Information entropy of Poisson noise

I would like to calculate the information entropy of the channel with additive Poisson noise.

The model consists of an input signal, an optical image, an image acquisition device, and the output signal I obtain on the screen of the device.

In general, the image acquisition device is supposed to be an electron-optical converter, so that there are several sources of noise in the system (photocathode, microchannel plate, etc.), but for the sake of simplicity, I want to model it as a simple photon counter, so that there is only one source of Poisson statistics, and calculate the information entropy of the noise of this device.

I have found a book in which this problem seems to be solved, however, the results look confusing to me.

The author assumes there is a Poisson noise in the channel, but instead of Poisson $$P(N=k)=\frac{\lambda^ke^{\lambda}}{k!}$$ he uses Laplacian statistics $$p(N)=\frac{1}{2 \alpha} e^{-\frac{N}{\alpha}}$$ (see the picture below).

So I just would like to know whether I can use the result of the author or there is a misconception. If the latter is true, then how can I calculate it myself.

If there are some useful links which you could share, I would be very grateful!

Thank you!