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Consider a light source enlighting a spherical object (which diffuses and reflects light) and a camera watching it. To reduce glare on the pictures grabbed from the camera, a first linear polarizer is inserted between the light and the spherical object and a second polarizer (analyzer) is inserted between the object and the camera.

Now my question is: how is light intensity reduced?

Simple case

If the polarizers were aligned and without the spherical object from Malus' law I know that polarized light passing through the second polarizer has intensity $$ I = I_1 cos^2 \theta$$ where $\theta$ is the angle between the polarization of light and the second polarizer and $I_1$ the intensity of light before the second polarizer.

Since the light emitted from the source is unpolarized, passing through the first polarizer, the intensity is the average on all the angles of polarization, so the intensity is halved: $$ I_1 = \frac{1}{2}I_0 $$

So the final intensity would be $$ I = \frac{1}{2}I_0 cos^2\theta$$

This case

What about the intensity of the light partially diffused and partially reflected on the sphere?

I tried considering the sphere as a second source. If in the situation without the polarizers, the camera receive light intensity $I_c$ (for sure $I_c < I_0$).

Is it true that if I insert the first polarizer $I_c$ is halved?

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The directly reflected light retains its polarisation so can be blocked by choosing $\theta=90^o$. The diffused light is unpolarised so half of it is lost in the second polariser. Of course, if your light source is unpolarised to begin with, you also lose half of the overall intensity in the first polariser.

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