MONOLITHIC MIRROR AND METHOD FOR DESIGNING SAME

Abstract

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.

Claims

1. A method for designing a mirror having total reflectance in a predefined vacuum wavelength range ([.sub.A, .sub.B]) for incident unpolarized radiation with an angle of incidence () lower than or equal to a predefined maximum angle of incidence (.sub.max), wherein the mirror comprises a plurality of one-dimensional photonic crystals forming layers, wherein each photonic crystal comprises a plurality of unit cells repeated identically a prescribed number of times, each unit cell comprising a layer of a first dielectric material and a layer of a second dielectric material, the first and second dielectric materials having different indices of refraction, wherein the reflectance of each photonic crystal as a function of vacuum wavelength (.sub.0) shows the shape of a rectangular pulse of unity height with rounded corners in an interval (.sub.0.sup.L, .sub.0.sup.T) between a leading edge wavelength value (.sub.0.sup.L) and a trailing edge wavelength value (.sub.0.sup.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 () and on the polarization of the incident radiation, wherein the method comprises the following steps for i=1, . . . m: (a) setting a leading edge wavelength value (.sub.0l,i.sup.L) of the total reflection band of the i-th photonic crystal for =0 and selecting the first and second dielectric materials to form the unit cell of said i-th photonic crystal, (b) determining a first thickness (h.sub.al,i) for the layer of first dielectric material of the i-th photonic crystal and a second thickness (h.sub.bl,i) for the layer of second dielectric material of the i-th photonic crystal as: $h_{\mathrm{al},i}=\frac{{}_{0l,i}^L}{4n_{\mathrm{al},i}}\left\{2-\arccos \left[\frac{{\left(1-\frac{n_{\mathrm{al},i}}{n_{\mathrm{bl},i}}\right)}^2-4\left(n_{\mathrm{al},i}/n_{\mathrm{bl},i}\right)}{{\left(1+\left(n_{\mathrm{al},i}/n_{\mathrm{bl},i}\right)\right)}^2}\right]\right\}$ $h_{\mathrm{bl},i}=h_{\mathrm{al},i}\left(n_{\mathrm{al},i}/n_{\mathrm{bl},i}\right)$ wherein n.sub.al,i and n.sub.bl,i are, respectively, the indices of refraction of the first dielectric material and the second dielectric material selected for the i-th photonic crystal; and (c) with the values of the first thickness (h.sub.al,i) and the second thickness (h.sub.bl,i) calculated in step (b), determining the trailing edge wavelength value (.sub.0l,i.sup.T) of the total reflection band of the i-th photonic crystal as ${}_{0l,i}^T=\frac{8n_{\mathrm{al},i}{h}_{\mathrm{al},i}\cos \left({}_{\mathrm{al},i}\right)}{X\left(1+r\right)}$ wherein parameter X is obtained by solving in X the equation .sub.TM+1=0 for the predefined maximum angle of incidence (.sub.max) and for transversal magnetic (TM) polarization, wherein said equation is solved by an iteration method whose initial value is X=1, wherein ${}_{\mathrm{TM}}=\cos \left(X/2\right)\frac{{\left(1+Z\right)}^2}{4Z}-\cos \left(\mathrm{rX}/2\right)\frac{{\left(1-Z\right)}^2}{4Z}Z=n_{\mathrm{bl},i}\cos \left({}_{\mathrm{ai},i}\right)/\left(n_{\mathrm{ai},i}\cos \left({}_{\mathrm{bl},i}\right)\right)r=\frac{{\cos }^2\left({}_{\mathrm{al},i}\right)-\left(\frac{{\mathrm{Zh}}_{\mathrm{bl},i}}{h_{\mathrm{al},i}}\right){\cos }^2\left({}_{\mathrm{bl},i}\right)}{{\cos }^2\left({}_{\mathrm{al},i}\right)+\left(\left({\mathrm{Zh}}_{\mathrm{bl},i}/h_{\mathrm{al},i}\right){\cos }^2\left({}_{\mathrm{bl},i}\right)\right)}{}_{\mathrm{al},i}=\arcsin \left(\frac{\sin \left({}_{\max }\right)}{n_{\mathrm{al},i}}\right){}_{\mathrm{bl},i}=\arcsin \left(\frac{\sin \left({}_{\max }\right)}{n_{\mathrm{bl},i}}\right)$ wherein in step (a) the leading edge wavelength value (.sub.0l,i.sup.L) is set to: a value equal to .sub.A, for i=1, and a value equal to the trailing edge wavelength value (.sub.0l,i-1.sup.T) of the total reflection band of the (i1)-th photonic crystal for =.sub.max and TM polarization, for i>1, wherein m is the number of the photonic crystal which fulfils that the trailing edge wavelength value (.sub.0l,m.sup.T) of the total reflection band of said m-th photonic crystal for =.sub.max and TM polarization is equal to or greater than .sub.B.

2. A method for designing a mirror having maximum reflectance in a predefined vacuum wavelength range ([.sub.A, .sub.B]) for incident unpolarized radiation with an angle of incidence () lower than or equal to a predefined maximum angle of incidence (.sub.max), wherein the mirror comprises a plurality of one-dimensional photonic crystals forming layers, wherein each photonic crystal comprises a plurality of unit cells repeated identically a prescribed number of times, each unit cell comprising a layer of a first dielectric material and a layer of a second dielectric material, the first and second dielectric materials having different indices of refraction, wherein the reflectance of each photonic crystal as a function of vacuum wavelength (.sub.0) shows the shape of a rectangular pulse of unity height with rounded corners in an interval (.sub.0.sup.L, .sub.0.sup.T) between a leading edge wavelength value (.sub.0.sup.L) and a trailing edge wavelength value (.sub.0.sup.T), said pulse in said interval being identified as total reflection band, the leading edge wavelength value and the trailing edge wavelength value being dependant on the angle of incidence () and on the polarization of the incident radiation, wherein the method comprises the following steps for i=1, . . . m: (a) setting a trailing edge wavelength value (.sub.0t,i.sup.T) of the total reflection band of the i-th photonic crystal for =.sub.max and TM polarization and selecting the first and second dielectric materials to form the unit cell of said i-th photonic crystal; (b) determining a first thickness (h.sub.at,i) for the layer of first dielectric material of the i-th photonic crystal and a second thickness (h.sub.bt,i) for the layer of second dielectric material of the i-th photonic crystal as: $$\begin{gathered} h_{\mathrm{at},i}=\frac{{}_{0t,i}^T}{4n_{\mathrm{at},i}} \\ \left\{\arccos \left[\frac{{\left(1-n_{\mathrm{bt},i}\cos \left({}_{\mathrm{at},i}\right)/\left(n_{\mathrm{at},i}\cos \left({}_{\mathrm{bt},i}\right)\right)\right)}^2-4n_{\mathrm{bt},i}\cos \left({}_{\mathrm{at},i}\right)/\left(n_{\mathrm{at},i}\cos \left({}_{\mathrm{bt},i}\right)\right)}{{\left(1+n_{\mathrm{bt},i}\cos \left({}_{\mathrm{at},i}\right)/\left(n_{\mathrm{at},i}\cos \left({}_{\mathrm{bt},i}\right)\right)\right)}^2}\right]\right\} \end{gathered}$$ $h_{\mathrm{bt},i}=h_{\mathrm{at},i}{n}_{\mathrm{at},i}\cos \left({}_{\mathrm{at},i}\right)/\left(n_{\mathrm{bt},i}\cos \left({}_{\mathrm{bt},i}\right)\right)$ wherein n.sub.at,i and n.sub.bt,i are, respectively, the indices of refraction of the first dielectric material and the second dielectric material selected for the i-th photonic crystal, wherein ${}_{\mathrm{at},i}=\arcsin \left(\frac{\sin \left({}_{\max }\right)}{n_{\mathrm{at},i}}\right)$ ${}_{\mathrm{bt},i}=\arcsin \left(\frac{\sin \left({}_{\max }\right)}{n_{\mathrm{bt},i}}\right)$ and (c) with the values of the first thickness (h.sub.at,i) and the second thickness (h.sub.bt,i) calculated in step (b), determining the leading edge wavelength value (.sub.0t,i.sup.L) of the total reflection band of the i-th photonic crystal as ${}_{0t,i}^L=\frac{8n_{\mathrm{at},i}{h}_{\mathrm{at},i}}{X\left(1+r\right)}$ wherein parameter X is obtained by solving in X the equation .sub.TM+1=0 for =0, wherein said equation is solved by an iteration method whose initial value is X=3, wherein ${}_{\mathrm{TM}}=\cos \left(X/2\right)\frac{{\left(1+Z\right)}^2}{4Z}-\cos \left(\mathrm{rX}/2\right)\frac{{\left(1-Z\right)}^2}{4Z}$ $Z=n_{\mathrm{bt},i}/n_{\mathrm{at},i}$ $r=\frac{1-\left(Z{h}_{\mathrm{bt},i}/h_{\mathrm{at},i}\right)}{1+\left(Z{h}_{\mathrm{bt},i}/h_{\mathrm{at},i}\right)}$ wherein in step (a) the trailing edge wavelength value (.sub.0t,i.sup.T) is set to: a value equal to .sub.B, for i=1, and a value equal to the leading edge wavelength value (.sub.0l,i-1.sup.L) of the total reflection band of the (i1)-th photonic crystal for =0, for i>1, wherein m is the number of the photonic crystal which fulfils that the leading edge wavelength value (.sub.0t,m.sup.L) of the total reflection band of said m-th photonic crystal for =0 is equal to or smaller than .sub.A.

3. The method according to claim 1, wherein .sub.A is comprised in the visible or near infrared range and/or .sub.B is comprised in the medium infrared range.

4. A method of manufacturing a mirror comprising m one-dimensional photonic crystals, with m>1, the method comprising the following steps: designing the mirror according to the method of claim 1, and forming m stacked one-dimensional photonic crystals, wherein each i-th photonic crystal is formed by stacking a plurality of alternate layers of a first dielectric material and a second dielectric material, the first dielectric material having an index of refraction (n.sub.al,i, n.sub.at,i) different to the index of refraction (n.sub.bl,i, n.sub.bt,i) of the second dielectric material, and wherein for each i-th photonic crystal the first thickness (h.sub.al,i, h.sub.at,i) for every layer of first dielectric material and the second thickness (h.sub.bl,i, h.sub.bt,i) for every layer of second dielectric material have the values determined in step (b) of claim 1, with i=1, . . . m.

5. A method according to claim 4, wherein the layers of the photonic crystals are deposited on a substrate.

6. A method according to claim 5, wherein the substrate is covered with a layer of a reflective metal and the photonic crystals are deposited on said layer.

7. A method according to claim 4, wherein the layers of the photonic crystals are covered of a protective thick transparent layer.

8. A method according to claim 4, wherein: (a) the photonic crystals are arranged in the mirror in the order defined by the position of their total reflection bands from .sub.A to .sub.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 .sub.A to .sub.B.

9. A method according to claim 5, 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 said another photonic crystal in the direction intended for incoming radiation.

10. A method according to claim 4, wherein the number of unit cells in each photonic crystal is greater than or equal to 5.

11. A mirror comprising m one-dimensional photonic crystals, with m>1, manufactured according to the method of claim 4, wherein the predefined maximum angle of incidence (.sub.max) is 0.99/2.

12. A photovoltaic cell comprising a mirror according to claim 11 deposited on a transparent substrate and coated with a metal layer, the photovoltaic cell being a thermo-photovoltaic cell.

13. A photovoltaic cell comprising a mirror according to claim 11 and a semiconductor substrate, the mirror being deposited on the back face of the semiconductor substrate and coated with a metal layer, the photovoltaic cell being a thermo-photovoltaic cell.

14. A thermal insulation for an incandescent body, wherein the thermal insulation comprises at least one mirror according to claim 11.

15. The method according to claim 2, wherein .sub.A is comprised in the visible or near infrared range and/or .sub.B is comprised in the medium infrared range.

16. A method according to claim 6, wherein the reflective metal is silver or gold.

17. A method according to claim 7, wherein the protective thick transparent layer is transparent in the range [.sub.A, .sub.B].

18. A method according to claim 10, wherein the number of unit cells in each photonic crystal is greater than or equal to 7.

19. A method according to claim 18, wherein the number of unit cells in each photonic crystal is greater than or equal to 10.

20. A method according to claim 1, wherein the predefined maximum angle of incidence (.sub.max) is </2.

21. A method according to claim 2, wherein the predefined maximum angle of incidence (.sub.max) is </2.

22. A method of manufacturing a mirror comprising in one-dimensional photonic crystals, with m>1, the method comprising the following steps: designing the mirror according to the method of claim 2, and forming m stacked one-dimensional photonic crystals, wherein each i-th photonic crystal is formed by stacking a plurality of alternate layers of a first dielectric material and a second dielectric material, the first dielectric material having an index of refraction (n.sub.al,i, n.sub.at,i) different to the index of refraction (n.sub.bl,i, n.sub.bt,i) of the second dielectric material, and wherein for each i-th photonic crystal the first thickness (h.sub.al,i, h.sub.at,i) for every layer of first dielectric material and the second thickness (h.sub.bl,i, h.sub.bt,i) for every layer of second dielectric material have the values determined in step (b) of claim 2, with i=1, . . . m.

23. A mirror comprising m one-dimensional photonic crystals, with m>1, manufactured according to the method of claim 22, wherein the predefined maximum angle of incidence (.sub.max) is 0.99/2.

24. The method according to claim 20, wherein the predefined maximum angle of incidence (.sub.max) is 0.99/2.

25. The method according to claim 21, wherein the predefined maximum angle of incidence (.sub.max) is 0.99/2.

Description

DESCRIPTION OF THE DRAWINGS

[0104] These and other characteristics and advantages of the invention will become clearly understood in view of the detailed description of the invention which becomes apparent from a preferred embodiment of the invention, given just as an example and not being limited thereto, with reference to the drawings.

[0105] FIG. 1 shows a schematic representation of a monolithic mirror made of multiple photonic crystals, according to an embodiment of the invention.

[0106] FIG. 2 shows the reflectance of a photonic crystal as a function of vacuum wavelength (in meters) for normal incidence (solid line) of the incident photons, for /4 rad incidence and TE polarization (dashed line) and for /4 rad incidence and TM polarization (dotted line). Furthermore, the absolute value of the Chebyshev arguments is shown for normal incidence (solid line), for /4 rad incidence and TE polarization (dashed line) and for /4 rad incidence and TM polarization (dotted line).

[0107] FIG. 3 shows vs. X for Y=0, for Z=3 (solid thick line), for Z=2 (dashed thick line), for Z=0.35 (solid thin line), and for Z=0.55 (dashed thin line).

[0108] FIG. 4 shows vs. X for Z=3, for Y=0 (solid line), for Y=0.5X (dashed line), and for X=0.45Y (dotted line).

[0109] FIG. 5 shows reflectance curves (in the plot upper part) and Chebyshev arguments (mainly in the plot lower part), as a function of the vacuum wavelength (in meters) of the incident photons for two photonic crystals and for different incidence angles.

[0110] FIG. 6 shows vs. X for the first photonic crystal of FIG. 5. Solid line for normal incidence with Y=0; dashed line for Y0. The X values of the leading and trailing edge of the hemispheric total reflection band are marked in the plot.

[0111] FIG. 7 shows the reflectance curves of a monolithic mirror made of multiple photonic crystals as a function of the vacuum wavelength (in meters) for different incidence angles.

[0112] FIG. 8 shows the spectral power in W/cm.sup.2 per meter of the blackbody at 1410 C. vs. the vacuum wavelength is in meters.

DETAILED DESCRIPTION OF THE INVENTION

[0113] FIG. 1 shows a schematic representation of a monolithic mirror comprising several one-dimensional photonic crystals (1, 2, 3), according to an embodiment of the invention, wherein all the photonic crystals are deposited on a single substrate (not shown). In this figure, a first (1), a second (2) and a third (3) photonic crystal are shown, but the blank space in the centre of the figure means that more photonic crystals might be present in the mirror. Each photonic crystal contains a plurality of unit cells (U1, U2, U3), each unit cell (U1, U2, U3) containing two dielectric layers (1.1, 1.2; 2.1, 2.2; 3.1, 3.2) of higher and lower indices of refraction and with different thicknesses repeated a number of times.

[0114] In FIG. 1 only some of the unit cells have been identified. The photonic crystals included in the mirror may have different characteristics, i.e. the dielectrics forming the unit cell of each photonic crystal may be different in each photonic crystal, thus having different index of refraction, and may have different thicknesses, the thicknesses of the layers being defined according to the method of the invention. The mirror thus formed has very high reflectance in a very broad range of wavelengths, in a broad range of directions (even hemispheric) and in different polarizations of the incident photons.

[0115] FIG. 2 shows the reflectance (R) as a function of vacuum wavelength (.sub.0, in meters) of a one-dimensional photonic crystal. The reflectance (R) has been plotted for normal incidence (solid line) of the incident photons, for /4 rad incidence and TE polarization (dashed line) and for /4 rad incidence and TM polarization (dotted line). As visible in the figure, the total reflection bands of the photonic crystal extend between points 7 and 8 for photons of normal incidence, between points 9 and 10 for TE photons of /4 incidence and between points 11 and 12 for TM photons of /4 incidence. It can be appreciated in FIG. 2 how the total reflection bands are shifted for different incidence angles and polarizations.

[0116] The forbidden band or bandgap of a photonic crystal occurs when the argument of the second-kind Chebyshev polynomials of the characteristic matrix of the photonic crystal exits the 1<<+1 range. The absolute value of the Chebyshev arguments is represented for the photonic crystal in FIG. 2 for normal incidence (solid line, denoted as 17 in the figure), for /4 rad incidence and TE polarization (dashed line, denoted as 18 in the figure) and for /4 rad incidence and TM polarization (dotted line denoted as 19 in the figure).

[0117] It is seen that the total reflection bands appear when the absolute value of the Chebyshev argument exceeds the value 1. The edges of the total reflection bands are the abscissas of the ends of the segment extending from point 7 to point 8 for normal incidence, of the segment extending from point 9 to point 10 for /4 rad incidence and TE polarization and of the segment extending from point 11 to point 12 for /4 rad incidence and TM polarization. According to the present invention, the wavelengths corresponding to the edges of the total reflection bands are calculated with an analysis of the Chebyshev argument, which can be totally analytical. This is much faster and simpler than calculating the reflectance curves.

[0118] It is clear from FIG. 2 that for a given 1D photonic crystal the total reflection bands vary in position and in width with the photon incidence angle and with its polarization. It can also be observed that in the band extending from point 7 to point 12, total reflection is produced for any incidence angle up to /4 rad and for any polarization. The same result would occur for a maximum incidence angle of /2 rad (levelling incidence), although in such case the band of hemispherical total reflection results narrower. Outside the total reflection bands represented by the segments above, the reflectance curves show a wavy behaviour, as already stated in this specification.

[0119] The photonic crystal represented in FIG. 2 contains 30 unit cells formed of a couple of dielectric layers of zinc sulphide, having an index of refraction of 2.3 and a thickness of 98 nanometers (nm) and magnesium fluoride, having an index of refraction of 1.35 and a thickness of 261 nm. All the layers are deposited on a glass substrate of index of refraction 1.52. This material of the substrate does not affect the forbidden bands (bandgaps) but it affects the reflections produced outside them. A slight rounding is produced at the corners of the total reflection bands. It is due to the finite number of layers in the photonic crystal (60 in this case). This rounding increases if the number of layers is reduced.

[0120] The present method is based on the study of the Chebyshev argument. For a given angle of incidence () of the radiation, the reflectance is wavy and below one when |.sub.TE|TM| <1 and is one (total reflectance) when |.sub.TE|TM|>1. The edges of the total reflection band, for a given angle of incidence, happen when |.sub.TE|TM|=1. For maximum angle of incidence (.sub.max), including .sub.max=/2, which corresponds to hemispheric radiation, the leading edge corresponds to normal radiation and the trailing edge corresponds to TM polarized .sub.max incidence (leveling for hemispheric radiation).

[0121] The present invention proposes the use of a plurality of photonic crystals which add their individual total reflection bands for the prescribed angular span (possibly hemispheric) until the desired wavelengths span is covered.

[0122] The present invention is based on a change of variables which allows the Chebyshev argument to be written in such a way that by selecting a leading edge, or alternatively a trailing edge, of the total reflection band of a photonic crystal, the first (h.sub.a) and second (h.sub.b) thicknesses of the photonic crystal's unit cell can be analytically determined. The other edge of the total reflection band inherent to each 1D-photonic crystal is obtained based on the calculated thicknesses.

[0123] Advantageously, the present method provides extremely high efficiency mirrors, with calculated efficiencies of up to 0.999999, with a broad span of the total reflectance band, for example from 1.77 to 20 m.

[0124] On the contrary, a configuration based on multilayer filters used in monochromatic mirrors (like the Carniglia's reference cited in the Background section), does not allow a total reflection band comprising tens of micrometers, thus including wavelengths from the visible to the mid infrared, nor does it provide the leading and trailing edges of the photonic crystal.

[0125] Albeit reference has been made herein to hemispheric or omnidirectional reflectance, the present invention can also be applied to cases where the total reflectance is assured within a maximum angle of incidence .sub.max</2.

[0126] To this end, the expression of the Chebyshev argument, .sub.TE|TM(.sub.0, , n.sub.a, n.sub.b, h.sub.a, h.sub.b), after some mathematical handling from the expression in the cited book of Born and Wolf, may be written as

[00015]$\begin{array}{cc}{}_{\mathrm{TE}.Math.\mathrm{TM}}=\cos \left[\frac{2\left(n_a{h}_a\cos {}_a+n_b{h}_b\cos {}_b\right)}{{}_0}\right]\frac{{\left(p_a+p_b\right)}^2}{4p_a{p}_b}& \left(1\right)\end{array}$$-\cos \left[\frac{2\left(n_a{h}_a\cos {}_a-n_b{h}_b\cos {}_b\right)}{{}_0}\right]\frac{{\left(p_a-p_b\right)}^2}{4p_a{p}_b}$$\mathrm{with}\left(\mathrm{for}\mathrm{non}\mathrm{magnetic}\mathrm{materials}\right)\{\begin{array}{c}\mathrm{TE}:p_{a,b}=n_{a,b}\cos \left({}_{a,b}\right)\\ \mathrm{TM}:p_{a,b}=\cos \left({}_{a,b}\right)/n_{a,b}\end{array}$

following the Snell law, .sub.a=arcsin()/n.sub.a, .sub.b=arcsin()/n.sub.b with being the angle of incidence with respect to axis z in the air and .sub.a, .sub.b being the angles of the photons (or planewave wavevectors) inside the layers of thickness h.sub.a, h.sub.b.

[0127] By making the following change of variables

[00016]$\begin{array}{cc}\frac{X}{2}=\frac{2\left(n_a{h}_a\cos {}_a+n_b{h}_b\cos {}_b\right)}{{}_0}& \left(2\right)\end{array}$$\frac{Y}{2}=\frac{2\left(n_a{h}_a\cos {}_a-n_b{h}_b\cos {}_b\right)}{{}_0}$$Z_{\mathrm{TE}.Math.\mathrm{TM}}=\{\begin{array}{c}{n}_a\cos \left({}_a\right)/\left(n_b\cos \left({}_b\right)\right)\mathrm{for}\mathrm{TE}\\ n_b\cos \left({}_a\right)/\left(n_a\cos \left({}_b\right)\right)\mathrm{for}\mathrm{TM}\end{array}$

.sub.TE|TM can be written as:

[00017]$\begin{array}{cc}{}_{\mathrm{TE}.Math.\mathrm{TM}}=\cos \left(X/2\right)\frac{{\left(1+Z_{\mathrm{TE}.Math.\mathrm{TM}}\right)}^2}{4Z_{\mathrm{TE}.Math.\mathrm{TM}}}-\cos \left(Y/2\right)\frac{{\left(1-Z_{\mathrm{TE}.Math.\mathrm{TM}}\right)}^2}{4Z_{\mathrm{TE}.Math.\mathrm{TM}}}& \left(3\right)\end{array}$

[0128] This change of variables provides big insight on the Chebyshev argument properties. Notice that the .sub.TE|TM (X, Y, Z.sub.TE|TM) function varies for different radiation incidence angles. In FIGS. 3 and 4 two plots of .sub.TE|TM (x, Y, Z.sub.TE|TM) vs. X are presented.

[0129] The plots in FIG. 3 correspond to cases where Y=0. When this happens .sub.TE|TM (X, Y, Z.sub.TE|TM) is periodic on variable X with period 4. In FIG. 3 the case for Z=3 is plotted in thick continuous line, and the case for Z=0.35 (almost the inverse of 3) is plotted in thin continuous line. This is to stress that a value of Z and its inverse give the same curve. The cases of Z=2 and Z=0.55 (almost ) are plotted in thick and thin dotted lines. The total reflection band takes place when is below the 1 gray line. If Z>1 it is broader when Z is larger. If Z<1 the total reflection band is broader when 1/Z is larger. In cases where Y=0 equality of the fractions of wavelengths in the high and low index of refraction layers is achieved.

[0130] In FIG. 4 it is shown the case of Z=3 and three values of Y. Y=0 (thick solid line) repeats one of the curves in FIG. 3. This case is the one with the largest total reflection band span. For the other cases shown in the figure, namely for Y=0.5 (dashed line) and Y=0.45 (dotted line), the curves are almost the same. This is to stress that opposite values of Y give the same . When Y0, a is not periodic anymore. Here the fraction of wavelength in the high and low index of refraction layers differs.

[0131] Many of the graphics in the present document are represented as a function of .sub.0. X and Y are inversely proportional to .sub.0 although their ratio Y/X=r.sub.TE|TM is independent of it, as it is also Z. This ratio is

[00018]$\begin{array}{cc}{r}_{\mathrm{TE}.Math.\mathrm{TM}}=\{\begin{array}{cc}\frac{\left(Z_{\mathrm{TE}}{h}_a/h_b\right)-1}{\frac{Z_{\mathrm{TE}}{h}_a}{h_b}+1}& \mathrm{for}\mathrm{TE}\\ \frac{{\cos }^2\left({}_a\right)-\left(\left(Z_{\mathrm{TM}}{h}_b/h_a\right){\cos }^2\left({}_b\right)\right)}{{\cos }^2\left({}_a\right)+\left(\left(\frac{Z_{\mathrm{TM}}{h}_b}{h_a}\right){\cos }^2\left({}_b\right)\right)}& \mathrm{for}\mathrm{TM}\end{array}& \left(4\right)\end{array}$

[0132] Furthermore, the first photonic bandgap which is the nearest to X=Y=0, occurring for .sub.0.fwdarw. is the most interesting one. This band (as visible in FIG. 3) is produced when .sub.TE|TM=1 and therefore the two first roots (respectively corresponding to the training and leading edges of the first total reflection band) that embrace the first photonic bandgap are the ones of interest.

[0133] Among the properties that can be extracted from the present analysis, it is found that the first photonic bandgap is largest when Y=0, that is, when the fraction of wavelength inside the high and low index of refraction materials is the same. The traditional use of the wavequarter, so much used in monochromatic optics, fulfils this condition. It is also found that the bandgap is larger when the ratio n.sub.a/n.sub.b or n.sub.b/n.sub.a is greater.

[0134] For Y=0, Equation (3) becomes a periodic function (with period 4 in X) ruled by cos(X/2) and the solution of .sub.TE/TM+1=0 is analytic. According to this, if a certain trailing edge wavelength .sub.0.sup.T is selected, the following expressions are obtained:

[00019]$\begin{array}{cc}{h}_{\mathrm{at}}=\frac{{}_0^T}{4n_a\cos {}_a}\left\{\arccos \left[\frac{{\left(1-Z\right)}^2-4Z}{{\left(1+Z\right)}^2}\right]\right\}& \left(5\right)\end{array}$$h_{\mathrm{bt}}=h_{\mathrm{at}}{n}_a\mathrm{Cos}\left({}_a\right)/\left(n_b\mathrm{Cos}\left({}_b\right)\right)$

wherein the subindex of Z has been dropped to mean that the equations are valid for both polarizations. If a certain leading edge is selected the equations are as follows:

[00020]$\begin{array}{cc}{h}_{\mathrm{al}}=\frac{{}_0^L}{4n_a\cos {}_a}\left\{2-\arccos \left[\frac{{\left(1-Z\right)}^2-4Z}{{\left(1+Z\right)}^2}\right]\right\}& \left(6\right)\end{array}$$h_{\mathrm{bl}}=h_{\mathrm{al}}{n}_a\mathrm{Cos}\left({}_a\right)/\left(n_b\mathrm{Cos}\left({}_b\right)\right)$

[0135] The arccos function has an infinity of solutions {, 2, 2+, 4, 4+ . . . }. The solution corresponds to the trailing edge of the total reflection band and the solution 2 corresponds to the leading edge of the total reflection band.

[0136] Once the thicknesses of the layers of the unit cell have been determined, either by setting the trailing edge wavelength or the leading edge wavelength, the photonic crystal is fully and uniquely defined, in default of the number N of unit cells forming the crystal, which as said above is not included in the Chebyshev argument. The more unit cells, the more squared is the total reflection band of the photonic crystal.

[0137] According to the present method, the unknown band edge, opposite to the band edge set at the beginning of the method, is obtained. For said purpose, Equation (1) is used and a .sub.0 root of +1=0 is obtained by numeric iterative resolution. There are several roots so that the initial .sub.0 set to start the iterations will determine the root found.

[0138] For it normalized Equation (3) is used to solve (X,rX,Z)+1=0 in X by numeric iterative resolution, starting with initial value 1 to obtain a trailing edge and with initial value 3 to find a leading edge. Being unnecessary, subindices have been dropped out. These initial values are found from inspection of FIG. 3. Once X has been extracted, the use of Equations (2) allows to write

[00021]$\begin{array}{cc}{}_{0,\mathrm{trail}}=\frac{8n_a{h}_{\mathrm{al}}\cos \left({}_a\right)}{X+Y}=\frac{8n_a{h}_{\mathrm{al}}\cos \left({}_a\right)}{X\left(1+r_{\mathrm{TM}}\right)}& \left(7\right)\end{array}$

[0139] Concerning .sub.a and .sub.b (also present in r.sub.TM), they can be 0 for a leading edge and their values may be derived from the Snell law relations starting with a vacuum (or air) angle of incidence .sub.max, that for hemispheric illumination is /2: .sub.a=ArcSin()/n.sub.a, .sub.b=ArcSin()/n.sub.b.

[0140] Concerning the indexes of refraction n.sub.a and n.sub.b, their values are defined by the material used and to a lesser extent, by the way of preparation of the material. In an embodiment the materials used for the layers of the unit cell are insulators, transparent to light. MgF.sub.2 or CaF.sub.2, having indexes of refraction 1.37397 and 1.4328 and electronic bandgaps of 12.2 and 10 eV, respectively, are preferred for the low index of refraction layers. ZnS, CdS and TiO.sub.2, having indexes of refraction 2.3677, 2.614 and 2.609 and electronic bandgaps of 2.54, 2.42 and 3.05 eV, respectively, are preferred for the high index of refraction layers. However, other materials, including polymers and organic materials, may be used.

[0141] In an embodiment, when arranging the photonic crystals in the mirror, layers of materials with small electronic bandgap are not located in the path of the radiation in order to avoid absorption of the photons before they reach the depth where they must suffer interference. For instance the vacuum wavelength corresponding to the CdS is, .sub.0=hc/2.42e=5.1210.sup.7 m, that renders this material not transparent below 512 nm, therefore opaque to the blue and UV radiation. For medium IR radiation the use of semiconductors is preferred. Si and Ge, with indices of refraction 3.42 and 4.04, are ideal as high index of refraction layers beyond .sub.0(Si)=hc/1.12e=1.10710.sup.6 m and .sub.0(Ge)=hc/0.67e=1.8505110.sup.6 m, since Si and Ge are not transparent to radiation in the visible range.

[0142] Thus, photonic crystals containing these semiconductors are preferably located in the mirror downstream photonic crystals having higher electronic bandgap dielectrics such that the high energy photons have been already reflected by said photonic crystals when the incoming radiation reaches the semiconductors.

[0143] For manufacturing the layers of the photonic crystals there is a bunch of possible technologies. Sputtering technology is interesting for price and reliability, but other techniques like MBE (molecular beam epitaxy) or MOVPE (metal organic vapor phase epitaxy) may be of high interest to explore high index of refraction layers.

[0144] FIG. 5 shows reflectance curves (in the upper part of the graph) and Chebyshev arguments (mainly in the lower part of the graph), as a function of the vacuum wavelength (in meters) of the incident photons for two photonic crystals. For both photonic crystals two cases are represented: under normal incidence (=0), and under levelling incidence (=/2 rad) and TM polarization; the solid lines represent the reflectance (R) and the Chebyshev argument () for the first photonic crystal under normal incidence (=0); the dotted lines represent the reflectance (R) and the Chebyshev argument () for the first photonic crystal under levelling incidence (=/2 rad for the Chebyshev argument and =0.99/2 rad for the reflectance) and TM polarization. The dashed lines represent the reflectance (R) and the Chebyshev argument () for the second photonic crystal under normal incidence (=0). The dot-dashed lines represent the reflectance (R) and the Chebyshev argument () for the second photonic crystal under levelling incidence (=/2 rad for the Chebyshev argument and =0.99/2 rad for the reflectance) and TM polarization. For the reflectance almost levelling (=0.99/2 rad) incidence has been used. The reason of using almost levelling rays in the reflectance is to avoid the presence of false total reflection bands formed by rays that do not actually enter the photonic crystals. This is not necessary in the Chebyshev arguments, for which levelling incidence (=/2 rad) has been used.

[0145] For the first photonic crystal a hemispheric total reflection band is formed between the point 24 (solid line) of diving of the Chebyshev argument of the first photonic crystal for normal incidence into the frame edge situated at =1, which defines its leading edge, and the point 25 (dotted line) of emergence into said frame of the Chebyshev argument of the first photonic crystal for levelling incidence and TM polarization, which defines its trailing edge; this is the hemispheric total reflection band (spanning from point 24 to point of the first photonic crystal, as explained in the discussion of FIG. 2. For the second photonic crystal a hemispheric total reflection band is formed between the point 26 (dashed line) of diving of the Chebyshev argument of the second photonic crystal for normal incidence, which defines its leading edge, and the point 27 (dot-dashed line) of emerging of the Chebyshev argument of the second photonic crystal for levelling incidence and TM polarization, which defines its trailing edge; this is the hemispheric total reflection band (spanning from point 26 to point 27) of the second photonic crystal.

[0146] The fact that the trailing edge wavelength value (25) of the first photonic crystal and the leading edge wavelength value (26) of the second photonic crystal coincide makes the two photonic crystals fit, if the two photonic crystals are deposited on the same substrate, forming a wider hemispheric total reflection band spanning from point 24 to point 27. The reflectance of the mirror comprising the two photonic crystals is not drawn in the figure and is squarer than those presented for the separated photonic crystals.

[0147] Once the leading edge (24) of the first photonic crystal for normal incidence is known, the thicknesses of the two layers of the unit cell are calculated using equation (6) particularized for normal incidence (also labelled leading in FIG. 6 of X abscissas). The trailing edge wavelength of the levelling radiation and TM polarization, is then calculated yielding the trailing edge of the hemispheric total reflection band of the first photonic crystal. As already explained, it is obtained by solving in X the equation (X,rX,Z)+1=0 starting in X=1 for levelling incidence and TM polarization. Once X is obtained, the trailing edge wavelength (25) is calculated with equation (7), again for levelling incidence and TM polarization (labelled trailing in FIG. 6). FIG. 6 shows a plot of the functions (X,rX,Z) X for normal incidence (solid line) and for levelling incidence and TM polarization (dashed line in FIG. 6). As shown, the dashed curve is slightly non-periodic, meaning that for it Y0. In the figure, the values of X for leading and training band edges are marked with thick dots.

[0148] For the second photonic crystal the trailing edge of the first photonic crystal obtained above becomes the leading edge (26), so leading to a perfect fitting of the two total reflection bands. The calculation method described for the first photonic crystal is repeated for the second photonic crystal. For a mirror comprising more than two photonic crystals, this process is to be repeated every two photonic crystals, until the trailing edge of the last photonic crystal equals or exceeds the highest vacuum wavelength at which the hemispheric total reflection band is desired to extend (.sub.B), that is, at the emergence point of the Chebyshev argument for levelling (.sub.max=/2 rad) incidence and TM polarization of the last photonic crystal. As for the initial wavelength of the hemispheric total reflection band of the mirror, it is situated at the leading edge of hemispheric total reflection band of the first photonic crystal (.sub.A), that is, at the diving point of the Chebyshev argument for normal incidence of the first photonic crystal.

[0149] In the example of FIG. 5 the high and low refraction indices for the firstly deposited photonic crystal are 3.43 (silicon) and 1.37 (magnesium fluoride) and the layer thicknesses are 166 and 413 nm, respectively. For the secondly deposited photonic crystal the refraction indices are 4.04 (germanium) and 1.37 (magnesium fluoride) and the layer thicknesses are 186 and 557 nm, respectively. The two photonic crystals are deposited in two separate glass substrates of index of refraction 1.52 and without any front protection (air). The leading edge (24) of the total reflection band of the mirror which includes the monolithic combination of the first and second photonic crystal is 1.77 m, corresponding to the electronic bandgap of a 0.7 eV photovoltaic cell; the obtained trailing edge (27) of the total reflection band of the mirror is 3.32 m. For a mirror including more than two photonic crystals conveniently fitted, the trailing edge would be much higher.

[0150] It can be appreciated in FIG. 5 that the total reflection bands at different incidence angles and polarizations extend well beyond the total reflection band for hemispheric total reflection, which extends from point 24 to point 27. This means that there is redundancy in the sense that many photons find more than one photonic crystal able to reflect them. The same happens with the levelling photons with TE polarization, whose reflectance in not drawn but forms a broader total reflection band and, in general, the same happens for all the photons. This explains the good results obtained even using very thin photonic crystals, having few unit cells. The results in FIG. 5 correspond to 10 unit cells per photonic crystal but only 7 unit cells often give good results and this number might be reduced.

[0151] FIG. 7 shows the reflectance (R) curves of a monolithic mirror made of multiple photonic crystals as a function of the vacuum wavelength in meters. The monolithic mirror is intended to reflect the radiation received hemispherically in the range of 1.77 to 20 m. In this embodiment the mirror is formed of a monolithic stack of 8 photonic crystals, each photonic crystal having 10 unit cells and, everything, monolithically deposited on the back face of a photovoltaic cell of 0.7 eV of electronic bandgap (close to that of germanium) and covered with a thick silver layer. In total, the stack has 160 layers of different dielectrics.

[0152] In this figure the solid line is the reflectance under normal incidence, whereas the dashed and dotted lines are the reflectance curves under =0.99/2 rad and TE and TM polarizations, respectively. The average reflectance under the radiation spectrum emitted by a blackbody at 1410 C. (the melting point of the metallurgical silicon), whose emission spectrum is presented in FIG. 8, in the range 1.77-20 m, unpolarized (as many TE as TM photons) and averaged by the energy spectrum, on all the angles of hemispheric incidence is 0.999999. Thus, a hemispheric total reflection band having a width of 18.24 m is achieved in this example with the averaged given energetic efficiency. It should be remarked that the best result obtained with the theoretical magnetic materials and the genetic algorithm of Qiang, H., Jiang, L., Li, X.: Design of broad omnidirectional total reflectors based on one-dimensional dielectric and magnetic Photonic Crystals, Optics and Laser Technology 42(1), 105-109 (2010), doi:10.1016/j.optlastec.2009.05.006 leads to a hemispheric total reflection band of 6.80 m, in contrast with the 18.24 m achieved in the example of FIG. 7 and no data of efficiency are given.

[0153] In this embodiment the high index of refraction material of the photonic crystals is zinc sulphide, silicon or germanium (depending on the specific photonic crystal) and the low reflection index material is magnesium fluoride, their thicknesses being different in each photonic crystal. The layers described in connection with the embodiment of FIG. 5 are a part of this mirror. As already said, the use of almost levelling incidence (=0.99/2 rad) is to avoid the apparent total reflection of photons not entering the mirror.

[0154] This invention also defines a thermal insulation for an incandescent body, wherein the thermal insulation comprises at least one mirror according to the invention. Preferably, the thermal insulation comprises a plurality of mirrors according to the invention. The incandescent body may be part of a furnace or of a system for energy storage, among others.

[0155] In the embodiment above a very good quality mirror has been designed according to the method of the invention by iterations from low to high wavelengths. With the present invention it is equally possible to design a similar structure starting with the high wavelengths and then iterate towards the shorter wavelengths. By knowing the trailing edge wavelength, equation (5) is readily used to obtain the thicknesses of the layers of the unit cell for levelling incidence and the successive leading edges are calculated with the solution in X of (X,rX,Z)+1=0 starting with X=3, and converting it into wavelength with equation (7).

[0156] A possible application of the present invention is the lining of a furnace for storing energy in molten silicon at 1410 C. The silicon is kept in a vessel heated by resistors, microwaves or by other means. This energy is eventually extracted as electrical power by means of thermo-photovoltaics.

[0157] In an embodiment of the present invention, the thermal insulation of the incandescent vessel is a wrapper comprising a plurality of monolithic mirrors of multiple photonic crystals. These mirrors reflect the photons emitted by the incandescent vessel with a very high efficiency. The averaged reflectance for a blackbody at 1410 C. between 0.6 and 35 m (outside this range the power is negligible) for unpolarized radiation in all hemispheric directions is 0.9998, constituting an exceptionally good thermal insulator. Indeed, the connection to the resistors heating the vessel and some pivots necessary for keeping the vessel in place leak heat, but they should be reduced to the strictly necessary to insure the electricity input and the mechanical stability.

[0158] In a preferred embodiment a mirror for thermal insulation of an incandescent body as described above contains 15 photonic crystals of 7 unit cells each, which are deposited monolithically on a metal covered with a thick layer of silver or gold. Several materials are used for the high index of refraction layers: zinc sulphide (2.614), silicon (3.42) and germanium (4.04). For the low index of refraction layers magnesium fluoride (1.374) is used in every photonic crystal. The mirror contains in total 210 layers. The mirror has been designed with the method of the present invention using the equation n.sub.ah.sub.a=n.sub.bh.sub.b (equation 6, normal incidence) for the ratio of high-low index thicknesses. Using the proceedings above, an averaged efficiency of 99.9899% is calculated, weighted by the radiation spectral power density of a blackbody at 1410 C. within the 0.6-20 m bandwidth and extending this weighted average to all hemispheric impinging angles and polarizations of the incident radiation. Following Stefan-Boltzmann's law, the radiation power of a blackbody at 1683K (1410 C.) is 45.5098 W/cm.sup.2, the reflected power in the range of 0.6-20 m, with the calculated average reflectance, is 45.3341 W/cm.sup.2; the reflectance outside this range is estimated by us in 70% that for the leftmost range of 0-0.6 m leads to a reflected power 0.0158632 W/cm.sup.2 and for the rightmost range of 20- m to a reflected power of 0.150173 W/cm.sup.2. The difference between the power incident and total three components of the power reflected is 0.00970275 W/cm.sup.2, this being the power absorbed and lost in the thermal insulation of the lining. This power is easily dissipated to the ambient with no substantial elevation of the mirror's temperature. The typical losses in a refractory/insulating state-of-the-art lining are of more than 1 W/cm.sup.2. Thus the present calculations give more than 100 times less thermal losses with the present mirror lining.

[0159] In an embodiment a part of the wrapper of mirrors are substituted by thermo-photovoltaic cells. In their manufacturing an integrated mirror of multiple photonic crystals may be deposited in the back face of the thermo-photovoltaic cell to reflect to the hot vessel many of the photons with energy too low as to produce photocurrent, and therefore almost not absorbed. In a thermo-photovoltaic cell adapted to convert the blackbody spectrum at the melting silicon temperature (1410 C.) the useless photons are of less than 0.7 eV, corresponding to a wavelength of 1.77 m. The mirror is deposited on the back face of the semiconductor cell during the cell manufacturing and it is convenient to finish it with a silver or gold layer to form the back electric contact. Reflectance curves appear in FIG. 7 and their behaviour has been previously described in the present specification. As said there, a 0.999999 averaged hemispheric reflectance is obtained between 1.77 and 20 m. However, the power reflected is today substantially less due to different losses in the thermo-photovoltaic cell but this result may stimulate important advances in thermo-photovoltaic efficiency.

[0160] In relation to coating with silver or gold the monolithic mirror of multiple photonic crystals, the zones of total reflection remain without variation but the zones outside them greatly increase their reflectance, although never as much as the zones of total reflection. This may have a practical interest in many applications.

[0161] In a different application, the monolithic mirror according to the invention can be used in parabolic mirrors of astronomical telescopes deposited in the hexagonal tesserae that usually constitute them and in which the small curvature of the tessera does not affect its manufacture. This mirror instead of only receiving light normal to the telescope within a moderate band of wavelengths, which can be achieved with a single photonic crystal, could operate receiving the light of the full firmament and with a very broad spectrum.

[0162] Many other applications may be envisaged for the mirrors designed and/or produced according to the present invention.