In this paper, a design for a multilayer dielectric cold mirror based on TiO2/ SiO2 and TiO2/MgF2 alternating layers is presented. A cold mirror is a specific dielectric mirror that reflects the complete visible light spectrum whereas transmitting the infrared wavelengths. These mirrors are designed for an incident angle of 45o, and are modeled with multilayer dielectric coatings similar to interference filters. Our designed mirror based on TiO2/SiO2 shows an average transmission of less than 5 % in the spectrum range of 425- 610 nm whereas it has an average transmission of 95 % in the spectrum range of 710-1500 nm.
Ts ( ) nnmo ts 2
, Rs ( ) rs 2 , where ts, rs — amplitude transmission and reflection coefficients of the multilayer interference system for s-polarized light whereas tp, rp — transmission and amplitude reflection coefficients for p-polarized light. Now, we will only consider spolarization because equations for both s and p polarization are related till equation (7). Amplitude coefficients are determined from the following equations: where n0, nm – the effective refractive indices of the substrate and the environment, respectively; mi, js – elements of the characteristic matrix Ms for s-polarized light:
M
m11s s im21s im12s
M1s M 2s M 3s .....M q2s M q1s M qs , m22s q – Number of layers. (2) (3) (4) (5) (6) (7) (8) (9) where hk – the physical thickness of the layers, nm; αk – the angles of refraction in the layers; nk – effective indexes refractive of the layers which depends on the wavelength. In this case, the angle of refraction in the layers is 45 degrees, relative to the normal.
The main difference in calculations between s- and p- polarized light is specified in (8) and (9) equations. n1s n1cos(1),
n2s n2cos( 2 ), n1p n1 / cos(1), n2 p n2 / cos( 2 ), The angle of refraction in the layers is calculated by the equations (10).
Tp ( )
, rs nom11s inonmm12s im21s nmm22s ,
nom11s inonmm12s im21s nmm22s In the expression (5) matrices Mk (k 1, q) determine the properties of each individual layer of the optical filter. Filter design needs layers with high and low refractive indices. Therefore, the spectral characteristics are described by matrices multiplying: cos(1) M1 in1sin(1) i
sin(1) n1 , cos(1) cos(2 ) M 2 in2sin(2 ) i
sin(2 ) n2 , cos(2 ) M q1
cos(q1) inq1sin(q1) nq1 i sin(q1)
, cos(q1) cos(q ) M p inqsin(q ) i nq
sin(q )
, cos(q ) where φk – phase thickness for s- polarized light, which is calculated by the following equations: 1 2 2 n1h1cos(1), 2
n2h2cos( 2 ), q1 nq1hq1cos( q1), q
nqhqcos( q ), 2 2 100 90 80 70 60 30 20 10 0 50 % 40
n2 n2 1 arccos 1 n1o2 sin2 ( o ) , 2 arccos 1 no22 sin2 ( o ) , Transmission of an unpolarized light is calculated as an average of Ts and Tp:
By using these equations, the transmission spectrum of the multilayer TiO2/MgF2 filter was plotted with the help of Java programing along with the transmission spectrum generated by commercially available open source filter Open filter. Their response is fairly comparable as shown in figure 2.
500 625 750 875 1000 1125
1250
Wavelength (nm)
In the designing of optical filters, the behaviour of the total multilayer system is estimated on the basis of the properties of the individual layers in the stack [8]. Therefore to achieve the optimum performance, it is significant to optically characterize and accurately determine the thickness of the individual layers. In this work, cold mirrors are designed in the wavelength range of 425-1500nm by using open source software Open Filter to selectively pass the wavelengths of interest and rejecting the undesired wavelengths in the visible spectrum. TiO2, SiO2 and MgF2 materials are carefully selected based on their high and low refractive indices, respectively. TiO2 is a vital dielectric material with a wide band-gap energy and high refractive index that can make it useful in the fabrication of multilayer thin films due to its high optical properties. For instance, its high transmittance and high refractive index in the visible region (380-760 nm) make it valuable to be employed in the production of the optical filter and window glazing [9, 10]. Layers made of oxides are harder than those made of fluorides, sulphides or semiconductors. Thus, they are ideal to be used on exposed surfaces. Semiconductor materials should be avoided in filters which have to be used over a wide range of temperatures because their optical constants can change considerably. The open filter uses transfer matrix method to analyze the transmission and reflection of light from layers based on thickness and type of materials. Designs are optimized to maximum the transmission required at wavelengths using needle synthesis method (addition of thin layers called needle and analyze transmission till the best results obtained) [11]. The thicknesses of the layers for cold mirror based on TiO2/MgF2 and TiO2/SiO2 are shown in table 1. Both mirrors have 20 layers with almost comparable total thickness. Special attention has been given to keep the thickness of the filters within economic limits.
Assuming the incident angle of un-polarized light equals 45o, these mirrors have reflective properties in the spectral range from 425-610 nm and 710-1500 nm up to 95 % and 5 %, respectively as shown in figure 3. For all dielectric stack filters, the transmission depends on the angle of incidence. The central wavelength of the FP filter shifts toward the smaller wavelengths as the angle of incidence is increased. When the incident angle of light decreases from 45o to 0o the transmission spectrum shifts from 710 to 750 nm.
In this work, we presented the modeling results of cold mirrors based on TiO2/MgF2 and TiO2/SiO2 for 45o of un-polarized incident light by using java programming and commercially available Open source software Open filter. Both mirrors show the reflection of 95% in the spectral range of 425-610 nm and 95% of transmission in the spectral range of 710-1500 nm. The designs are optimized to maximum the transmission required at wavelengths using needle synthesis method. We observed a right shift in a spectrum when the angle of incidence of light was reduced from 45o to 0 .
o
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