Tutorial : source and screen conjugated by an optical system - lambertian source at infinity - aperture defined with F number

Etendue conservation

Let consider a small light source of area dS_o conjugated by a perfect optical system in Gauss conditions.

The etendue in the object space is :

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n_o is the refraction index in the object space, dΩ_o is the solid angle defined by the aperture angle i_o in the object space.

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The etendue in the image space is :

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n_o is the refraction index in the image space, dΩ_i is the solid angle defined by the aperture angle i_i in the image space and dS_i is the area of the source conjucgated.

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The paraxial formulas give the following relations :

and .

m is the magnification of the optical system. Therfore :

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The etendue is thus conserved through the optical system.

Considering the transmission T of the optical system and the flux in the object space d²F_o, the flux in the image space is :

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Therefore, as , the radiance of the secondary source (conjugated of the initial source by the optical system) is :

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If the initial and final media have the same refractive index, the ratio between the radiance of the conjugated source and the radiance of the initial source is equal to the transmission.

Irradiance produced by a source and an optical system

The irradiance on a screen can be anaytically calculated in some cases where light go through a perfect optical system ( in Gauss conditions).

In the case where a punctual isotropic light source ( intensity I ) is in the focal plane of a system ( focal length f_i ), the irradiance on a screen, whatever its position, is :

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D is the aperture diameter and T is the system transmission.

In the case of a small Lambertian source ( area S and radiance L ) located in the front focal plane of the slightly opened system, the beam is slightly diverging after being transmitted trough the optical system. Therefore, the irradiance depends on the position z_i of the screen. Indeed, the flux after the system is :

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Considering the beam area S_i on the screen : .

Therefore, the irradiance E on the screen is :

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In the case where the screen is conjugated with a small Lambertian source ( area S_o and radiance L ) and considering that the system has a small aperture, the flux on the screen is :

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x_o is the algebric distance of the source.

where x_i is the algebric distance of the source conjugate.

The magnification is given by :

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If S_i is the area on the screen and m is the magnification :

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Then, the irradiance on the screen is :

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In the case where the light source is at infinity and the screen is located in the back focal plane of the system :

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In this case, when defining the aperture with the aperture numbre N, the ratio between the irradiance on the screen with and without the optical system is ( the angular size α of the source is supposed to be small ) :

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