Optical lens physical parameters

January 12, 2024

The general industrial lens does not have the magnification factor because the general industrial lens has a different magnification when used at different working distances. At this time, we need to calculate the focal length (f) of the lens and the working distance (WD) of the lens. .

1.1 Magnification (X)

The magnification X of the optics is used to describe the ratio of the image size (h') to the object size (h):

X=h'/h

Generally when imaging with a camera with an industrial camera, the image size is the physical size of the camera chip (h*v)

h = number of horizontal cells on the chip * side length of pixels

v = number of vertical cells on the chip * side length of the pixel

Object size (H*V) is the field of view (FOV) of the entire lens with the camera imaging

H = h/X

V = v/X

A useful relationship between working distance WD, magnification (X) and focal length (f) is as follows: WD=f(X-1)/X

1.2 Focal length (f)

Focal length, also known as focal length, is a measure of the concentration or divergence of light in an optical system, and refers to the distance from the lens center to the focal point of light gathering. It is also the distance from the optical center of the lens to the imaging plane such as the CCD or CMOS in the camera. An optical system with a short focal length has a better ability to collect light than a long focal length optical system.

The general industrial lens has a fixed focal length parameter, which is the most important indicator of the lens.

The types of focal lengths commonly used in the industry are: 4mm6mm8mm12mm16mm25mm35mm50mm75mm100mm, etc. According to different use distances, and with the needs of different types of cameras and different field of view (FOV), we can calculate the focal length that needs to be used. The calculation method is as above.

Different focal lengths, different object distances and the same camera can appear the same field of view. How to choose in this case?

Generally, it is not recommended to use the imaging method with small focal length in the small object state. This method will cause the picture to have a relatively large physical distortion.

1.3 Depth of Field (DOF)

Depth of Field (DoF) is the range between the closest position and the farthest position of the object when it is allowed to focus.

A rough estimate of the depth of field is given by the following formula:

DoF[mm]=WF/# ?P[μm]?k/M^2

Where p is the pixel size of the sensor, M is the lens magnification, and k is a dimensionless parameter depending on the specific application.

As can be seen from the above formula, the depth of field of the lens is closely related to the aperture, and the depth of field of the lens is directly proportional to F#. It can be seen that when the lens has a relatively low amount of light, it will have a relatively large depth of field. ,vice versa.

1.4 resolution

Resolution is an important parameter to measure the sharpness of the lens imaging.

In general, the resolution is determined by the frequency, and the frequency is measured by the logarithm per millimeter (lp/mm), but the resolution of the lens is not an absolute value. The relationship between alternating black and white squares is often referred to as a line pair. The ability to display two squares as separate entities at a given resolution depends on the gray level. The greater the gray distance between the squares and the space (as shown below), the stronger the ability to parse squares. This gray separation is called contrast (at the specified frequency). The given spatial frequency is in lp/mm. Therefore, it is useful to calculate the resolution in lp/mm when comparing lenses and determining the best choice for a given sensor and application.

The sensor is the starting point for calculating the resolution of the system. Starting from the sensor, it is easier to determine the lens performance needed to meet the needs of the sensor or other application. The highest frequency the sensor can resolve, the Nyquist frequency, is actually two pixels or a pair of lines.

The following table shows the Nyquist limits associated with the size of pixels seen on some common sensors. The resolution of the sensor (image spatial resolution) can be calculated by multiplying the pixel size (μm) by 2 (creating the pair) and dividing the product by 1000 to convert mm:

Sensor Resolution (lp/mm) = Image Space Resolution (lp/mm) = 1000/2 × Pixel Size (μm)

Larger pixels have lower limit resolution. The smaller pixel sensor has a higher limit resolution. The sensor size refers to the size of the camera sensor's effective area and is usually specified by the sensor format size. However, the exact sensor ratio will vary depending on the aspect ratio, and the nominal sensor format should only be used as a guide, especially for telecentric lenses and high-magnification objectives. The sensor size can be calculated directly from the pixel size and the number of active pixels on the sensor.

Horizontal sensor size (mm) = [(horizontal pixel size, μm) × (number of active horizontal pixels)]/1000 μm/mm

Vertical sensor size (mm) = [(vertical pixel size, μm) × (number of active vertical pixels)]/1000 μm/mm

In general, the lens imaging has an object and an image, and the resolution of the lens is also divided into the object resolution and the image resolution. Generally, the lens and camera matching are based on the image resolution and the pixel size. The accuracy of the assessment is based on the resolution of the object. What is the relationship between these two resolutions?

Object spatial resolution (lp/mm) = image spatial resolution (lp/mm) × X

In general, when developing an application, the system's resolution requirements are not given in lp/mm but in μm or inches. There are two ways to convert:

Object spatial resolution (μm) = 1000 (μm/mm)/[2 × object spatial resolution (lp/mm)]

Or object spatial resolution (μm) = pixel size (μm) / system magnification

1.5 Contrast (Sharpness)

Contrast describes the degree of discrimination between black and white at a given object resolution. To make the image look sharp, black details need to be displayed in black and white details must be displayed in white (as shown below). The more the black and white information tends to the middle gray, the lower the contrast at this frequency. The greater the difference in intensity between light and dark lines, the higher the contrast.

It can be seen from the figure that the transition from black to white is a high contrast and the gray in the middle indicates a low contrast.

The contrast at a given frequency can be calculated according to the following formula. Among them, Imax is the maximum intensity (usually pixel gray value is used if the camera is used), Imin is the minimum intensity:

%Contrast=[(Imax-Imin)/(Imax+Imin)]×100

The contrast (sharpness) of a lens directly determines the distinguishing accuracy of boundary features when visual contours are detected. Generally, the visual contour detection uses the backlight illumination to capture the object. The level of the contrast directly determines the accuracy of the edge extraction by the image algorithm, which ultimately determines the accuracy of the output result.

1.6 Aperture (F#) / Numerical Aperture (NA)

The F/# setting on the lens controls a number of lens parameters: total luminous flux, depth of field, and the ability to produce contrast at a given resolution. Fundamentally speaking, F/# is the ratio between the effective focal length (EFL) and effective aperture diameter (DEP) of the lens:

F/#=EFL/ DEP

Typical F/# values are F/1.0, F/1.4, F/2.0, F/2.8, F/4.0, F/5.6, F/8.0, F/11.0, F/16.0, F/22.0, and so on. For each increase in F/#, the incident light is reduced by a factor of two. As shown below.

Most lenses are set F/# by turning the iris adjustment ring, which in turn opens and closes the inner iris aperture. The number marked on the adjustment circle indicates the luminous flux and its associated aperture diameter. These numbers often increase in multiples of 21/2. Increasing the F/# by a 21/2-bit coefficient will halve the aperture area, effectively reducing the lens' luminous flux by a factor of two. Lower F/# lenses are considered faster and allow more light to pass through the system, while higher F/# lenses are considered slower and have lower luminous flux.

The following table shows examples of F/#, aperture diameter, and effective opening size for a 25mm focal length lens. When the setting is changed from F/1 to F/2 and then from F/4 to F/8, the lens aperture for each interval will be reduced in half. This describes the reduction in flux associated with the increase in lens F/#.

Aperture has a direct bearing on the brightness of the imaging surface of the lens, but it is closely related to the image contrast, resolution, and depth of field. When we adjust the lens aperture, we must consider its impact on the entire image. Specifically, F/# is directly related to theoretical resolution and contrast limits as well as depth of field (DOF) and lens depth of focus. In addition, it also affects the aberrations of the lens design. As the pixel size continues to decrease, F/# will become the most important factor limiting system performance because it is inversely proportional to depth of field and resolution. In the equations for the calculation work F/#, X represents the paraxial magnification of the objective lens (the ratio of the image to the height of the object). Note that the closer X is to 0 (the closer the object is to infinity), the closer the working distance F/# is to the infinite F/#. In the case of a small working distance, it is important to keep in mind that F/# changes as the working distance changes.

The F/# in the equation “F/#=EFL/DEP” is defined at an infinite working distance, where the magnification is actually 0. In this sense, the definition of F/# is limited. In most machine vision applications, the length of the object and the lens is much shorter than the wireless distance, and F/# is more accurately expressed as the working F/# in the following equation.

(F/#)w = (1+|m|)× F/#

The numerical aperture (NA), like F#, is a way of describing the lens aperture. It is often easier to talk about the total luminous flux from the perspective of the lens cone angle or numerical aperture (NA). The numerical aperture of the lens is defined as the sine of the marginal ray angle in the image space. (As shown below)

Relationship between F/# and numerical aperture NA:

NA=1/[2×(F/#)]

The following table shows the typical F/# layout of the lens (each subsequent digit is incremented by a factor of 21/2) and its relation to the numerical aperture.

Numerical apertures are often noted in microscopes, not F/#, but numerical apertures assigned to microscope objectives are specified in object space because light collection is easier at this point. In another case, infinite conjugation can be thought of as the opposite machine vision objective (focusing on infinity).

The next issue of BTSOS will continue to share the relevant introduction of optical lens aberration parameters. There are related questions and we welcome you to leave messages on Wechat!

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