Wanted: (simplified/good-enough) analytical formula for quarter-wave AR coating

[Matthias_H]

Dear all,

For a computer graphics rendering routine, I'm looking for an efficient way of computing the net Fresnel reflectivity of a glass (lens) surface with a quarter-wave coating for a wavelength range of 400--700 nm, and incident angles between 0 and 90 degrees to the normal. Incident light is assumed to be unpolarized. Since the implementation will be tailored to GPU, look-up tables are a no-go (due to scattered memory access), but computation is as cheap as it gets. Still, simplicity is more important than accuracy, as long as the results are not too far off. A polynomial, for instance, would be nice to have.

float reflectivity(float lambda, float theta, float n1, float n2)  {

// anyone? :)

}

Regards,
Matthias

 

[eljose79]

Dear Matthias,

Thank you for your inquiry about computing the net Fresnel reflectivity for a glass surface with a quarter-wave coating. This is an interesting problem that has practical applications in computer graphics rendering.

One approach to this problem is to use the Fresnel equations, which describe the reflection and transmission coefficients for a light wave at an interface between two media with different refractive indices. In this case, the interface is between air (n1=1) and the glass surface (n2) with a quarter-wave coating.

To start, we can define the angle of incidence (theta) and the wavelength (lambda) as inputs to the function. We can also define the refractive indices for air and glass as constants in the function.

Next, we can use the Fresnel equations to calculate the reflection coefficient (R) for the incident light at the glass surface. This can be done for both parallel and perpendicular polarizations of the incident light, and the results can be combined using the unpolarized assumption.

R = ((n1*cos(theta) - n2*sqrt(1-(n1/n2*sin(theta))^2)) / (n1*cos(theta) + n2*sqrt(1-(n1/n2*sin(theta))^2)))^2

Note that this equation assumes that the quarter-wave coating has a refractive index equal to the square root of the refractive index of the glass. This is a simplification, but it should provide reasonable results for your purposes.

Finally, we can use a polynomial fit to approximate the reflection coefficient over the desired wavelength range. This can be done by collecting data points of R for different wavelengths and angles of incidence, and fitting a polynomial function to these points. The polynomial function can then be used to calculate the net reflectivity for any given wavelength and angle of incidence.

I hope this helps you in your implementation for GPU. If you have any further questions or need clarification, please don't hesitate to ask.