Lens

definition:
It is a transparent optical device that affects the wavefront curvature of light.

 

Figure 1: Focusing and defocusing of lenses.  

An optical lens contains a transparent medium in which light enters from one side and exits from the other side. The function of a lens is to change the wavefront curvature of light, that is, to focus or defocuse the light. For example:
 

• A collimated beam with a wavefront that is approximately planar is transformed into a curved wavefront, and the beam is focused at the focal point. At this point, the lens acts as a focusing lens, as shown in Figure 1 (a).
• The same lens as above can also convert divergent beams into collimated beams, in which case the lens acts as a collimating lens. In Figure 1 (a), it is assumed that the light beam is incident from the right side.
• A lens with a concave surface can transform a collimated or converging beam into a diverging beam, as shown in Figure 1 (b). This lens can also be used to convert originally diverging beams into collimated beams.

Although the change in beam radius is usually regarded as a lens equation, the basic equation of a lens is actually the change in wavefront curvature, because the change in wavefront curvature causes the change in beam radius after light passes through the lens. The direction of energy propagation is usually perpendicular to the direction of the wavefront, as demonstrated in Figure 2.  

Figure 2: Changes in wavefront curvature of focusing lens. Red and blue represent the strength and polarity of the electric field at a certain moment. The assumed wavelength is greater than the actual wavelength.  

catalogue

1. The physical origin of wavefront changes
2. focal length
3. Lensmaker equation
4. Thin lens and thick lens
5. Lens equation
6. Double convex, flat convex, double concave, flat concave, and crescent shaped lenses
7. Cylindrical mirror and astigmatic lens
8. Astigmatism caused by lenses
9. aspheric lens
10. Achromatic lens
11. Lens surface coating
12. The application of optical lenses

The physical origin of wavefront changes
Most wavefront changes caused by lenses originate from the curvature of the surface. Figure 2 shows a typical biconvex lens (with two convex surfaces), where the phase delay of light passing near the lens axis is greater than the phase delay of light on both sides.
Because the refractive index of the lens material is higher than that of the nearby medium (usually air). The radial variation of phase also represents the variation of wavefront curvature.  
Another physical explanation is the refraction of the lens surface. Especially for thick lenses, using refraction theory to calculate results is more accurate than using radial phase delay, which ignores the variation of beam size inside the lens.  
There are also gradient refractive index lenses (GRIN lenses), which means that the refractive index in the lens material varies.
For a focused GRIN lens, the refractive index is highest at the center and gradually decreases on both sides, while as the radial position increases, the refractive index changes approximately in a parabolic shape. The surface of GRIN lenses is usually flat, similar to a regular plate or cylinder.  

Focal length
When the collimated beam is directed towards the lens, the focal length f of the focusing lens is the distance between the lens and the focal point behind it (as shown in Figure 1 (a)). For defocusing lenses, the focal length is a negative value, which is the sign of the distance between the lens and the real focal point (see Figure 1 (b)).  
The refractive power (or focusing power) of a lens is the reciprocal of its focal length.  
The focal length of lenses commonly used in laser technology ranges from 10mm to several meters. Small non spherical lenses can easily reach several millimeters, sometimes up to 1mm or less.  

Lensmaker equation
The following equation is called the Lensmaker's equation, which can be used to calculate the focal length of a lens material with a refractive index of n and two surface curvature radii of R1 and R2, respectively

  

The curvature radius of a convex surface is positive, while the curvature radius of a concave surface is negative. The latter item only needs to be considered in the case of thick lenses with large curvature radii on both surfaces. This equation holds for paraxial rays and assumes that the refractive index of the surrounding medium is close to 1 (air).  
There are different symbol rules in writing. For example, a common rule is that if the surface is concave, then the radius of the second surface is considered positive. This is the opposite of the symbol rule used above.  

Thin lens and thick lens
In many practical situations, lenses are very thin, so it can be assumed that their beam radius remains constant within the lens. A lens with a small surface curvature (i.e. a large curvature radius) satisfies this condition. The third term in Lensmaker's equation can be ignored, and a simple thin lens equation can be obtained.  
When high focusing power is required, thick lenses are needed. The thickness d of the lens (the distance between the surfaces of the two lenses in the axial direction) has a significant impact on the focal length, as can be seen from the lensmaker's equation. The definition of the position and focal length of a thick lens is not very clear, at least when the lens is asymmetric.  
The difference between thick lenses and thin lenses lies in the approximation used in the calculation.  

Lens equation

Figure 3: Schematic diagram of lens equation.
When the diverging beam is directed towards the focusing lens, the distance from the lens to the focal point will be greater than the focal length f (see Figure 3). It can be calculated by the lens equation:

 

Where a is the distance between the initial focal point of the beam and the lens. This indicates that when a>>f, b ≈ f, while in other cases b>f. The above relationship can be understood as follows: the required focusing power is 1/a to collimate the incident beam (i.e. eliminate the beam divergence angle), so only 1/f-1/a of power is needed for focusing.  
When a ≤ f, the above equation does not hold, so the lens cannot focus the beam.  
When the paraxial approximation is satisfied, i.e. the angle between the lens and the optical axis is relatively small, the lens equation also applies to rays.  

==Numerical aperture and aperture number of lens==(f-number) The definition of numerical aperture (NA) of a lens is the product of the sine value of the edge fiber angle at the focal point and the refractive index of the medium produced by the incident beam.
The NA (not focal length) of the lens will limit the size of the beam waist. The lenses of storage media players and recorders require a large numerical aperture (0.5-0.9), such as CDs, DVDs, Blu ray discs, etc.  
Colliding a small aperture laser beam requires a high numerical aperture lens. For example, the light emitted by low-power single-mode laser diodes falls under this category. When the numerical aperture of the lens is very low, the resulting collimated beam may be distorted or even truncated.  
Obviously, if the focal length of a high NA lens is large, the size of the lens will be relatively large.  
The numerical aperture of a defined lens may be smaller than that of a lens with an open aperture in geometry, as additional aberrations may occur in the edge region.  
The aperture number (f-number) of the lens in a camera is very clear. For example, an f/4 lens refers to an aperture with a diameter that is one fourth of the focal length. Here, 'f' refers to the aperture book, not the focal length! ）The edge of the lens is also utilized, so the numerical aperture is approximately sin (1/4) ≈ 0.247, which may be slightly smaller in practice.  

Double convex, flat convex, double concave, flat concave, and crescent shaped lenses
The lenses given above are all biconvex lenses, meaning that both sides are convex. A plano convex lens has one side as a plane and the other side as a convex surface. Double concave lenses with different curvature radii on both sides can also be made. Similarly, defocusing lenses are either biconcave lenses or plano concave lenses.  

 

Figure 4: Different types of optical lenses.  
According to the lensmaker's equation, different refractive powers can be obtained through different lens designs. However, the aberrations of different lens designs are also different. When imaging a small dot as another light spot of the same size, it is best to use a symmetrical biconvex lens.
In asymmetric applications, such as focusing a collimated beam or collimating a strongly divergent beam, using a plano convex lens is more suitable. The lens needs to be rotated to position the curved surface on one side of the collimated beam. Both surfaces of the lens contribute to the focusing process.  
A crescent shaped lens is concave convex, meaning that one side of the lens is concave and the other side is convex. The contribution of two surfaces to refractive power is equal; The focal length of a lens can be positive (focusing) or negative (defocusing). Curved crescent lenses are commonly used as corrective lenses for objective lenses; Their main function is to correct aberrations. It can also be used as a concentrator for lighting systems.  
Double lens is the process of sticking two lenses together, each made of a different material. The most common achromatic dual lens.  

Cylindrical mirror and astigmatic lens
The curvature of the lens surface can exist only in the horizontal direction and have no curvature in the numerical direction. Cylindrical mirrors only focus or defocus light in the horizontal direction and do not affect the wavefront curvature in its numerical direction.  
Cylindrical mirrors can obtain elliptical beam focal points or be used to generate or compensate for beam or optical system astigmatism, which is relatively difficult to manufacture.  
If there is curvature in both directions, but with different curvature sizes, an astigmatic lens is obtained. It can be used to correct the astigmatism of the light source.  

Astigmatism caused by lenses
Lenses can produce various types of astigmatism:

• The surface of most lenses is spherical simply because it is the easiest shape to make. However, a spherical surface is not ideal, which can cause astigmatism (especially in the edge region) or decrease the quality of the laser beam. This is called spherical aberration. Non spherical lenses can greatly reduce spherical aberration.  
• When a collimated beam is incident on the lens at a certain angle to the axis of symmetry of the lens, the resulting focal point will be deformed. This type of aberration is called coma. Two curvature radii can be adjusted to minimize them.  
• The chromatic dispersion of lens materials can cause chromatic aberration. The direct result is that the focal length is wavelength dependent, so light cannot be well focused because the focal points of different wavelength components are located at different positions. The chromatic aberration of achromatic lenses is greatly reduced.  
• When the beam radius of the laser beam incident on the lens is very large, the beam cross-section will be truncated at the edge of the lens. This will result in significant beam distortion.
• This type of hole diffraction can also occur in imaging applications; The limited size of the lens will limit the imaging resolution of the optical system. But if the optical components and design do not have very high quality, the quality of the image is not limited by diffraction effects.  

Aberrations can be greatly reduced by using a combination of different lenses. That's also why objective lenses usually contain multiple lenses.  

Non spherical lens
Although a reasonable combination of different lenses can greatly compensate for spherical aberration, sometimes it is better to use non spherical lenses whose surface is not spherical. By using only one lens, high imaging quality (low spherical aberration) can be achieved. However, non spherical lenses are more difficult to manufacture and therefore more expensive.  

Achromatic lens
The most commonly used method to obtain an achromatic lens (a lens with greatly reduced chromatic aberration) is to place two lenses (of different materials) together (see Figure 4 on the right).
For example, a biconvex low refractive index crown glass can be combined with a flat concave high refractive index flint glass to obtain an achromatic doublet lens. The curvature radius of the adhesive layer needs to be calculated to obtain the minimum dispersion and must be strictly equal.  

Lens surface coating
Many lens surfaces have anti reflective coatings, which greatly reduce reflections caused by changes in surface refractive index. This only works within a specific wavelength range. A balance needs to be struck between high reflectivity and wide operating bandwidth.  
The presence of wear-resistant coating makes the lens more wear-resistant.  

The application of optical lenses
The application of lenses is versatile:

• A single lens is commonly used as corrective glasses, which can compensate for visual impairments to some extent.  
• A single lens can also be used to magnify an image, usually by selecting a combination of multiple lenses. For example, photographic objectives, microscope objectives, and lens objectives.  
• In laser technology, lenses are usually used to focus or collimate the laser beam. Especially when high NA lenses are used to focus large diffraction limited beams, the focal size obtained is very small (the waist radius may be less than 1 micron).  

Lenses can also be used to generate modes in laser resonators, with curved mirrors being more commonly used. Compared to curved mirrors, the disadvantage of lenses is their reflection loss and chromatic aberration. However, they can produce tight focusing and have no astigmatism. This is very important when using lenses to focus ultra short pulses.