Incoherent multiple internal reflection

The following figure illustrates the effects of incoherent multiple internal reflections on the energy flow through a finite thickness substrate. For a given optical substrate of finite thickness (t), absorption coefficient (α), and reflectivity from front and rear surfaces of (R_a^-) and (R_b⁺), the amount of transmitted radiation traversing the first interface T_a⁺ will be (1-R_a⁺)I_o, where I_o is the intensity of the incident wave. After travelling through a distance (t) of the substrate the incident radiation is attenuated by

e^-αt. The radiation therefore reaching the second interface (T_a^-) is (1-R_a⁺)I_oe^-αt. Due to reflection (R_b^-) at the interface of the second surface, only the fraction given by

(1-R_a⁺)(1-R_b^-)I_oe^-αt will emerge. The amount internally reflected as R_b^-(1-R_b^-)I_oe^-αt will become progressively more attenuated as a result of the multiple internal reflections becoming absorbed within the material or emerging as a negligible contribution compared to the first order reflectivity.

Analysis of the energy flow through a substrate of finite thickness inclusive of multiple internal reflections

Variation in transmission (T_ab) vs absorption coefficient (α) for different refractive indices between 1.0-6.0 for a 1mm substrate

Variation in transmission (T_ab) vs absorption coefficient for different thicknesses of Germanium in the range (0.5 - 3.0mm) at 10µm

Germanium absorption topography - plan view (thickness 1-6mm at 293K)

Fz Silicon absorption topography - plan view (thickness 1-6mm at 293K)

Zinc Sulphide absorption topography - plan view (thickness 1-6mm at 293K)

Zinc Selenide absorption topography - plan view (thickness 1-6mm at 293K)

Cadmium Telluride absorption topography - plan view (thickness 1-6mm at 293K)

Substrate optical theory

• Introduction
• Absorption and extinction coefficient theory
• Loss-free incoherent internal reflection
• Incoherent multiple internal reflection
• Reduced substrate-temperature effects
• Angle of incidence effects
• Total internal reflection
• Coherence of multiple internal reflections