Polarization dependence of gratings

Light and its behaviour and properties
Din
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Joined: Thu Mar 12, 2015 4:47 pm

Re: Polarization dependence of gratings

Post by Din »

lobaz wrote:surprisingly, power in the +1 order is about 1.6x higher than for the sinusoidal grating. Although I can calculate it, I still cannot imagine why it happens (without any equation).
Because the edge is sharp. Diffraction occurs in any slit because of oscillations of electrons at the slit edge. If the edge is sharp, the E field is greater, and so electrons have more energy to oscillate.
lobaz
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Joined: Mon Jan 12, 2015 6:08 am
Location: Pilsen, Czech Republic

Re: Polarization dependence of gratings

Post by lobaz »

Partly true. Sharp edges mean that more light is diffracted away from 0 order. It does not tell how much light gets to +1 order. Usually, diffraction efficiency is defined as the ratio of the intensity of a selected diffraction order (say, +1) and the incident intensity.

Imagine, for example, a sinusoidal amplitude grating, i.e. amplitude transmittance is given as

c(x) = cos(2*pi*x)/2 + 0.5

If the grating is illuminated by a plane wave at normal incidence and unit amplitude, the average intensity just behind the grating is

integrate(c(x)^2, x, 0, 1) = 3/8.

It can be easily found that intensity of the 0 order is 0.25x the incident intensity, and intensity of the 1st order is 0.0625x the incident intensity. Thus, the diffraction efficiency is 6.25%.

For comparison, take a flat amplitude grating, i.e. amplitude transmittance is given as

f(x) = squareWave(x) * sqrt(3/4).

Here, squareWave is periodic with period 1, and squareWave = 0 for 0<x<1/2, otherwise 1. The average intensity just behind the grating is again

integrate(f(x)^2, x, 0, 1) = 3/8.

(If we did not introduce the factor sqrt(3/4) to f(x), the average intensity would be 1/2 instead of 3/8, and it would be unfair to compare diffraction efficiencies.)

Now, intensity of the 0 order is approximately 0.19x the incident intensity, which confirms the idea - a grating with sharp edges diffracts more light away from the 0 order.

Intensity of the 1st order is approximately 0.079x the incident intensity, which means that the diffraction efficiency of the "sharp" grating is higher. However, this "sharp" grating produces more diffraction orders (all odd orders), for example 3rd diffraction order is approximately 0.013x the incident intensity.

To conclude:
1. A "sharp" grating diffracts more light, which could indicate higher diffraction efficiency.
2. On the other hand, a "sharp" grating produces more diffraction orders, which could indicate smaller diffraction efficiency.

My point is that I cannot guess which conclusion is valid without a rigorous analysis.
Din
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Joined: Thu Mar 12, 2015 4:47 pm

Re: Polarization dependence of gratings

Post by Din »

Without actually doing the analysis, I would think that you could decompose the 'sharp' grating into a set of sinusoidal gratings. Then the efficiency of each sinusoidal component will be proportional to the zero order Bessel function of the amplitude of the sinusoidal grating. In other words, each sinusoidal component would be proportional to a zero order Bessel function whose argument would be proportional to the coefficient of the decomposition. So, the first order would be J₀(α*4/π), the second order would be J₀(α*4/(3π)) etc.
lobaz
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Location: Pilsen, Czech Republic

Re: Polarization dependence of gratings

Post by lobaz »

Why Bessel? I thought that Bessel functions emerge just in radially symmetric structures due to properties of the Hankel transform.
Din
Posts: 402
Joined: Thu Mar 12, 2015 4:47 pm

Re: Polarization dependence of gratings

Post by Din »

lobaz wrote:Why Bessel? I thought that Bessel functions emerge just in radially symmetric structures due to properties of the Hankel transform.
For simplicity, assume one-dimensional model. Thus, there is a phase variation, as a function of x, φ(x). Therefore, the transmissivity of the hologram, after recording, is:

t = exp[i*φ(x)]

The developed hologram will have a phase variation:

φ(x) = k{a² + r² + 2a*r*cos(2πξx) = φ(0) + φ(1)*cos(2πξx)

So, the transmissivity of the hologram becomes

t = exp[i*φ(x)]*exp[i*φ(1)*cos(2πξx)] = k'*exp[i*φ(1)*cos(2πξx)] = k'*Σiⁿ Jn[φ(1)]*exp[i*n*(2πξx)]

(ignoring the constant phase factor, summation over n, Jn is the Bessel function of the first kind, nth order)

Thus, the amplitude of a thin phase grating is proportional to the Bessel function.
lobaz
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Joined: Mon Jan 12, 2015 6:08 am
Location: Pilsen, Czech Republic

Re: Polarization dependence of gratings

Post by lobaz »

Oh, I see, you assume phase holograms. I usually assume amplitude modulation :)
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