Physics of Reflections and Refractions Snell's Laws Angles of Incidence, Reflection & Refraction Snell's Law of Reflection: The angle of reflection equals the angle of incidence.
θ < e m > i = θ < / e m > r \theta<em>i = \theta</em>r θ < e m > i = θ < / e m > r ( 8.28 a ) (8.28a) ( 8.28 a ) Snell's Law of Refraction: Relates the sine of the angles of incidence and refraction to the ratio of phase velocities in the two media.
sin θ < e m > t sin θ < / e m > i = μ < e m > 1 ϵ < / e m > 1 μ < e m > 2 ϵ < / e m > 2 = v < e m > p 1 v < / e m > p 2 \frac{\sin \theta<em>t}{\sin \theta</em>i} = \sqrt{\frac{\mu<em>1 \epsilon</em>1}{\mu<em>2 \epsilon</em>2}} = \frac{v<em>{p1}}{v</em>{p2}} s i n θ < / e m > i s i n θ < e m > t = μ < e m > 2 ϵ < / e m > 2 μ < e m > 1 ϵ < / e m > 1 = v < / e m > p 2 v < e m > p 1 ( 8.28 b ) (8.28b) ( 8.28 b ) Index of Refraction (n): Defined as the ratio of the speed of light in vacuum (c) to the phase velocity (v p v_p v p
n = c v < e m > p = μ < / e m > r ϵ < e m > r = μ ϵ μ < / e m > 0 ϵ 0 n = \frac{c}{v<em>p} = \sqrt{\mu</em>r \epsilon<em>r} = \sqrt{\frac{\mu \epsilon}{\mu</em>0 \epsilon_0}} n = v < e m > p c = μ < / e m > rϵ < e m > r = μ < / e m > 0 ϵ 0 μ ϵ ( 8.29 ) (8.29) ( 8.29 ) Rewriting Snell's Law of Refraction: Using the index of refraction.
sin θ < e m > t sin θ < / e m > i = n < e m > 1 n < / e m > 2 = μ < e m > r 1 ϵ < / e m > r 1 μ < e m > r 2 ϵ < / e m > r 2 \frac{\sin \theta<em>t}{\sin \theta</em>i} = \frac{n<em>1}{n</em>2} = \sqrt{\frac{\mu<em>{r1} \epsilon</em>{r1}}{\mu<em>{r2} \epsilon</em>{r2}}} s i n θ < / e m > i s i n θ < e m > t = n < / e m > 2 n < e m > 1 = μ < e m > r 2 ϵ < / e m > r 2 μ < e m > r 1 ϵ < / e m > r 1 ( 8.30 ) (8.30) ( 8.30 ) For Nonmagnetic Materials (μ < e m > r 1 = μ < / e m > r 2 = 1 \mu<em>{r1} = \mu</em>{r2} = 1 μ < e m > r 1 = μ < / e m > r 2 = 1
sin θ < e m > t sin θ < / e m > i = n < e m > 1 n < / e m > 2 = ϵ < e m > r 1 ϵ < / e m > r 2 \frac{\sin \theta<em>t}{\sin \theta</em>i} = \frac{n<em>1}{n</em>2} = \sqrt{\frac{\epsilon<em>{r1}}{\epsilon</em>{r2}}} s i n θ < / e m > i s i n θ < e m > t = n < / e m > 2 n < e m > 1 = ϵ < / e m > r 2 ϵ < e m > r 1 ( 8.31 ) (8.31) ( 8.31 ) Refraction Behavior Inward Refraction:
Occurs when a wave enters a more dense medium (n < e m > 1 < n < / e m > 2 n<em>1 < n</em>2 n < e m > 1 < n < / e m > 2 The angle of refraction is smaller than the angle of incidence (θ < e m > t < θ < / e m > i \theta<em>t < \theta</em>i θ < e m > t < θ < / e m > i Outward Refraction:
Occurs when a wave enters a less dense medium (n < e m > 1 > n < / e m > 2 n<em>1 > n</em>2 n < e m > 1 > n < / e m > 2 The angle of refraction is larger than the angle of incidence (θ < e m > t > θ < / e m > i \theta<em>t > \theta</em>i θ < e m > t > θ < / e m > i Total Internal Reflection Critical Angle: The angle of incidence at which the refraction angle is 90 degrees.
Total Internal Reflection: When the angle of incidence exceeds the critical angle, the wave is totally reflected back into the original medium.
Surface waves exist on the transmission side. The reflection coefficient (Γ \Gamma Γ Γ = 1 \Gamma = 1 Γ = 1 Optical Fiber Components:
Fiber core (index of refraction n f n_f n f Cladding (index of refraction n c n_c n c Surrounding medium (index of refraction n 0 n_0 n 0 Acceptance Cone: Defines the range of angles within which light can enter the fiber and be guided through it via total internal reflection.
Waveguiding: Waves can be guided along optical fibers due to successive internal reflections.
Example: Light Beam Passing Through a Slab Setup: A dielectric slab (index of refraction n < e m > 2 n<em>2 n < e m > 2 n < / e m > 1 n</em>1 n < / e m > 1
Condition: If the incidence angle (θ < e m > i \theta<em>i θ < e m > i θ < / e m > c \theta</em>c θ < / e m > c
Solution:
At the upper surface: sin θ < e m > 2 sin θ < / e m > 1 = n < e m > 1 n < / e m > 2 \frac{\sin \theta<em>2}{\sin \theta</em>1} = \frac{n<em>1}{n</em>2} s i n θ < / e m > 1 s i n θ < e m > 2 = n < / e m > 2 n < e m > 1 ( 8.33 ) (8.33) ( 8.33 ) At the lower surface: sin θ < e m > 3 sin θ < / e m > 2 = n < e m > 2 n < / e m > 1 \frac{\sin \theta<em>3}{\sin \theta</em>2} = \frac{n<em>2}{n</em>1} s i n θ < / e m > 2 s i n θ < e m > 3 = n < / e m > 1 n < e m > 2 ( 8.34 ) (8.34) ( 8.34 ) Combining the equations: sin θ < e m > 3 = n < / e m > 2 n < e m > 1 sin θ < / e m > 2 = n < e m > 2 n < / e m > 1 ⋅ n < e m > 1 n < / e m > 2 sin θ < e m > 1 = sin θ < / e m > 1 \sin \theta<em>3 = \frac{n</em>2}{n<em>1} \sin \theta</em>2 = \frac{n<em>2}{n</em>1} \cdot \frac{n<em>1}{n</em>2} \sin \theta<em>1 = \sin \theta</em>1 sin θ < e m > 3 = n < e m > 1 n < / e m > 2 sin θ < / e m > 2 = n < / e m > 1 n < e m > 2 ⋅ n < / e m > 2 n < e m > 1 sin θ < e m > 1 = sin θ < / e m > 1 Therefore: θ < e m > 3 = θ < / e m > 1 \theta<em>3 = \theta</em>1 θ < e m > 3 = θ < / e m > 1 Conclusion: The slab displaces the beam's position, but the beam's direction remains unchanged.
Oblique Incidence: Snell's Law Inward Refraction:
Occurs when n < e m > 1 < n < / e m > 2 n<em>1 < n</em>2 n < e m > 1 < n < / e m > 2 θ < e m > t < θ < / e m > i \theta<em>t < \theta</em>i θ < e m > t < θ < / e m > i Outward Refraction:
Occurs when n < e m > 1 > n < / e m > 2 n<em>1 > n</em>2 n < e m > 1 > n < / e m > 2 θ < e m > t > θ < / e m > i \theta<em>t > \theta</em>i θ < e m > t > θ < / e m > i EM Polarized Waves Traveling and standing waves in different mediums. Medium 1: (\mu1, \epsilon 1, \sigma1, \eta 1, \beta_1) Medium 2: (\mu2, \epsilon 2, \sigma2, \eta 2, \beta_2) Oblique Incidence Plane of Incidence: Defined by the normal to the boundary and the direction of propagation of the incident wave (x-y plane).Perpendicular/TE Polarization Electric field is perpendicular to the plane of incidence.
Magnetic Field components: H < e m > i H<em>i H < e m > i H < / e m > r H</em>r H < / e m > r H t H_t H t
Electric Field components: E < e m > i E<em>i E < e m > i E < / e m > r E</em>r E < / e m > r E t E_t E t
Parallel/TM Polarization Electric field is parallel to the plane of incidence.
Magnetic Field components: H < e m > i H<em>i H < e m > i H < / e m > r H</em>r H < / e m > r H t H_t H t
Electric Field components: E < e m > i E<em>i E < e m > i E < / e m > r E</em>r E < / e m > r E t E_t E t
General Polarization Electric and Magnetic Field components. k ⃗ = k ( cos ϕ x ^ − sin ϕ y ^ ) \vec{k} = k(\cos\phi \hat{x} - \sin\phi \hat{y}) k = k ( cos ϕ x ^ − sin ϕ y ^ ) r ⃗ = x x ^ + y y ^ + z z ^ \vec{r} = x\hat{x} + y\hat{y} + z\hat{z} r = x x ^ + y y ^ + z z ^ E ⃗ = E < e m > x + E < / e m > y \vec{E} = E<em>x + E</em>y E = E < e m > x + E < / e m > y H ⃗ \vec{H} H Perpendicular Polarization Detailed illustration of incident, reflected, and transmitted waves with perpendicular polarization.
Relationships between electric and magnetic fields, angles, and media properties.
Incident Wave: E < e m > i ⃗ = y ^ E < / e m > i 0 e − j k < e m > 1 ( x sin θ < / e m > i + z cos θ < e m > i ) \vec{E<em>i} = \hat{y} E</em>{i0} e^{-jk<em>1(x\sin\theta</em>i + z\cos\theta<em>i)} E < e m > i = y ^ E < / e m > i 0 e − jk < e m > 1 ( x s i n θ < / e m > i + z c o s θ < e m > i ) H < / e m > i ⃗ = ( − x ^ cos θ < e m > i + z ^ sin θ < / e m > i ) E < e m > i 0 η < / e m > 1 e − j k < e m > 1 ( x sin θ < / e m > i + z cos θ i ) \vec{H</em>i} = (-\hat{x} \cos\theta<em>i + \hat{z} \sin\theta</em>i) \frac{E<em>{i0}}{\eta</em>1} e^{-jk<em>1(x\sin\theta</em>i + z\cos\theta_i)} H < / e m > i = ( − x ^ cos θ < e m > i + z ^ sin θ < / e m > i ) η < / e m > 1 E < e m > i 0 e − jk < e m > 1 ( x s i n θ < / e m > i + z c o s θ i )
Reflected Wave: E < e m > r ⃗ = y ^ E < / e m > r 0 e − j k < e m > 1 ( x sin θ < / e m > r − z cos θ < e m > r ) \vec{E<em>r} = \hat{y} E</em>{r0} e^{-jk<em>1(x\sin\theta</em>r - z\cos\theta<em>r)} E < e m > r = y ^ E < / e m > r 0 e − jk < e m > 1 ( x s i n θ < / e m > r − z c o s θ < e m > r ) H < / e m > r ⃗ = ( x ^ cos θ < e m > r + z ^ sin θ < / e m > r ) E < e m > r 0 η < / e m > 1 e − j k < e m > 1 ( x sin θ < / e m > r − z cos θ r ) \vec{H</em>r} = (\hat{x} \cos\theta<em>r + \hat{z} \sin\theta</em>r) \frac{E<em>{r0}}{\eta</em>1} e^{-jk<em>1(x\sin\theta</em>r - z\cos\theta_r)} H < / e m > r = ( x ^ cos θ < e m > r + z ^ sin θ < / e m > r ) η < / e m > 1 E < e m > r 0 e − jk < e m > 1 ( x s i n θ < / e m > r − z c o s θ r )
Transmitted Wave: E < e m > t ⃗ = y ^ E < / e m > t 0 e − j k < e m > 2 ( x sin θ < / e m > t + z cos θ < e m > t ) \vec{E<em>t} = \hat{y} E</em>{t0} e^{-jk<em>2(x\sin\theta</em>t + z\cos\theta<em>t)} E < e m > t = y ^ E < / e m > t 0 e − jk < e m > 2 ( x s i n θ < / e m > t + z c o s θ < e m > t ) H < / e m > t ⃗ = ( − x ^ cos θ < e m > t + z ^ sin θ < / e m > t ) E < e m > t 0 η < / e m > 2 e − j k < e m > 2 ( x sin θ < / e m > t + z cos θ t ) \vec{H</em>t} = (-\hat{x} \cos\theta<em>t + \hat{z} \sin\theta</em>t) \frac{E<em>{t0}}{\eta</em>2} e^{-jk<em>2(x\sin\theta</em>t + z\cos\theta_t)} H < / e m > t = ( − x ^ cos θ < e m > t + z ^ sin θ < / e m > t ) η < / e m > 2 E < e m > t 0 e − jk < e m > 2 ( x s i n θ < / e m > t + z c o s θ t )
Applying Boundary Conditions Tangential E Continuous:
Tangential H Continuous:
Solution of Boundary Equations Parallel Polarization Reflection coefficient: r < e m > ∥ = η < / e m > 2 cos θ < e m > t − η < / e m > 1 cos θ < e m > i η < / e m > 2 cos θ < e m > t + η < / e m > 1 cos θ i r<em>\parallel = \frac{\eta</em>2 \cos \theta<em>t - \eta</em>1 \cos \theta<em>i}{\eta</em>2 \cos \theta<em>t + \eta</em>1 \cos \theta_i} r < e m >∥= η < / e m > 2 c o s θ < e m > t + η < / e m > 1 c o s θ i η < / e m > 2 c o s θ < e m > t − η < / e m > 1 c o s θ < e m > i
Transmission coefficient: t < e m > ∥ = 2 η < / e m > 2 cos θ < e m > i η < / e m > 2 cos θ < e m > t + η < / e m > 1 cos θ i t<em>\parallel = \frac{2\eta</em>2 \cos \theta<em>i}{\eta</em>2 \cos \theta<em>t + \eta</em>1 \cos \theta_i} t < e m >∥= η < / e m > 2 c o s θ < e m > t + η < / e m > 1 c o s θ i 2 η < / e m > 2 c o s θ < e m > i
Relationship: t < e m > ∥ = ( 1 + r < / e m > ∥ ) cos θ < e m > i cos θ < / e m > t t<em>\parallel= (1 + r</em>\parallel) \frac{\cos \theta<em>i}{\cos \theta</em>t} t < e m >∥= ( 1 + r < / e m >∥ ) c o s θ < / e m > t c o s θ < e m > i
General Polarization General case with both perpendicular and parallel components.
E < e m > i ⃗ = E < / e m > ∥ i ⃗ + E < e m > ⊥ i ⃗ \vec{E<em>i} = \vec{E</em>{\parallel i}} + \vec{E<em>{\perp i}} E < e m > i = E < / e m > ∥ i + E < e m > ⊥ i E < / e m > r ⃗ = E < e m > ∥ r ⃗ + E < / e m > ⊥ r ⃗ \vec{E</em>r} = \vec{E<em>{\parallel r}} + \vec{E</em>{\perp r}} E < / e m > r = E < e m > ∥ r + E < / e m > ⊥ r E < e m > t ⃗ = E < / e m > ∥ t ⃗ + E ⊥ t ⃗ \vec{E<em>t} = \vec{E</em>{\parallel t}} + \vec{E_{\perp t}} E < e m > t = E < / e m > ∥ t + E ⊥ t
Example 8-6: Wave Incident Obliquely on a Soil Surface Example 8-6 (cont.) Example 8-6 (cont.) Wavelength in air (medium 1):
λ 1 = 2 π π = 2 m \lambda_1 = \frac{2\pi}{\pi} = 2 \quad m λ 1 = π 2 π = 2 m Wavelength in soil (medium 2):
λ < e m > 2 = λ < / e m > 1 ϵ r 2 = 2 4 = 1 m \lambda<em>2 = \frac{\lambda</em>1}{\sqrt{\epsilon_{r2}}} = \frac{2}{\sqrt{4}} = 1 \quad m λ < e m > 2 = ϵ r 2 λ < / e m > 1 = 4 2 = 1 m Wave number in medium 2:
k < e m > 2 = 2 π λ < / e m > 2 = 2 π 0.5 = 4 π ( r a d / m ) k<em>2 = \frac{2\pi}{\lambda</em>2} = \frac{2\pi}{0.5} = 4\pi \quad (rad/m) k < e m > 2 = λ < / e m > 2 2 π = 0.5 2 π = 4 π ( r a d / m ) Incidence angle:
sin θ < e m > i = π k < / e m > 1 = π 2 π = 0.5 \sin \theta<em>i = \frac{\pi}{k</em>1} = \frac{\pi}{2\pi} = 0.5 sin θ < e m > i = k < / e m > 1 π = 2 π π = 0.5 θ i = 30 ∘ \theta_i = 30^\circ θ i = 3 0 ∘ Transmission angle:
sin θ < e m > t = k < / e m > 1 k < e m > 2 sin θ < / e m > i = 2 π 4 π sin 30 ∘ = 0.25 \sin \theta<em>t = \frac{k</em>1}{k<em>2} \sin \theta</em>i = \frac{2\pi}{4\pi} \sin 30^\circ = 0.25 sin θ < e m > t = k < e m > 2 k < / e m > 1 sin θ < / e m > i = 4 π 2 π sin 3 0 ∘ = 0.25 θ t = 14.5 ∘ \theta_t = 14.5^\circ θ t = 14. 5 ∘ Reflection and transmission coefficients (perpendicular polarization):
r < e m > ⊥ = cos θ < / e m > i − ( ϵ < e m > 2 / ϵ < / e m > 1 ) − sin 2 θ < e m > i cos θ < / e m > i + ( ϵ < e m > 2 / ϵ < / e m > 1 ) − sin 2 θ i = − 0.38 r<em>\perp = \frac{\cos \theta</em>i - \sqrt{(\epsilon<em>2/\epsilon</em>1) - \sin^2 \theta<em>i}}{\cos \theta</em>i + \sqrt{(\epsilon<em>2/\epsilon</em>1) - \sin^2 \theta_i}} = -0.38 r < e m >⊥= c o s θ < / e m > i + ( ϵ < e m > 2/ ϵ < / e m > 1 ) − s i n 2 θ i c o s θ < / e m > i − ( ϵ < e m > 2/ ϵ < / e m > 1 ) − s i n 2 θ < e m > i = − 0.38 t < e m > ⊥ = 1 + r < / e m > ⊥ = 0.62 t<em>\perp = 1 + r</em>\perp = 0.62 t < e m >⊥= 1 + r < / e m >⊥= 0.62 Example 8-6 (cont.) Example 8-6 (cont.) Electric field in medium 2 (soil):
E < e m > t ⃗ = y ^ t < / e m > ⊥ E < e m > i 0 e − j k < / e m > 2 ( x sin θ < e m > t + z cos θ < / e m > t ) = y ^ 62 e − j ( π x + 3.87 π z ) \vec{E<em>t} = \hat{y} t</em>\perp E<em>{i0} e^{-j k</em>2 (x \sin \theta<em>t + z \cos \theta</em>t)} = \hat{y} 62 e^{-j(\pi x + 3.87 \pi z)} E < e m > t = y ^ t < / e m >⊥ E < e m > i 0 e − jk < / e m > 2 ( x s i n θ < e m > t + z c o s θ < / e m > t ) = y ^ 62 e − j ( π x + 3.87 π z ) Instantaneous electric field in medium 2:
E < e m > t ( x , z , t ) = R e [ E < / e m > t ⃗ e j ω t ] = y ^ 62 cos ( ω t − π x − 3.87 π z ) ( V / m ) E<em>t(x, z, t) = Re[\vec{E</em>t} e^{j \omega t}] = \hat{y} 62 \cos(\omega t - \pi x - 3.87 \pi z) \quad (V/m) E < e m > t ( x , z , t ) = R e [ E < / e m > t e jω t ] = y ^ 62 cos ( ω t − π x − 3.87 π z ) ( V / m ) Average power density in medium 2:
η < e m > 2 = η < / e m > 0 ϵ r 2 = 120 π 4 = 60 π ( Ω ) \eta<em>2 = \frac{\eta</em>0}{\sqrt{\epsilon_{r2}}} = \frac{120 \pi}{\sqrt{4}} = 60 \pi \quad (\Omega) η < e m > 2 = ϵ r 2 η < / e m > 0 = 4 120 π = 60 π ( Ω ) S < e m > a v = ∣ E < / e m > t 0 ∣ 2 2 η 2 = ( 62 ) 2 2 ⋅ 60 π ≈ 10.2 ( W / m 2 ) S<em>{av} = \frac{|E</em>{t0}|^2}{2 \eta_2} = \frac{(62)^2}{2 \cdot 60 \pi} \approx 10.2 \quad (W/m^2) S < e m > a v = 2 η 2 ∣ E < / e m > t 0 ∣ 2 = 2 ⋅ 60 π ( 62 ) 2 ≈ 10.2 ( W / m 2 ) Reflection Coefficient vs. Angle Plots of the magnitude of the reflection coefficient (Γ \Gamma Γ
Brewster angle is shown for each surface, where Γ ∥ = 0 \Gamma_{\parallel} = 0 Γ ∥ = 0
Brewster Angle Summary For Reflection and Transmission Table summarizing reflection coefficient (Γ \Gamma Γ
Normal Incidence:
θ < e m > i = θ < / e m > t = 0 \theta<em>i = \theta</em>t = 0 θ < e m > i = θ < / e m > t = 0 Γ = η < e m > 2 − η < / e m > 1 η < e m > 2 + η < / e m > 1 \Gamma = \frac{\eta<em>2 - \eta</em>1}{\eta<em>2 + \eta</em>1} Γ = η < e m > 2 + η < / e m > 1 η < e m > 2 − η < / e m > 1 t = 2 η < e m > 2 η < / e m > 2 + η 1 t = \frac{2 \eta<em>2}{\eta</em>2 + \eta_1} t = η < / e m > 2 + η 1 2 η < e m > 2 t = 1 + Γ t = 1 + \Gamma t = 1 + Γ R = ∣ Γ ∣ 2 R = |\Gamma|^2 R = ∣Γ ∣ 2 T = ∣ t ∣ 2 η < e m > 1 η < / e m > 2 T = |t|^2 \frac{\eta<em>1}{\eta</em>2} T = ∣ t ∣ 2 η < / e m > 2 η < e m > 1 T = 1 − R T = 1 - R T = 1 − R Perpendicular Polarization:
Γ < e m > ⊥ = η < / e m > 2 cos θ < e m > i − η < / e m > 1 cos θ < e m > t η < / e m > 2 cos θ < e m > i + η < / e m > 1 cos θ t \Gamma<em>\perp = \frac{\eta</em>2 \cos \theta<em>i - \eta</em>1 \cos \theta<em>t }{\eta</em>2 \cos \theta<em>i + \eta</em>1 \cos \theta_t} Γ < e m >⊥= η < / e m > 2 c o s θ < e m > i + η < / e m > 1 c o s θ t η < / e m > 2 c o s θ < e m > i − η < / e m > 1 c o s θ < e m > t t < e m > ⊥ = 2 η < / e m > 2 cos θ < e m > i η < / e m > 2 cos θ < e m > i + η < / e m > 1 cos θ t t<em>\perp = \frac{2 \eta</em>2 \cos \theta<em>i}{\eta</em>2 \cos \theta<em>i + \eta</em>1 \cos \theta_t} t < e m >⊥= η < / e m > 2 c o s θ < e m > i + η < / e m > 1 c o s θ t 2 η < / e m > 2 c o s θ < e m > i t < e m > ⊥ = 1 + Γ < / e m > ⊥ t<em>\perp = 1 + \Gamma</em>\perp t < e m >⊥= 1 + Γ < / e m >⊥ R < e m > ⊥ = ∣ Γ < / e m > ⊥ ∣ 2 R<em>\perp = |\Gamma</em>\perp|^2 R < e m >⊥= ∣Γ < / e m >⊥ ∣ 2 T < e m > ⊥ = ∣ t < / e m > ⊥ ∣ 2 η < e m > 1 cos θ < / e m > t η < e m > 2 cos θ < / e m > i T<em>\perp = |t</em>\perp|^2 \frac{\eta<em>1 \cos \theta</em>t }{\eta<em>2 \cos \theta</em>i} T < e m >⊥= ∣ t < / e m >⊥ ∣ 2 η < e m > 2 c o s θ < / e m > i η < e m > 1 c o s θ < / e m > t T < e m > ⊥ = 1 − R < / e m > ⊥ T<em>\perp = 1 - R</em>\perp T < e m >⊥= 1 − R < / e m >⊥ Parallel Polarization:
Γ < e m > ∥ = η < / e m > 2 cos θ < e m > t − η < / e m > 1 cos θ < e m > i η < / e m > 2 cos θ < e m > t + η < / e m > 1 cos θ i \Gamma<em>\parallel = \frac{\eta</em>2 \cos \theta<em>t - \eta</em>1 \cos \theta<em>i}{\eta</em>2 \cos \theta<em>t + \eta</em>1 \cos \theta_i} Γ < e m >∥= η < / e m > 2 c o s θ < e m > t + η < / e m > 1 c o s θ i η < / e m > 2 c o s θ < e m > t − η < / e m > 1 c o s θ < e m > i t < e m > ∥ = 2 η < / e m > 2 cos θ < e m > i η < / e m > 2 cos θ < e m > t + η < / e m > 1 cos θ i t<em>\parallel = \frac{2 \eta</em>2 \cos \theta<em>i}{\eta</em>2 \cos \theta<em>t + \eta</em>1 \cos \theta_i} t < e m >∥= η < / e m > 2 c o s θ < e m > t + η < / e m > 1 c o s θ i 2 η < / e m > 2 c o s θ < e m > i t < e m > ∥ = ( 1 + Γ < / e m > ∥ ) cos θ < e m > i cos θ < / e m > t t<em>\parallel = (1 + \Gamma</em>\parallel) \frac{\cos \theta<em>i}{\cos \theta</em>t} t < e m >∥= ( 1 + Γ < / e m >∥ ) c o s θ < / e m > t c o s θ < e m > i R < e m > ∥ = ∣ Γ < / e m > ∥ ∣ 2 R<em>\parallel = |\Gamma</em>\parallel|^2 R < e m >∥= ∣Γ < / e m >∥ ∣ 2 T < e m > ∥ = ∣ t < / e m > ∥ ∣ 2 η < e m > 1 cos θ < / e m > t η < e m > 2 cos θ < / e m > i T<em>\parallel = |t</em>\parallel|^2 \frac{\eta<em>1 \cos \theta</em>t}{\eta<em>2 \cos \theta</em>i} T < e m >∥= ∣ t < / e m >∥ ∣ 2 η < e m > 2 c o s θ < / e m > i η < e m > 1 c o s θ < / e m > t T < e m > ∥ = 1 − R < / e m > ∥ T<em>\parallel = 1 - R</em>\parallel T < e m >∥= 1 − R < / e m >∥