Optical power collected by a detector coming from a Lambertian disk object in an active illuminated camera

I would like to estimate reduction of optical power in an active illuminated camera when a flat lambertian target is moved away from the camera. Below is formal description of the problem, I’m just not sure if I came to the right conclusion at the end.

Find the reduction of optical power collected by a pixel of an image sensor (detector) coming from
a circular Lambertian disk when the disk is moved from position $r_1$ to position $r_2 = 2 r_1$. The radius of the disk is $a$ and $2 a$ for the two positions as in the picture below.
I assume constant radiant intensity illumination, cone half angle $θ$, that produces uniform incident irradiance $E_i$ on the disk.
For simplicity everything is positioned on (single) optical axis and there are no transmission losses (I am assuming field of illumination (IFOV) and detector FOV are the same and have the same origin).

Assuming source intensity is $ I = phi / Ω $

Irradiance incident on Lambertian disk at $r_1$ (area $A_1$) and $r_2$ (area $A_2$) is

$E_{in1} = phi / A_1 = I*Ω / A_1 = I*Ω / (a^2* π) $

$ E_{in2} = phi / A_2 = I*Ω / A_2 = I*Ω / (4a^2* π) $

Using radiation transfer relation for a circular lambertian disk (p.15 H.Gross Part 6:
Photometry)

$ displaystyle E = π L frac{ a^2}{(a^2 + r^2)} $, and $ L= E_{in}/π $

where $E$ is irradiance at the position of the lens, $L$ is radiance of lambertian disk
and $E_{in}$ is incident irradiance on the disk.
Irradiance reduction at the position of the lens coming from the disk at $r_1$ and $r_2$ is then

$ displaystyle E_1 = πL_1 frac {a^2} { (a^2 + r_1^2)} $

$ displaystyle E_2 = πL_2 frac{(2a)^2}{((2a)^2 + (2r_2)^2)} = πL_2 frac{a^2}{(a^2 + r_2^2)} $

By using $ L = E_{in}/π $, we have

$E_2 / E_1 = L_2/L_1 = 1/4$

Irradiance at the lens is reduced by a factor of 4.

If there are no transmission losses the optical power $P$ transferred to the pixel (detector) is also reduced by a factor of 4.

(assuming $P=E_{lens}/A_{lens}$, where $E_{lens}$ is irradiance at the lens and $A_{lens}$ is
entrance pupil area of the lens).