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2. Fresnel Equations The Fresnel equations are typically derived for homogeneous, lossless dielectric media. One form of the Fresnel expressions for lossless media is given by Equations 1 and 2. These equations express the single surface power reflectance and single surface power transmittance in terms of the indices n, and n₂ and the angle of incidence, 0₁. RTE (0₁)= RTM (0₁)= n₁.cos (0,)-√√n-n sin² (0₁) n, -cos (0,)+√√n-n sin² (0,) n cos (0,)-n,√√n-n sin² (0₁) n cos (0₁)+n√√n-n sin² (0₁) n, R(0°) = n₁-n₂ n₁ +n₂ (1) At incident angles near zero degrees, the equations can be simplified. The reflectance and transmittance equations are no longer a function of polarization state and are given as (2) n = n+ik (3) The Brewster angle is defined as the angle at which the reflectance of the TM-polarization, RTM goes to zero. It occurs when 0₁ +0₂= 90 degrees. When this occurs, the reflected and refracted ray directions are 90 degrees apart and the refracted ray's TM-state electric field vibrational direction is parallel to the direction of the reflected ray path. Therefore, the electromagnetic wave cannot propagate along the reflected ray path. The expression for the Brewster angle is given by Đg =arctan| A more general form of the Fresnel equations can be derived for the homogeneous, lossy dielectric media. The complex index of refraction for a lossy media is given by (5)
where n is the real part of the index as described above and k is the index of extinction. The following expressions for the single surface power reflectance have been derived assumin n₁ = 1 (air) and n₂ = n+ik (lossy media) and are given by and where and b² RTE RTM = RTE ² = ½ [√(n² = (a-cos (0₁))² + b² (a+cos (0₁))² + b² (a-sin (0₁) tan (0,₁))² + b² (a+sin (0,) tan (0₁)) + b² (n² −k² − sin² (0₁))² +4n²k² + (n² −k² −sin² -sin- (0,))] 2 −k² −sin³² (0, ))² + 4n³k² − (n² −k² – sin² (0,))] (6) (7) (8) (9)
These expressions are obtained by substituting the expression for the complex index of refraction (Equation 5) into the Fresnel equations for lossless dielectric media (Equations 1-2). Again, at an angle of incidence near zero degrees, the expression for the reflectance simplifies and is given by Material Fused Silica Zinc Selenide R(0°) = Germanium Aluminum (n-1)² +k² (n+1)² +k² A Brewster angle does not exist for a lossy media since the reflectance RTM does not go to zero. But, it does dip down to a minimum value. The angle at which RTM drops to the minimum value is called the principle angle of incidence. n 1.5 2.4 3.6 1.55 k 0.0 (10) 0.0 1.5 7.0 a) Plot the power reflectance RTM and RTE for all four materials as a function of incident angle from 0 to 90 degrees. b) Calculate the Brewster angle or principle angle of incidence for all four materials.