# What happens to the power of the beam passing through a lens?

Sorry for the fundamental and basic question, Does the power of the beam that is falling on lens is equal to the power of the beam after passing through lens. Suppose If I used convex lens to converge a divergent beam of power P then the power of the beam at focus is also equal to p? My understanding is only intensity changes but when I tried calculating this power for a gaussian beam my math seem to be off. Lets take power density of sun and strip away the spectral properties for ease in calculations then power falling on the lens would be

$$P=0.14w/cm^2(power density)*Pi*s^2$$ s-size of lens

power after focusing would be

$$P=0.14*pi*W^2$$ w-beam waist

These two would be different. So, where am I going wrong.

• I think I made a mistake in the formula for the power after focusing.It should be equal to the power before focusing. So, we can't take the same power density. May 1, 2021 at 22:10

The term for $$W/cm^2$$ is intensity.

$$I = \frac{Power}{Area}$$

All the light that hits the lens must be transmitted through the lens, reflected, or absorbed. Given that absorption and reflection are negligable, we have $$Power_{before} = Power_{after}$$. So

$$\frac{I_{before}}{Area_{before}} = \frac{I_{after}}{Area_{after}}$$

You can calculate the ratio of the intensities from this.

• Absorption and reflection are only negligible if they are. They certainly are not in the general case. Also, I see W/cm2 called "irradiance" much more than "intensity".
– Matt
May 2, 2021 at 1:18
• @Matt - They are typically negligible in lenses, particularly if the lenses have anti reflection coatings. Intensity and irradiance are related and have the same unit, but they are not the same thing. See this for more on it. May 2, 2021 at 3:16
• Well yes. Whether or not you have an ARC matters quite a bit. Example lens: thorlabs.com/newgrouppage9.cfm?objectgroup_id=8898 Only slightly better than 90% transmission at any wavelength without ARC. At select wavelengths the ARC gets closer to 100% transmission, but depending on your application, this may or may not be close enough to call 100%. I also just checked my radiometry book and it talks about an intensity as power per solid angle, and I don't think it ever gives a name to the quantity discussed by your link. I guess some of these terms are more context/field dependent.
– Matt
May 2, 2021 at 12:48