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Methods for measuring a lens focal length
December 2011 Sophie Morel OPTI – 521: Tutorial
Introduction
The focal length is one of the most important parameter of a lens. Sometimes the focal length is slightly different from the datasheet value given by the manufacturer, having an impact on the predicted optical system performances. Many methods have been elaborated to measure with more or less accuracy a lens’ focal length. Some methods are easy to implement, other require more material and analyses, with more accurate measurements.
- By definition, the effective focal length EFL is the distance between the rear principal point P’, and the rear focal point F’ of the lens.
- The back focal length BFL is the distance between the rear vertex V’ of the lens, and the rear focal point F’.
- The front focal length FFL is the distance between the front focal point F and the front vertex V.
Measuring the back focal distance is usually easy. Determining the distance between P’ and V’ is a bigger issue. Besides, the result accuracy depends on several parameters, such as the glass index inhomogeneity, surfaces errors, radiuses of curvature error… The lens application defines the accuracy needed on its focal length knowledge.
This tutorial first describes in detail methods which are easy to implement in student labworks sessions. The last methods are usually more accurate, but require more equipment and analyses.
Figure 1 : focal length definition
1. Quick estimation
The easiest and fastest way to roughly evaluate a lens focal length is to hold it under a ceiling light. A projection screen (the floor, or the surface table) is under the lens. The optical element is moved vertically back and forth until a sharp image of the ceiling light is projected on the screen. The distance between the lens and the projection surface gives an idea of the focal length. Of course, there is no precision with this technique, but it can be used to check the results obtained with other methods.
2. Y’/tan(θ) method
a. Principle
This method uses the paraxial optic formula which relates the focal length with the image height and the angle of incidence:
^ = −. tan = . tan ′
Assuming that the lens is in the air, its nodal points are coincident with its principal points, and the angle in the image space is equal to the angle of incidence in the object space: θ = θ’.
b. Measurement
Here is the setup drawing:
Figure 3: y’/ tan(θ) setup
The reticle is located at the front focus of the collimated lens. It is illuminated by a light source. Thus, the reticle appears to be at infinity for the test lens, and its image is located at the rear focal plane of the lens. The angular size of the reticle is known with high accuracy.
Figure 2: y' = f'.tan(θ)
The lens focal length is deduced from two configurations:
- The lens rear vertex is on the rotation axis, and the microscope is positioned such that it focuses on this rear vertex.
- The lens nodal point is on the rotation axis, and the microscope focuses on the rear focal point. The lens is at the right position if the image does not shift when the lens is rotated.
The microscope position is marked for both configurations, and the lens focal length results from the difference in position of the microscope.
c. Uncertainty analysis
The uncertainty of this measurement is mainly due to the microscope and lens position error. It also depends on the accuracy of the position reading device. The depth of focus depends on the microscope objective aperture, because of diffraction:
∆ (^) !""~ (^) %&$', where ()~ (^) +/#* is the microscope objective’s numerical aperture.
A short focal length lens has a short depth of focus, and position reading is more precise than with a longer focal length lens. The observer should repeat the measurements many times in order to evaluate the positioning error. A root sum square of the different sources of errors also gives an estimation of the measurement uncertainty, since all sources of error are independent.
4. Cornu method
a. Principle
This technique is based on Newton’s formula: × ^ = /)^0000 × /′)′^00000 = /′1^00000 × /1′^00000 , Where A is the front vertex of the lens, A’ is the image of A through the lens, B is the rear vertex, and B’ is the image of B through the lens.
b. Measurement
o The first step is to measure the reticle image position which is located at the test lens focal plane. Then the microscope is translated in order to focus on the lens rear vertex B, and on the image of the front vertex A’.
Figure 6: Cornu method first measurement.
The microscope position is marked for each measurement. At this stage, /′1^00000 and /′)′^00000 are known.
o The lens is flipped, and similar measurements are performed, in order to locate the front focal point F, the front vertex A, and the image of the rear vertex B’.
Figure 7 : Cornu method second measurement
Now /)^0000 and /1′^00000 are known, and Newton’s formula can be used to calculate the lens focal length.
c. Uncertainty analysis
Sources of error are the same as with the nodal slide method: o Diffraction
The depth of focus is related to the diffraction such that: ∆ (^) !""~ (^) %&$', where NA is the microscope objective’s numerical aperture. o Reading error
5. Moiré deflectometry
Interference between two waves diffracted by two different gratings creates a Moiré pattern. For a converging lens, two gratings are placed between the lens and its rear focal plane. It can be shown that the focal length is functions of the second grating pitch p, the aperture size a, the distance between the two gratings ∆, and the number of observed fringes N: = (^) 3%2∆
The error measurement comes from the counting
of fringes. Assuming that (^) 4* of a fringe is be resolved, the focal length can be
measured with about 0.8% accuracy, according to the authors of this method.
Figure 8: Moiré deflectometry
Figure 10 : Fizeau interferometry setup
The reference beam reflects at R, L is the test lens, and the mirror M is placed slightly away from the lens focal point, at distance U1. Measurements are made for two positions of the mirror. The mirror displacement x is known. It results in 2x displacement of the virtual object: U2-U1=2x The Gaussian lens formula is applied for these two positions:
1 61
:*;< +^
=> =^
" And 61 = −? + ?² + 2? (^) =>B=>=>.=>
Circular fringes are observed. The wavefront radius Rv1 is related to the spacing between the nth^ and (n+p)th^ fringes, and the distance S between the lens vertex and the screen:
891 =
CD;3^ − CD
8FG
+ H
Knowing U1 and Rv1, the focal length can be computed using the Gaussian lens formula.
The measurement error is about 2.5% according to the authors. It depends mainly on the error in measuring dU and the fringes diameter.
8. Shack-Hartmann wavefront analysis
Figure 11 : interference pattern
The Shack-Hartmann wavfront sensor uses a lenslet array to measure the wavefront slope coming from the optical system under test. Polynomial expansion can be used to retrieve the corresponding wavefront curvature. For this method, a Shack- Hartmann sensor is used to determine the radius of curvature R of the wavefront coming out of the test lens.
Figure 12 : Schack-Hartmann wavefront sensor
Z is the source position, Z 0 is the source position for a collimated beam, L is the distance between the wavefront sensor and the principal plane of the lens. The focal length is deduced from the following equation:
1
+ I − IJ
8 + K
The lens power is measured for different positions of the source. The author of this method claims that routinely measurements are made with less than 0.5% accuracy.
Conclusion
This tutorial describes methods for measuring a lens focal length. Most of them use paraxial optic formula (Gaussian equation, Newton equation,…) Others use diffraction theory with gratings. Measurements are performed with a microscope translated on an optical rail, or by using interferometers for more precision. All techniques can be applied to positive or negative lenses, although it may require some adjustments.
This list of methods is not all-comprehensive, and probably many other techniques have been and will be developed.
