定义：
衡量光学系统聚焦或者使光散焦程度的量。

很多类型的光学系统（例如，显微镜物镜和弯曲的激光器反射镜）都可以聚焦或者使光散焦，焦距就是量化这些效应的量。
最简单的情形为薄透镜（图1a）。如果入射到透镜的光是一束准直光束，经过透镜后光束会发生聚焦，这时焦距就是从透镜到焦点的距离（假设棱镜是处于真空或者空气中，而不是处于折射率很高的物质中）。
而对于散焦透镜（图1b），焦距则是从透镜到虚焦点（采用虚线表示）的距离，是负值。 

图1：聚焦和散焦透镜的焦距。散焦透镜的焦距为负值。 

目录 

1. 透镜的焦距
2. 曲面镜的焦距
3. 光学系统的焦距
4. 照相物镜的有效焦距
5. 焦距可调的光学系统
6. 与波长有关的焦距
7. 屈光本领
8. 发散光束的聚焦
9. 束腰半径

透镜的焦距 
可以采用下列方程计算透镜的屈光度和焦距，透镜材料的折射率为n，两表面的曲率半径分别为R1和R2： 

凸面的曲率半径为正数，而凹面的曲率半径则是负值。聚焦透镜得到的结果为正值，结果为负值则对应的为散焦透镜。后面的一项只有当棱镜很厚，两面曲率半径都很大时才需要考虑。 
这一方程适宜于傍轴光线情形，即光线离对称轴不远的情况。 

曲面镜的焦距 
经常采用曲面镜聚焦或者散焦光束。例如，在激光器谐振腔，具有电介质涂层的激光反射镜比普通的透镜更适用，主要是因为反射镜损耗更小。 
当光轴垂直于反射镜表面时，表面曲面半径为R的反射镜的焦距f = R / 2。（正号代表凸面和聚焦反射镜。）如果光轴与表面法线方向夹角 θ 不为0，那么切向（入射平面）的焦距为 ftan = (R / 2) • cos θ，矢状方向的焦距为 fsag = (R / 2) / cos θ 。 
激光反射镜的曲面半径通常在10 mm到5 m之间。如果曲面半径非常小，制备介质涂层会非常困难，但是采用精细的工艺也可以得到仅为几毫米焦距的反射镜，这在一些微型激光器中会用到。 

光学系统的焦距 
一个光学系统可能包含多个透镜和其它光学元件，因此不能采用上面定义的焦距，因为不能明确应计算何处到焦点的距离：
是从光学系统的起点、终点、中点还是其它位置？原则上可以采用任意的参考点，但是这时有些原理不能使用，这些原理在具有特定焦距的透镜焦点处光束束腰半径是适用的，或者望远镜的放大倍数等。有人采用前焦距来表示焦点与光学元件的前表面之间的距离。 

照相物镜的有效焦距 
在照相术中，有效焦距具有非常不同的意义，下面我们详细解释一下。 
照相机的视角由胶片上像的尺寸与焦距的比值决定。胶片相机一直采用35 mm胶片（根据ISO标准1007也称为135胶片），胶片上像的尺寸为标准的36 mm × 24 mm。
（胶卷轴的宽度为35mm，比24 mm大是为了使图像不会扩展到卷轴的边缘。）这时物镜的焦距为标准的50 mm。然而现在的数码相机（尤其是尺寸小的）通常包含尺寸小于36 mm × 24 mm的像传感器，因此为了得到相同的视场，对应的物镜焦距也比较小（例如32 mm）。
许多摄影师仍然习惯采用常用的焦距与视角的比值，因此常用有效焦距来表征数码相机的焦距，此时的焦距数码相机的视场与与普通35 mm胶片的视场相同。
例如，实际焦距为32 mm也可以说成标准物镜的有效焦距为50 mm。 
随着越来越少的人采用35 mm胶片，这一转换过程以后可能不会采用。 

焦距可调的光学系统 
在有些系统中，尤其是聚焦成像系统，需要光学系统的焦距是可调的。可以采用下面的原理： 

• 如果透镜是由可变形的材料做成的，施加机械应力会改变其形状，因此可以改变焦距。这一原理可用于接目镜中。需要聚焦附近的物体时焦距会变短。 
• 当光学系统包含多个光学元件时（例如，透镜），可以调节光学元件之间的相对距离来调节焦距。这一原理可以用于照相机的变焦物镜中。 

与波长有关的焦距 
普通透镜是利用光的折射，由于折射率与波长有关（色散），因此焦距也与波长相关。这一效应会使成像系统产生色差，在工作于宽波长区域的光学系统中也存在类似的问题。
可以设计采用多个透镜（例如，照相机的物镜）来使色差最小化。最常用的做法是采用消色差双合透镜，即由两种材料组成的透镜，这样总体色差很大部分的被抵消了。 
也可以只采用包含反射镜的光学系统来消除色差。曲面半径为R的曲面镜的焦距为 f = R / 2 ，只由几何形状决定，而与波长无关。但是，在非正交入射的情况下，切平面的焦距与入射角余弦成正比，矢状面与入射角余弦的倒数成正比。因此这种系统会产生像散。 

屈光本领 
透镜的屈光本领等于焦距的倒数。表明强聚焦的透镜焦距小，但是屈光本领大。屈光本领的单位为m−1，也称为屈光度（dpt）。对于验光眼镜，常用屈光本领来表征，这时焦距指的是标准透镜，显微镜物镜和照相物镜的焦距。 
很多情况下，屈光本领是比焦距更常用的量。例如，激光晶体中热透镜的屈光本领正比于耗散功率。
采用热透镜屈光本领表示的激光器谐振腔稳定区域的宽度与激光晶体的最小模式半径和光波长都有关系，而用焦距表征的稳定区域与参数之间的关系更加复杂。 

发散光束的聚焦 

 图2：透镜方程的图解。
发散光束入射到聚焦透镜上时，焦点距离透镜的距离比f大（图2）。透镜方程可写为： 

其中a是入射光束的焦点与透镜之间的距离。当a >> f时，有b ≈ f，而其它情况下，则有b > f 。这一关系可以这样直观的理解：使入射光束准直（即消除光束发散）需要聚焦能力为1 / a，这样只需聚焦能力为 1 / f − 1 / a可将光束聚焦。 
如果a ≤ f，以上方程不成立，透镜则不能使光束聚焦。 
在射线情况下，傍轴近似满足时透镜方程成立。 

束腰半径 
光束半径为 w0的准直高斯光束入射到焦距为f的透镜上，经过透镜后束腰处的半径满足方程： 

这时假设焦点处光束半径远小于初始光束半径w0。（当光束入射半径很小时该条件不满足，这时焦点比根据上面方程得到的值要大。）同时，还需要假设光束半径远大于波长 λ，这样傍轴近似是满足的。 
根据上面方程发现决定最小光束半径的并不只是焦距f，还有f与透镜孔径半径的比值，这一比值限制了最大光束半径w0。该比值称为透镜的数值孔径。 
这一原理能否用于焦距为f扩展的光学系统取决于采用的焦距的定义。有时需要定义有效焦距来满足这一关系。

Definition: a measure of how strongly an optical system focuses or defocuses light

Category: general optics

Formula symbol: f

Units: m

Various types of optical systems (e.g. microscope objectives and curved laser mirrors) can focus or defocus light, and the focal length is used for quantifying such effects. The simplest case is that of a thin focusing lens (Figure 1a). If a sufficiently large collimated beam of light is incident on the lens, the beam will be focused, and the focal length is the distance from the lens to that focus (assuming that the lens is surrounded by vacuum or air, not by some dense substance with a significant refractive index). For a defocusing lens (Figure 1b), the focal length is the distance from the lens to the virtual focus (indicated by the dashed lines), taken as a negative value. Some authors use different sign conventions, however, in particularly concerning the front and back focal length (see below).

Figure 1: Focusing and defocusing lenses and their focal length. For the defocusing lens, the focal length is specified as a negative value.

Focal Length of a Thin or Thick Lens

A lens with a given focal length f (taken as positive in case of a focusing lens) creates a radially varying phase delay for a laser beam according to the following equation:

This formula ignores the constant part of the optical phase change as well as optical aberrations. Note that depending on the function of the lens – for example, focusing collimated input beams or refocusing divergent light –, higher-order terms in the phase profile may be required to avoid optical aberrations.

The following equation allows one to calculate the dioptric power and thus the focal length of lens made of a material with refractive index n and with curvature radii R₁ and R₂ on the two surfaces:

The curvature radii are taken as positive values for convex surfaces and negative for concave surfaces. Positive results are obtained for focusing lenses, negative results for defocusing lenses. The last term is relevant only for thick lenses with substantial curvature on both sides. The formula delivers the focal length within the paraxial approximation, not considering spherical aberrations, for example.

The equation holds for paraxial rays, not too far from the symmetry axis.

Focal Length of a Curved Mirror

Curved mirrors are often used for focusing or defocusing light. For example, within laser resonators curved laser mirrors with dielectric coatings are more commonly used than lenses, mainly because they introduce lower losses.

A mirror with a curvature radius R of the surface has a focal length f = R / 2, if the beam axis is normal to the mirror surface. (We take positive signs for concave curvatures and focusing mirrors.) If there is some non-zero angle θ between the beam axis and the normal direction, the focal length is f_tan = (R / 2) · cos θ in the tangential direction (i.e., within the plane of incidence) and f_sag = (R / 2) / cos θ in the sagittal direction.

Curved laser mirrors usually have a curvature radius somewhere between 10 mm and 5 m. The fabrication of dielectric mirror coatings can be more difficult for very strongly curved mirror substrates, but with refined techniques it is possible to reach focal lengths of only a few millimeters, as required for some miniature lasers.

Focal Length of an Extended Optical System

It is not obvious how to define the focal length of an extended system – there are different possibilities!

For an optical system, which may consist of multiple lenses and other optical elements, the above definition of the focal length cannot be used, as it is not clear a priori for an extended system where to measure the distance to the focus: from the entrance into the optical system, from the exit, the middle, or some other position? In principle, an arbitrary definition of a reference point (e.g. the entrance or the middle) could be used, but that would in general mean that some common rules can not be applied, which e.g. hold for the radius of the beam waist at a focus behind some lens with a given focal length (see below), or the possible magnification of a telescope containing that optical system.

A common (but not universally used) approach for the definition of focal lengths of extended systems is based on geometrical optics. For finding the front focal point, one calculates rays which are horizontal on the back side (see Figure 2), using the paraxial approximation. The optical system is considered as a “black box”, where one does not care about the actual ray paths; instead, one works with internal rays which are extrapolated from the outer rays. Based on those extrapolated rays, one can define the front principal plane (or first principal plane). The front focal length is then the distance between the front focal point (in the front focal plane) and the front principal plane (see Figure 2). Some authors define the focal length to be negative in the situation of Figure 2, because the focal point is located before the front principal plane; others take the absolute value.

Figure 2: The front focal point is related to parallel rays on the back side (= output side). The optical input is considered to be on the left side. The ray path within the device is only extrapolated from the outer ray path.

For a defocusing system, the front focal plane can be located on the output side; it contains a virtual focal point. Again, the focal length is the distance between principal plane and focal plane.

Similarly, one can define the back focal plane (or second focal plane) and back principal plane (or second principal plane), where horizontal rays occur on the left side, while on the right side one has converging rays for a focusing system and diverging rays for a defocusing system. If the refractive index is the same on the input and output side (e.g. ≈1 for air), the front focal length and back focal length are identical (apart from possible sign differences used by some authors) and can thus simply called the focal length. The two principal planes, however, generally do not coincide for thick lenses, and they can even lie outside a lens.

The explained definition delivers a focal length which can also be used in equations for the size of the focus (see below), for example.

Note that the locations of the left and right edges of the optical system (e.g. positions of outer lens surfaces, optical windows etc.) or its housing are not relevant for those definitions.

Different sign conventions for focal lengths are used in the literature. For example, one may have a negative front focal length if the front focal point lies before the front principal plane. Obviously, any equations involving focal lengths should be used with the assumed sign conventions.

See also the article on principal planes.

Focal Distances

In contrast to focal lengths, focal distances are related not to the principal planes but rather to the vertex points of lenses (not caring about a housing, which may be further extended). The front focal distance is thus the distance between front focal point and the entrance surface of the optics, while the back focal distance is the distance between the back surface and the back focal point.

Focal distances are sometimes confused with focal lengths!

Unfortunately, the terms are also used differently by other authors. For example, it happens that a focal distance is assumed to be the same as a focal length. Therefore, some product catalogs specify focal lengths, which should actually be called focal distances, and in addition the effective focal length.

The focal distance should also not be confused with the working distance, which is the distance between a specimen and the lens housing. Note that a specimen is not necessarily placed in the focal plane, e.g. when the input light to an objective is not collimated.

Effective Focal Length of a Photographic Objective

Considerable confusion arises from the fact that in the context of photo cameras the term effective focal length is also used with a completely different meaning, as explained in the following.

The angle of view of the camera is determined by the ratio of the image size on the film and the focal length. Film-based cameras have for a long time mostly used 35-mm film (also called 135 film according to ISO Standard 1007), where the image size on the film is typically 36 mm × 24 mm. (The width of the film spool is 35 mm, somewhat larger than 24 mm, as the picture does not extend to the edges of the spool.) A standard objective then has a focal length of 50 mm. However, modern digital cameras (particularly the more compact ones) often contain image sensors which are smaller than 36 mm × 24 mm, so that an objective lens with a correspondingly smaller focal length (e.g. 32 mm instead of 50 mm) is required for obtaining the same field of view. As many photographers are still used to the previously valid relation between focal length and angle of view, it has become common to specify the effective focal length of an objective of a digital camera as that focal length which would give the same angle of view in combination with ordinary 35-mm film. For example, an objective with a true focal length of 32 mm may then be said to have an effective focal length of 50 mm and thus function as a standard objective, rather than e.g. a macro or tele objective.

It can be expected that this type of conversion will be abandoned as 35-mm film becomes less and less common.

Optical Systems with an Adjustable Focal Length

For some applications, in particular for focusing of imaging systems, it is essential than the focal length of an optical system can be fine adjusted. The following physical principles can be used:

• If a lens is made of deformable material, applying some mechanical pressure can alter its shape, and this can change the focal length. This principle is used in the eye's lens. The focal length is somewhat reduced for focusing on nearby objects.
• When an optical system contains multiple optical elements (e.g. lenses), the focal length may be tuned by adjusting the relative distances between the optical elements. This principle is used e.g. in photographic zoom objectives.

Wavelength Dependence of the Focal Length; Using Curved Mirrors

Ordinary lenses, working on the basis of refraction, have a focal length which is slightly wavelength-dependent due to the wavelength dependence of the refractive index (–> chromatic dispersion). This effect leads to chromatic aberrations of imaging systems and similar problems in other applications where an optical system is used for a wide range of optical wavelengths. Lens combinations (e.g., objectives for photographic cameras) can be designed such that chromatic aberrations are minimized. Most common is the use of achromatic doublets, i.e., lenses consisting of two different glass materials chosen such that the overall chromatic aberrations are largely canceled.

One may eliminate chromatic aberrations altogether by using optical systems with mirrors only. A curved mirror with radius of curvature R has a focal length f = R / 2 (for normal incidence), determined only by the geometry and thus independent of the wavelength. On the other hand, for non-normal incidence the focal length in the tangential direction is decreased by the cosine of the angle of incidence, and increased by the inverse cosine of that angle in the sagittal direction. Therefore, such mirrors can introduce astigmatism.

Dioptric Power

The dioptric power (also called focusing power) of a lens is defined as the inverse of the effective focal length (which is the same is the front and back focal length if the median on both sides of the optics is the same). This means that a strongly focusing lens has a small focal length, but a large dioptric power. For prescription glasses, it is common the specify the dioptric power, whereas the focal length is specified for standard lenses, microscope objectives, and photographic objectives.

Focusing of Divergent Beams

Figure 3: Illustration of the lens equation.

If a divergent (rather than collimated) beam hits a focusing lens, the distance b from the lens to the focus becomes larger than f (Figure 2). The lens equation states that

where a is the distance from the original focus to the lens. This shows that b ≈ f if a >> f, but b > f otherwise. That relation can be intuitively understood: a focusing power 1 / a would be required to collimate the incident beam (i.e. to remove its beam divergence), so that only a focusing power 1 / f − 1 / a is left for focusing.

If a ≤ f, the equation cannot be fulfilled: the lens can then not focus the beam.

Note that the lens equation applies for rays, assuming that the paraxial approximation is valid, i.e., all angles relative to the beam axis remain small.

Achievable Beam Waist Radius

If a collimated Gaussian beam with beam radius w₀ hits a focusing lens with focal length f, the beam radius of the beam waist (focus) after the lens can be calculated with the equation

where it is assumed that the beam radius at the focus is much smaller than the initial beam radius w₀. (This condition is violated for beams with a too small incident radius; the focus is then larger than according to the given equation.) Also, it is assumed that the beam radius is significantly larger than the wavelength λ, so that the paraxial approximation is valid.

The equation shows that what determines the minimum possible beam radius is not the focal length f alone, but rather the ratio of f to the radius of the open aperture of the lens, which sets a maximum to the input beam radius w₀. For a focusing or collimation lens, that ratio is essentially the numerical aperture of the lens.

Whether this rule can also be applied to an extended optical system with focal length f depends on the applied definition of f. It is useful to specify an effective focal length which is valid for such relations.