt
,
i
/
h
at
,
i
)
1
+
(
Z
h
bt
,
i
/
h
at
,
i
)
wherein the trailing edge wavelength value
(
λ
0
t
,
i
T
)
is set to:
a value equal to λ B , for i=1, and
a value equal to the leading edge wavelength value of the total reflection band of the (i−1)-th photonic crystal for θ=0, for i>1,
wherein m is the number of the photonic crystal which fulfils that the leading edge wavelength value
(
λ
0
t
,
m
L
)
of the total reflection band of said m-th photonic crystal for θ=0 is equal to or smaller than λA.
20 . The method according to claim 19 , wherein λ A is comprised in the visible or near infrared range and/or λ B is comprised in the medium infrared range.
21 . The method according to claim 19 , wherein the predefined maximum angle of incidence (θ max ) is <π/2.
22 . The method according to claim 21 , wherein the predefined maximum angle of incidence (θ max ) is 0.99×π/2.
23 . The mirror manufactured according to the method of claim 19 , wherein m>1 and the predefined maximum angle of incidence (θ max ) is 0.99×π/2.