Gaussian beam formulations in different textbooks

[KFC]

In the book "Fundamentals of Photonics", the form of the Gaussian beam is written as
$$I(\rho,z) = I_0 \left(\frac{W_0}{W(z)}\right)^2\exp\left[-\frac{2\rho^2}{W^2(z)}\right]$$
where $$\rho = \sqrt{x^2 + y^2}$$

However, in some books (I forgot which one), the author use the following form
$$I(R) = I_0 \exp\left[-\frac{R^2W_0^2}{W^2}\right]$$
where
$$R = \rho/W_0, \qquad \rho=\sqrt{x^2+y^2}$$

In the second expression, I don't know why there is no $$\left(W_0/W(z)\right)^2$$ in the amplitude and why he want to define R instead of using $$\rho$$ directly? And what about $$W_0$$ and $$W$$ in the second expression? Are they have some meaning as in the first one?

I forgot which book using such form, if you know any information, could you please tell me the title and author of the book? Thanks.

 

[mathman]

Off hand (I'll admit I know nothing about the physics involved) it looks like the two expressions are equivalent, except for the 2 in the numerator of the first expression. W and W₀ look like they are the same in both. The coeficient of both is I₀. In one case the argument seems to be expressed, while the other may be implicit - again I don't know what any of this is supposed to be physically.

 

[KFC]

Off hand (I'll admit I know nothing about the physics involved) it looks like the two expressions are equivalent, except for the 2 in the numerator of the first expression. W and W₀ look like they are the same in both. The coeficient of both is I₀. In one case the argument seems to be expressed, while the other may be implicit - again I don't know what any of this is supposed to be physically.

Thanks. I am thinking one aspect on physics. Since the energy is conserved (the total energy of the input beam should be conserved after transported to some distance), so if the intensity is not inverse proportional to the waist, how to make the energy conserved? Please show me if I am wrong :)

 

[Bob S]

I would first write the cross section of the beam traveling in the z direction in cartesian coordinates:

I(z) = I₀ exp[-x²/2σ_x(z)²] exp[-y²/2σ_y(z)²]

where σ_x(z) and σ_y(z) are the rms widths of the Gaussian beam in the x and y directions at z.

This may be rewritten as

I(z) = I₀ exp[-x²/2σ_x(z)²-y²/2σ_y(z)²]

and finally as

I(z) = I₀ exp[-ρ²/2σ(z)²]

if σ_x(z) = σ_y(z), where ρ² = x² + y².

Bob S