### Fixed Focal Length Lenses

A **fixed focal length lens**, also known as a conventional or entocentric lens, is a lens with a fixed **angular field of view** (AFOV). By focusing the lens for different working distances (WDs), differently sized field of view (FOV) can be obtained, though the viewing angle is constant. AFOV is typically specified as the full angle (in degrees) associated with the horizontal dimension (width) of the sensor that the lens is to be used with.

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**Note**: Fixed focal length lenses should not be confused with **fixed focus lenses**. Fixed focal length lenses can be focused for different distances; fixed focus lenses are intended for use at a single, specific WD. Examples of fixed focus lenses are many telecentric lenses và microscope objectives.

The focal length of a lens defines the AFOV. For a given sensor kích thước, the shorter the focal length, the wider the AFOV. Additionally, the shorter the focal length of the lens, the shorter the distance needed khổng lồ obtain the same FOV compared to a longer focal length lens. For a simple, thin convex lens, the focal length is the distance from the baông chồng surface of the lens lớn the plane of the image formed of an object placed infinitely far in front of the lens. From this definition, it can be shown that the AFOV of a lens is related to lớn the focal length (*Equation 1*), where $ smallf $ is the focal length & $ smallH $ is the sensor size (*Figure 1*).

**(1)**$$ extAFOV = 2 imes an ^-1 left( fracH2fight) $$

**Figure 1:** For a given sensor kích cỡ, H, shorter focal lengths produce wider AFOV’s.

In general, however, the focal length is measured from the rear **principal plane**, rarely located at the mechanical baông chồng of an imaging lens; this is one of the reasons why WDs calculated using **paraxial** equations are only approximations và the mechanical design of a system should only be laid out using data produced by computer simulation or data taken from lens specification tables. Paraxial calculations, as from lens calculators, are a good starting point khổng lồ tốc độ the lens selection process, but the numerical values produced should be used with caution.

When using fixed focal length lenses, there are three ways khổng lồ change the FOV of the system (camera and lens). The first and often easiest option is lớn change the WD from the lens to the object; moving the lens farther away from the object plane increases the FOV. The second option is khổng lồ swap out the lens with one of a different focal length. The third option is khổng lồ change the kích thước of the sensor; a larger sensor will yield a larger FOV for the same WD, as defined in *Equation 1*.

While it may be convenient khổng lồ have sầu a very wide AFOV, there are some negatives to lớn consider. First, the level of distortion that is associated with some short focal length lenses can greatly influence the actual AFOV and can cause variations in the angle with respect khổng lồ WD due to lớn distortion. Next, short focal length lenses generally struggle to lớn obtain the highest cấp độ of performance when compared against longer focal length options (see Best Practice #3 in Best Practices for Better Imaging). Additionally, short focal length lenses can have sầu difficulties covering medium to lớn large sensor sizes, which can limit their usability, as discussed in Relative sầu Illumination, Roll-Off, & Vignetting.

Another way to change the FOV of a system is to lớn use either a **varifocal lens** or a **zoom lens**; these types of lenses allow for adjustment of their focal lengths & thus have variable AFOV. Varifocal & zoom lenses often have form size & cost drawbacks compared khổng lồ fixed focal length lenses, and often cannot offer the same cấp độ of performance as fixed focal length lenses.

### Using WD and FOV to Determine Focal Length

In many applications, the required distance from an object và the desired FOV (typically the form size of the object with additional buffer space) are known quantities. This information can be used lớn directly determine the required AFOV via *Equation 2. Equation 2* is the equivalent of finding the vertex angle of a triangle with its height equal lớn the WD và its base equal lớn the horizontal FOV, or HFOV, as shown in *Figure 2*. **Note:** In practice, the vertex of this triangle is rarely located at the mechanical front of the lens, from which WD is measured, và is only lớn be used as an approximation unless the entrance pupil location is known.

**(2)**eginalign extAFOV & = 2 imes an ^-1 left( fracextFOV 2 , imes , extWDight) & ext or extFOV & = 2 , imes , extWD imes an left( frac extAFOV 2ight) endalign

Once the required AFOV has been determined, the focal length can be approximated using *Equation 1* & the proper lens can be chosen from a lens specification table or datasheet by finding the closest available focal length with the necessary AFOV for the sensor being used.

The 14.25° derived in Example 1 (see White box below) can be used to determine the lens that is needed, but the sensor size must also be chosen. As the sensor form size is increased or decreased it will change how much of the lens’s image is utilized; this will alter the AFOV of the system & thus the overall FOV. The larger the sensor, the larger the obtainable AFOV for the same focal length. For example, a 25mm lens could be used with a ½” (6.4milimet horizontal) sensor or a 35milimet lens could be used with a 2/3” (8.8mm horizontal) sensor as they would both approximately produce a 14.5° AFOV on their respective sensors. Alternatively, if the sensor has already been chosen, the focal length can be determined directly from the FOV và WD by substituting *Equation 1* in *Equation 2*, as shown in *Equation 3*.

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**(3)**$$ f = frac left( H imes extWDight) extFOV $$

As previously stated, some amount of flexibility to lớn the system’s WD should be factored in, as the above sầu examples are only first-order approximations và they also vì not take distortion into lớn tài khoản.

**Figure 2:** Relationship between FOV, sensor kích thước, và WD for a given AFOV.

### Calculating FOV Using a Lens with a Fixed Magnification

Generally, lenses that have sầu fixed magnifications have sầu fixed or limited WD ranges. While using a telecentric or other fixed magnification lens can be more constraining, as they vì not allow for different FOVs by varying the WD, the calculations for them are very direct, as shown in *Equation 4*.

**(4)**$$ extFOV = frac H m $$

Since the desired FOV and sensor are often known, the lens selection process can be simplified by using *Equation 5*.

**(5)**$$ m = frac H extFOV $$

If the required magnification is already known and the WD is constrained, *Equation 3* can be rearranged (replacing $ small fracHextFOV $ with magnification) và used to determine an appropriate fixed focal length lens, as shown in *Equation 6*.

**(6)**$$ m = frac f extWD $$

Be aware that *Equation 6* is an approximation và will rapidly deteriorate for magnifications greater than 0.1 or for short WDs. For magnifications beyond 0.1, either a fixed magnification lens or computer simulations (e.g. Zemax) with the appropriate lens Mã Sản Phẩm should be used. For the same reasons, lens calculators commonly found on the mạng internet should only be used for reference. When in doubt, consult a lens specification table.

**Note:** Horizontal FOV is typically used in discussions of FOV as a matter of convenience, but the sensor aspect ratio (ratio of a sensor’s width to lớn its height) must be taken into lớn trương mục khổng lồ ensure that the entire object fits inlớn the image where the aspect ratio is used as a fraction (e.g. 4:3 = 4/3), *Equation 7*.

**(7)**$$ extHorizontal FOV = extVertical FOV imes extAspect Ratio $$

While most sensors are 4:3, 5:4 và 1:1 are also quite comtháng. This distinction in aspect ratio also leads khổng lồ varying dimensions of sensors of the same **sensor format**. All of the equations used in this section can also be used for vertical FOV as long as the sensor’s vertical dimension is substituted in for the horizontal dimension specified in the equations.

## LENS FOCAL LENGTH EXAMPLES

### Using WD và FOV khổng lồ Determine Focal Length

**Example 1:** For a system with a desired WD of 200mm and a FOV of 50mm, what is the AFOV?

eginalignextAFOV & = 2 imes an^-1 left( frac50 extmm2 imes 200 extmmight) extAFOV & = 14.25° endalign

### Calculating FOV Using a Lens with a Fixed Magnification

**Example 2:** For an application using a ½” sensor, which has a horizontal sensor form size of 6.4milimet, a horizontal FOV of 25mm is desired.

eginalign m & = frac6.4 extmm25 extmm m & = 0.256 extX endalign

By reviewing a menu of fixed magnification or telecentric lenses, a proper magnification can be selected.

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**Note:** As the magnification increases, the kích cỡ of the FOV will decrease; a magnification that is lower than what is calculated is usually desirable so that the full FOV can be visualized. In the case of *Example 2*, a 0.25X lens is the closest comtháng option, which yields a 25.6milimet FOV on the same sensor.

Chuyên mục: ĐỊNH NGHĨA