Vector representation of reflection and refraction of light at material interfaces
Expressing the reflection of light at the interface of a material and the refraction of light based on Snell's law as vectors is extremely useful when dealing geometrically with the propagation of light. Here, we will derive it.
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Vector representation of reflection at material interfaces
Each direction vector is defined as follows.
: Direction vector of incident light
: Direction vector of reflected light
: Normal vector of the reflecting surface
The absolute values of all these are defined as 1.
In this case, k can be expressed as follows, where k is a positive real number.
Transforming this into the formula,
Squaring both sides,
Since the absolute value of each direction vector and normal vector is 1,
By substituting this into ①', we finally get the following equation.
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Vector representation of refraction at material interfaces
Define a direction vector as shown in the following diagram.
: Direction vector of incident light
: Direction vector of transmitted light
: Normal vector of material interface
: A direction vector parallel to the material interface and in the same plane as the incident and transmitted light
The absolute values of all of these are set to 1.
In this case, the relationship between the vectors can be described as follows:
From ①,
Substituting ② and ③ into this,
Next, substitute ① and ③ into ⑤,
Substituting ④ into this,
By substituting ①' and ② into this, we finally obtain the following equation.
