1 Focal Length of Optical Systems

Focal length is a critical parameter of an optical system. Most readers have some basic understanding of this concept, which we will review here. The focal length of an optical system is defined as the distance from the optical center of the optical system to the beam focal point when parallel light is incident. It quantifies the ability of an optical system to converge or diverge light. This concept is illustrated in the diagram below.

In the diagram, parallel light beams incident from the left converge at the image‑side focal point F’ after passing through the optical system. If the converging light rays are extended backward to intersect with the extensions of the corresponding incident parallel rays, a plane perpendicular to the optical axis passing through this intersection is known as the rear principal plane. The rear principal plane intersects the optical axis at point P₂, called the principal point (or optical center). The distance from the principal point to the image‑side focal point is the focal length we usually refer to, formally named the image‑side effective focal length. As also shown in the diagram, the distance from the last surface of the optical system to the image‑side focal point F’ is the back focal length (BFL). Correspondingly, if parallel light beams are incident from the right side, there are matching concepts of object‑side effective focal length and front focal length (FFL).

2 Focal Length Test Methods

In practice, numerous methods are available to measure the focal length of optical systems. Based on underlying principles, they fall into three categories: methods based on image plane position, methods deriving focal length from the relationship between magnification and focal length, and methods calculating focal length using the wavefront curvature of converging beams. This section introduces commonly used methods for optical system focal length measurement:

2.1 Collimator Method

The measurement principle of the collimator method is shown in the schematic diagram below.]

In the diagram, a test pattern is placed at the focal point of the collimator. The height of the test pattern y and the collimator focal length \(f'_c\) are known. Parallel light emitted by the collimator converges through the optical system under test and forms an image on the image plane. By measuring the height \(y'\) of the pattern image on the image plane, the focal length of the optical system under test can be calculated using the corresponding formula.

2.2 Gaussian Method

The schematic for focal length measurement using the Gaussian method is shown below.

In the diagram, the front and rear principal planes of the optical system under test are P and P’ respectively, with a separation distance \(d_P\) between them. This method assumes \(d_P\) is known or sufficiently small to be neglected. An object and a receiving screen are placed at the left and right ends with a separation distance L, where L must be greater than four times the focal length of the system under test. There exist two positions for the optical system at which the object on the left forms a sharp image on the receiving screen: Position 1 and Position 2. The separation D between these two positions can be measured. Based on conjugate relations:

Let the object distances at the two positions be \(s_1\) and \(s_2\), so \(s_2-s_1=D\). Formula derivation yields the focal length of the optical system.

2.3 Focimeter Method

Focimeters are particularly suitable for measuring optical systems with long focal lengths. Its working principle is shown in the schematic diagram below.

First, remove the lens under test from the optical path. The observation target on the left is collimated into parallel light by the collimating objective lens. The parallel beam is converged by a converging lens of focal length \(f_2\) to form a sharp image on the reference image plane. After optical path calibration, insert the lens under test into the optical path at a distance \(f_2\) from the converging lens. Due to the effect of the lens under test, the light beam refocuses and the image plane shifts, forming a sharp image at the new image plane in the diagram. The distance between the new image plane and the converging lens is x. The focal length of the lens under test can be deduced from object‑image relations.

Focimeters are widely used to measure the vertex power of spectacle lenses, featuring simple operation and reliable accuracy.

2.4 Abbe Focimeter Method

The Abbe focimeter provides another approach to measure the focal length of optical systems. Its schematic diagram is shown below.

Two scales of different heights, Scale 1 and Scale 2 with heights \(y_1\) and \(y_2\), are placed on the object side of the lens under test. The separation between the two scales is e, and the angle between the line connecting the scale tops and the optical axis is u. The scales are imaged by the lens under test of focal length f. A microscope is installed on the image side. By moving the microscope forward and backward, locate the images of the tops of the two scales. The distance from the microscope to the optical axis at this time is recorded as y. From object‑image relations:

The formula above can be used to solve for the focal length f of the lens under test.

2.5 Moiré Deflectometry

Moiré deflectometry employs two sets of Ronchi rulings in a parallel light beam. A Ronchi ruling is a grating formed by depositing grid‑shaped metallic chromium film on a glass substrate, commonly used for optical system performance testing. This method measures focal length by detecting changes in Moiré fringes generated by the two gratings. The schematic diagram is shown below.

In the diagram, the observation target passes through a collimating lens and becomes a parallel light beam. Without the lens under test in the optical path, the parallel beam passes through two gratings with an offset angle \(\theta\) and grating pitch d, generating a set of Moiré fringes on the image plane. Insert the lens under test into the optical path. The original collimated light is refracted by the lens and acquires optical power. The radius of curvature of the light beam can be calculated by the corresponding formula.

Normally, the lens under test is placed very close to the first grating. In this case, the calculated radius R corresponds to the focal length of the lens. A major advantage of this method is its capability to measure both positive and negative focal length systems.

2.6 Fiber Optic Autocollimation Method

The principle of fiber optic autocollimation for lens focal length measurement is shown in the schematic diagram. A fiber emits a diverging beam, which passes through the lens under test and strikes a plane mirror. The three optical paths shown correspond to three scenarios: the fiber tip located inside the focal point, exactly at the focal point, and outside the focal point. Move the lens under test forward and backward to find the position where the fiber tip sits at the focal point. At this position, autocollimation occurs: most of the energy from the emitted beam returns to the fiber tip after reflection by the plane mirror. This method features a simple principle and easy implementation.

3 Conclusion

Focal length is an important parameter of optical systems. This article elaborates on the concept and test methods of optical system focal length. Supported by schematic diagrams, we explained the definition of focal length, including image‑side and object‑side focal length, front focal length and back focal length. A variety of practical methods exist for focal length measurement. This paper introduced the principles of the collimator method, Gaussian method, focimeter method, Abbe focimeter method, Moiré deflectometry and fiber optic autocollimation method.