Focal length measurement based on Fresnel diffraction from a phase plate
PS: This is a review of the paper Focal length measurement based on Fresnel diffraction from a phase plate by Masoomeh Dashtdar, S Mohammad-Ali Hosseini-Saber
One of the measures of how strong an optical system diverges or converges light is commonly called focal length and can also be understood as the inverse of it’s optical power. Hence in all imaging systems, focal length is a very important parameter to measure. Over the years, we have come up with many ways to measure the focal length( like image magnification method, nodal slide method and the many interferometry-based approaches) but they either have limited accuracy or require complicated setups which are extremely mechanical and optical noise sensitive. Hence, the recent advances in research based on Fresnel diffraction has enabled us to come up with a method based on the same to calculate the EFL(effective focal length) and BFL(back focal length-the distance between the image plane and the lens’s last surface) of optical imaging systems and has been explored in this paper.
From the concept of Fresnel’s diffraction, we know that when a coherent light beam which has only one wavelength(monochromatic) is reflected from an object or transmitted through a plane parallel plate(which is a homogeneous layer that is transparent for certain wavelength ranges of optical radiation), the beam will suffer an abrupt phase change at the boundaries of the plate/edges. Thus, we can write the equation for phase difference between the two beams(one making theta angle and passing through while the other travels in the surrounding medium) in terms of wavelength used, the thickness of the plate used and the ratio of the the refractive index of the plate to that of its surrounding medium. Thus, we will also be able to obtain the diffracted intensity in terms of the phase difference(which we already calculated), amplitude transmission coefficient(t) and two parameters(called Fresnel’s cosine and sine integrals). For simplicity, we then consider the case of edges which would mathematically mean that we take both the parameters(Fresnel’s cosine and sine integrals) as 0 which further reduces the equation. It is known that changing the plate’s thickness and changing the angle of incidence both affect the phase difference. We already have an equation relating angle of incidence and effective focal length in terms of aperture diameter. Hence we combine these two mathematical equations to finally derive an equation for phase difference involving a term for focal length. Hence we can effectively measure focal length in terms of the position of the consecutive extremas since the phase difference is an integral multiple of pi.
If we mount the lens system just before the aperture, we will also be able to calculate the back focal length. When we carry out this experiment, practically using He-Ne laser and a lens whose effective focal length is to be determined, we find that the results are extremely accurate with an uncertainty as low as 0.04% when compared to the values provided by the lens manufacturer. The same was observed while calculating the back focal length.
We can conclude from the above observations and results that this method for calculating EFL and BFL is more precise and effective than the classical and/or existing methods to do the same. Another advantage of the technique is that it does not require sophisticated equipment or accessories to produce results that have very low levels of optical or mechanical noises. Also, even for imaging systems that are large in size, we can use sample plates of very small sizes since it can be kept very close to the focal plane. We can thus conclude that this method of finding EFL and BFL is more effective, widely applicable and produces consistent and accurate results with very low levels of mechanical and optical noise. Thus, this proves to be an exciting prospect to explore & apply in imaging systems in the near future.
