Designing High Reflectivity Omnidirectional Coating of Mirrors for Near Infrared Spectrum ( 700-2500 nm )

In this paper, a high reflection coating is designed depending on the variable of refractive indices for NIR spectral region (700-2500 nm) by the use of the computer program MATLAB version 7. We could find the reflective 99.62% for seven layers at the incident angels (90°, 40°) in the wavelength (1064 nm) for coatings (Si, MgF2), substrata BK7 (relatively hard borosilicate crown glass with high homogeneity), which is used for laser application such as the ND:YAG laser (1060 nm), and R=98.37% for coatings (SbSe, Na3AIF6) and substrata glass for eleven layers at θ=90° which covers the wavelength from (955.6 nm) to (1622 nm) and represents the complete range for optical telecommunication band (short (S) 1460-1530 nm, conventional (C) 1530-1560 nm and long (l)1560-1620 nm). The results show that the reflectivity of the stack increases with the number of layers in the stack, the best layer number is nine which has a reflective of 99.62% at (1060 nm), as shown in Figure 4a. Also the reflective changes with incident angel; the best angel is (40°) which gives the convergent reflective for electric and magnetic polarization 99.91% and 99.36%, respectively for the wavelength (1060 nm).


Introduction
Infrared waves are electromagnetic waves with frequencies lower than visible light.The lowest frequencies of visible light are red, so one call the lower frequencies infrared, meaning beyond red.This type of electromagnetic radiation is widely used for local communications (Daniel et al., 1996).The IR spectral region of the electromagnetic spectrum extends from the red end of the visible spectrum to the microwave region; it includes radiation with wave numbers ranging from about 14,000 to 20 cm -1 , or wavelengths from 700 to 50000 nm.The near-IR (NIR) region extends from the visible region at (700 nm) to the mid-IR region at (2500 nm) (Michele et al., 1995).
When the electromagnetic wave is incident on plane surface bonding two substances of indices n and n¯, this wave suffer absorption, reflection, dispersion and transmittance , according to Equation (1), as shown in Figure 1 (Stenzel, 2005):

T+R+A+S=1
(1) Figure 1.The definitions of T, R, and S. ϕ are the incidence angle (Stenzel, 2005) Where transmittance is T, the specula reflectance R, the optical scatter S, the absorptions A, as conservation of energy.But as we work with homogenous materials and flat surfaces, scatter does not occur and Equation ( 1) becomes (Stenzel, 2005): Multilayer optical thin film coating has extensively been used for reflectivity modulation in various optical and optoelectronic components (Kheraj et al., 2008).It's has a wide range of applications, such as displays; applications of near-infrared, such as camera lenses, mirrors, and filters; eyeglasses; mirrors, laser windows, and polarizes; optics of photocopiers and compact disks; optical communications; home appliances, such as heat reflecting oven windows ( (Kheraj et al., 2008;Sophocles, 2010) Multilayer optical thin film coating include anti reflection (AR) and high reflection (HR) coating on leaser diode facets (Kheraj et al., 2008).Laser and fabry-perot resonators require high reflectivity that is unattainable from metallic films which also suffer from high absorption in a lesser degree (Sharma, 2006).AR coating is used on lenses of camera and telescope and fabrication of polarizing beam splitters and various optical filters (Kheraj et al., 2008).
The main interest in dielectric mirrors is that they have extremely low losses at optical and infrared frequencies, as compared to ordinary metallic mirrors.On the other hand, metallic mirrors reflect over a wider bandwidth than dielectric ones and from all incident angles.A dielectric mirror consists of identical alternating layers of high and low refractive indices, as shown in Figure 2(a).The optical thicknesses are typically chosen to be quarter-wavelength long, n l n l ₒ at some operating wavelength λ 0 (center wavelength).The standard arrangement is to have an odd number of layers, with the high index layer being the first and last layer (Sophocles, 2010).
The Fresnel amplitude reflection coefficient (r) for an interface between two Non-absorbing media at normal incidence can be represented by the following equation (Ronald, 2002): where ₁ and ₂ are the (real) indices of refraction of the two media.For the more general case of absorbing media with complex refractive indices 1,2, … .the reflection intensity coefficient (R) is the amplitude reflection coefficient (r) times its complex conjugate (r*) (Wwillely, 2002)  Figure 2b shows the geometry of this situation.Notice that reflected and transmitted waves have been introduced.The plane containing the incident wave-vector k, the reflected and transmitted wave-vector and the normal n is the (x, z) plane, and the vector n is along the z-direction.We denote the amplitudes (electric field magnitudes) of the incident, reflected and transmitted waves by I, R and T respectively.The magnitudes of the wave-vectors in the two media are K₁ and K₂ and clearly k₁/n₁ = k₂/n₂ = k 0 since both waves have the same frequency.At normal incidence, the reflection and transmission coefficients for the two polarizations are equal and are given by (Lipson et al., 2010): where n is the relative refractive index between the two media.We define reflection and transmission coefficients for polarization TE or ⊥ (denoted by the subscript ⊥ ) (A. LIPSON et al., 2010): The coefficients for the││ plane of polarization can be worked out similarly (Lipson et al., 2010): These functions are known as Fresnel coefficients (Lipson et al., 2010).
A high reflectance coating can be designed by using dielectric quarter-wave stack of alternate high-and lowrefractive index materials.If and are the indices of the high-and low-index layers and if the stack is arranged, the high-index layers are outermost at both sides.The transformation matrix for a stack of N pairs of quarter-wave layers of high and low refractive index materials can be expressed in the form (Sharma, 2006) where the normal incidence was assumed to be : where (t) is the thickness, cos and cos in the process of wave propagation through an N layer stack.n H , n L are indices of the high and low index layers, where θ, are the angle of reflection and angle of refraction, , , and as shown in Figure 3 (Sharma,2006).
Figure 3.The N layer stack; φ j and n j are the angles of refraction and index of refraction of the jth layer (Sharma, 2006) The coefficients , can be calculated for either state of polarization from the knowledge of indices of refraction of the films and angles of refraction in each film.The latter can be obtained from: giving the reflectance of the stack as (Sharma, 2006): the reflectance in air or free space is then (Al-Dujely, 2000;Macleod, 1986Macleod, , 2001):

R = (17)
The reflectivity of the stack increases with the number of layers in the stack and the spectral bandwidth of high reflectance increases with the ratio.Multi-layer broad band high reflectance dielectric mirrors are now readily available (Sharma, 2006).The reflectance of such a film depends on the constructive or destructive interference of light reflected at successive boundaries of different layers of the multilayer stack (Kheraj et al., 2008).
High reflectance coating usually consists of alternate quarter-wave layer of materials with high and low indices of refraction ( ₒ ) (Wwillely, 2002).The standard arrangement is to have an odd number of layers, with the high index layer being the first and last layer (Orfanidis, 2010).

Results and Discussion
We design a computer program using MATLAB version 7 to calculate and optimize the high-reflectivity for different materials (coatings) within near infrared wave.This program depends on refractive index of materials (coating), number of layers and incident angle.5) show that the coating consists of (Si) as high refraction index (3.42)(Farlow & Boatener, 1997), (MgF 2 ) as low refraction index (1.37)(Al-Dujely, 2000) and the substrate (BK7) with the refractive index (1.505)(BK7 used because it has low losses and high stability, Al-Dujely, 2000), for the central wavelength ( 0 =1200 nm).The results are shown in Table 1.

Figures (4) and (
Figures 6 and 7 show that the coating consists of (SbSe) as high refraction index (3.23 (Kohoutek et al., 2009)), (Na3AlF6 as low refraction index (1.35(Kim & Hwangbo, 2002)) and the substrate (glass) with the refractive index (1.52 (Lusk & Placido, 2005)), for the central wavelength ( 0 =1200 µm).The results are shown in Table 2 The reflectivity of (BK7) is 4.064 %, and when coating one layer from the Si, the reflective becomes R=59.6 %, and if one uses three layers [Si MgF2 Si], the reflective becomes R=92.07 % at incident angle θ=90˚ at λ 1200 nm .This shows that the reflectivity of the stack increases with the number of layers in the stack, according to equation ( 16).The spectral bandwidth of high reflectance increases with the n H /n L ratio (Sharma, 2002).When incident angle θ=90, this reflects that both polarization TE and TM are corresponding.
To obtain a high reflectivity by reducing the number of layers, one can use incident angle as a variable (Kim & Hwangbo, 2002), as the incident angle increases, the admittance of TE polarization increases and that of TM polarization decreases.So the reflection bandwidth of TE polarization is wider than that at normal incidence, and that of TM polarization is narrower.Therefore, at a high incident angle and reference wavelength, the reflectance of TM-polarized light may be quite low in contrast to the high reflectance of TE-polarized light (Kim & Hwangbo, 2002).It appears that the best coating is when R=99.79 %at incident angle θ=90˚ and R=99.90 % R=99.31 % for TE, TM respectively at incident angle θ=40˚ for seven layers for coating (MgF 2 , Si and substrate BK7), and R=99.21 %, R=99.13 for TE, TM respectively at incident angle θ=40˚ for nine layers for coating (SbSe, Na3AlF6 and substrate glass) to obtain the higher reflectivity and lowest number of layers.
As the incident angel increases the center wavelengths of both reflection bands shift to shorter wavelength region as show in Figures 6(b, c) and 7(b, c) in Table (2).This coating which used BK7 as substrate is used in laser applications as coating for the ND:YAG laser (1060 nm) (Al-Dujely, 2000).In this paper, the reflectivity for the coating was R=98.07 % at incident angle θ =90˚, R=99.31 for TE and R=96.92 % for TM, θ=45˚, R=99.45 % for TE and R=96.35 % for TM for N=2, as shows in Figure (8a, b, c) when N=3, R=99.61 % at incident angle θ =90˚, R=99.91 for TE and R=99.35 % for TM for θ=40˚ and when θ=45˚ R=99.93% for TE and R=99.2% for TM, as shows in Figure 4 (a ,b, c).It appears that the best coating was R=99.91% ,R=99.35% at θ=90˚, θ=40˚, respectively for N=3, the best coating was when obtaining the higher reflectivity and lower number of layers, and the best incident angle was θ=40˚ because the beam attenuation increases with increasing the angle of incidence beyond the critical angle (Sophocles, 2010).
When comparing these results with the results of AL-Dujely Wasfi Hammed Rasheed (2000), who used thin-film ZnS, MgF 2 as coating and BK7 as the substrate and got the reflectivity R =82.5 % at a center wavelength of 1060 nm in five layers, it seems that the coating one can design when N=3 is the best.
The substrate (glass) which has the reflectivity R=4.258 % with the coating (SbSe) and (Na3AlF6) has a reflectivity greater than 98.37 % for the range (955.6-1622nm) for N=5 at θ=90˚ as shows Figure 7(a).This coating is used in laser applications and optical communications.So that it covers bands S, C and L for optical telecommunication where (C band is 1530-1560 nm), (S band is 1460-1530 nm) and (L band is 1560-1620 nm) (Kim & Hwangbo, 2002).The coating one can design when N=5 is best compared with that of Kim and Hwangbo used glass as the substrate and (TiO₂, SiO₂) as coating and obtained a reflectivity greater than 98 % at TM , θ=85˚, θ=0˚ for ( N=30) covering the S, C and L from 1286.8 nm to 1629.1 nm (Kim & Hwangbo, 2002).
The reflectivity of (BK7) is 4.064%, and when coating one layer from the Si, the reflective becomes R=59.6%, and if one uses three layers [Si MgF2 Si], the reflective becomes R=92.07 % at incident angle θ=90˚ at λ 1200 nm).This shows that the reflectivity of the stack increases with the number of layers in the stack, according to Equation ( 16).The spectral bandwidth of high reflectance increases with the n H /n L ratio (Sharma, 2002).When incident angle θ=90˚, this reflects that both polarization TE and TM are corresponding.
To obtain a high reflectivity by reducing the number of layers, one can use incident angle as a variable (Kim & Hwangbo, 2002), as the incident angle increases, the admittance of TE polarization increases and that of TM polarization decreases.So the reflection bandwidth of TE polarization is wider than that at normal incidence, and that of TM polarization is narrower.Therefore, at a high incident angle and reference wavelength, the reflectance of TM-polarized light may be quite low in contrast to the high reflectance of TE-polarized light (Kim & Hwangbo, 2002).It appears that the best coating is when R=99.79% at incident angle θ=90˚ and R=99.90 % R=99.31% for TE, TM respectively at incident angle θ=40˚ for seven layers for coating (MgF 2 , Si and substrate BK7), and R=99.21 %, R=99.13 for TE, TM respectively at incident angle θ=40˚ for nine layers for coating (SbSe, Na 3 AlF 6 and substrate glass) to obtain the higher reflectivity and lowest number of layers.
As the incident angel increases the center wavelengths of both reflection bands shift to shorter wavelength region as show in Figures 6 (b, c) and 7 (b, c) in Table 2.This coating which used BK7 as substrate is used in laser applications as coating for the ND:YAG laser (1060nm) (Al-Dujely, 2000).In this paper, the reflectivity for the coating was R=98.07 % at incident angle θ=90˚, R=99.31 for TE and R=96.92 % for TM, θ=45˚, R=99.45 % for TE and R=96.35 % for TM for N=2, as shows in Figure 8 (a, b, c) when N=3, R=99.61 % at incident angle θ =90˚, R=99.91 for TE and R=99.35 % for TM for θ=40 ̊ and when θ=45 ̊ R=99.93 % for TE and R=99.2 % for TM, as shows in Figure 4 (a, b, c).It appears that the best coating was R=99.91 %, R=99.35 % at θ=90˚, θ=40˚, respectively for N=3, the best coating was when obtaining the higher reflectivity and lower number of layers, and the best incident angle was θ=40˚ because the beam attenuation increases with increasing the angle of incidence beyond the critical angle (Sophocles, 2010).
When comparing these results with the results of AL-Dujely Wasfi Hammed Rasheed", 2000, who used thin-film ZnS, MgF 2 as coating and BK7 as the substrate and got the reflectivity R=82.5 % at a center wavelength of 1060 nm in five layers, it seems that the coating one can design when N=3 is the best.The substrate (glass) which has the reflectivity R=4.258 % with the coating (SbSe) and (Na3AlF6) has a reflectivity greater than 98.37 % for the range (955.6-1622nm) for N=5 at θ=90˚ as shows Figure 7a.This coating is used in laser applications and optical communications.So that it covers bands S, C and L for optical telecommunication where (C band is 1530-1560 nm), (S band is 1460-1530 nm) and (L band is 1560-1620 nm) (Kim & Hwangbo, 2002).The coating one can design when N=5 is best compared with that of Kim and Hwangbo used glass as the substrate and (TiO₂, SiO₂) as coating and obtained a reflectivity greater than 98% at TM, θ=85˚, θ=0˚ for (N=30) covering the S,C and L from 1286.8 nm to 1629.1 nm (Kim & Hwangbo, 2002).

Conclusion
From the above result, one can conclude that the reflectivity for coating within near infrared waves (700-2500 nm) depending on the coating type, number of layers and angle of incidence.The addition of layers does not affect the width of zone of high reflectance, but increases the reflectivity within.The best coating used in laser applications as coating for the ND:YAG laser (1060nm) is (Si ,MgF 2 ) on the substrata BK7 which gives the best reflection R=99.61 % at θ=90˚, R=99.91 for TE and R=99.35 % for TM for θ=40˚ at N=3.The best coating used in optical communications is (SbSe, Na3AlF6) on the substrata which gives the best reflection (greater than 98.37 %) for the range( 955.6-1622 nm) at =90˚ for N=5.