Cylindrical lens | optics

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Author: Dr. Rüdiger Paschotta, a specialist in photonics.

An optical lens comprises a transparent medium (commonly optical glass or polymer) that enables light to enter from one side and exit from the opposite. Typically, lenses include at least one curved surface aimed at altering the wavefront curvature of light, resulting in either a focused or defocused beam. Lenses categorized as focusing or defocusing can also be described as converging or diverging lenses, although it may be more intuitive to apply those terms to beams instead of the lenses themselves. It is also common to refer to lenses as positive (when focusing) or negative (when defocusing).

In more precise terms, a lens is defined as a singular optical element, although assemblies containing multiple lenses within a shared housing are frequently referred to as lenses themselves. These systems are discussed in more detail in articles focusing on objectives or specific categories like photographic objectives.

Some common examples of optical lens functions include:

• A collimated beam, featuring approximately flat wavefronts, is transformed by a lens into a beam where the wavefronts curve, directing the light to a focal point. This lens acts as a focusing lens; see Figure 1 (a).
• Similarly, the same lens can convert a diverging beam into a collimated one, functioning as a collimating lens; this is depicted in Figure 1 (a) if the incoming beam originates from the right.
• Alternatively, lenses with concave surfaces can convert a collimated or convergent beam into a divergent one; see Figure 1 (b). Such lenses may also be utilized to collimate an originally convergent beam.
• Often, a single lens or a combination of lenses (an objective) is employed for imaging applications. For example, camera objectives are used to capture scenes on photographic film or electronic image sensors, while microscope objectives serve to image smaller objects.

Figure 1:

Focusing and Defocusing Lenses.

While the alteration in beam radius is frequently seen as the primary function of a lens, the core purpose lies in the modification of wavefront curvature, which directly causes changes in beam radius during propagation after passing through the lens. (It's important to note that optical energy will always propagate perpendicularly to the wavefronts.) This is illustrated in Figure 2; between the lens and the beam's focus, light converges due to wavefront curvature, and post-focus, it diverges as a result of wavefront curvature in the opposite direction.

Figure 2:

Change of Wavefront Curvature at a Focusing Lens.

The red and blue areas illustrate the intensity and polarity of the electric field at a specific moment in time, assuming a wavelength significantly larger than actual conditions.

Physical Origins of Wavefront Changes

Wavefront changes produced by most lenses stem from the curvature of at least one of their surfaces. For typical biconvex lenses (i.e., lenses featuring two convex surfaces), as depicted in Figure 2, light passing through the lens' center experiences a greater optical phase delay than light traveling near the perimeter, where the lens is thinner. This is attributed to the lens material's refractive index being higher than that of the surrounding medium (often air). The resultant radial variation in phase delay corresponds directly to the change in wavefront curvature.

Another explanation focuses on refraction occurring at lens surfaces. In scenarios involving thick lenses, a thorough calculation based on refraction tends to be more precise than one that relies solely on radially varying phase delay, which does not consider beam size variations within the lens.

Additionally, gradient-index lenses (GRIN lenses) exhibit a refractive index that varies systematically throughout the lens material. In a focusing GRIN lens, the refractive index peaks at the center and decreases outward, creating an essentially parabolic index gradient. The surfaces of a GRIN lens are generally flat, resembling either a standard plate or more commonly, a cylindrical rod.

Innovative optical solutions are also emerging from photonic metasurfaces, allowing for customization of arbitrary transverse phase change profiles and facilitating the avoidance of specific optical aberrations. Furthermore, such applications can be incredibly flat and thin.

Focal Length

The focal length <$f$> of a thin lens is defined as the distance from the lens to its focus when receiving a collimated beam (refer to Figure 1 (a)). For defocusing lenses, the focal length appears negative, representing the negative distance to the virtual focus (as depicted in Figure 1 (b)).

When considering a thick lens, the situation becomes more complicated. Such a lens typically features two principal planes, with distinct focal lengths for each side based on the distance to their corresponding focal points. Provided the lens's surrounding medium is uniform on both sides, the two focal lengths will be equal. For comprehensive information, refer to the article on focal length.

The dioptric power (or focusing capability) of a lens equals the inverse of its focal length or the refractive index of the surrounding medium divided by the focal length itself.

Conventional lenses found in laser technology have focal lengths ranging from 5 mm to several meters. Compact ball lenses and tiny aspheric lenses can be considerably smaller, sometimes measuring below 1 mm.

A lens with a particular focal length <$f$> delivers a radially varying phase delay for a laser beam based on the following equation:

$$\Delta \varphi (r) = - \frac{\pi }{{\lambda f}}{r^2}$$

This equation disregards the constant component of phase alteration and potential aberrations. Depending on the lens's function—such as focusing collimated beams or refocusing divergent light—higher-order terms in the phase profile may be necessary to prevent optical aberrations.

For further insights, consult the article on focal length.

Lensmaker's Equation

The following formula, termed the lensmaker's equation, allows for the calculation of a lens's focal length formed from material with refractive index <$n$> and curvature radii <$R_1$> and <$R_2$> across its two surfaces:

$$\frac{1}{f} = (n - 1)\left( {\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}} \right) - \frac{{{{(n - 1)}^2}\;d}}{{n\;{R_1}\;{R_2}}}$$

The curvature radii are assigned positive values for convex surfaces and negative ones for concave surfaces. The last term is only relevant for thick lenses (as mentioned above) that possess significant curvatures on both sides. This equation holds for paraxial rays that remain close to the optical axis and assumes a surrounding medium with a refractive index approaching 1 (like air).

Be aware that various sign conventions surface in the literature; for instance, a prevalent method designates the radius for the second interface as positive for concave surfaces, which contrasts with the convention applied here.

Thin and Thick Lenses

In many practical scenarios, a lens's thinness results in negligible beam radius alterations within the component. This phenomenon is frequently seen in lenses with mild surface curvatures (i.e., large curvature radii). The lensmaker's equation can be simplified by omitting the third term in such cases, resulting in a streamlined thin lens equation. Numerous other optical equations feature simplified versions applicable solely to thin lenses; the criteria for a lens's classification as thin can differ across these equations.

If thick lenses are deployed, particularly where strong focusing power is essential, the thickness <$d$> (the distance between lens surfaces as measured along the optical axis) significantly influences the focal length, as shown in the lensmaker's equation. It's worth noting that defining the exact position of a thick lens and consequently its focal length can be ambiguous, especially in asymmetrical designs.

The Lens Equation

Figure 3:

Illustration of the Lens Equation.

In instances where a divergent beam (rather than a collimated one) strikes a focusing lens, the distance <$b$> from the lens to its focus exceeds the focal length <$f$> (see Figure 3). This situation is quantifiable via the lens equation:

$$\frac{1}{a} + \frac{1}{b} = \frac{1}{f}$$

where <$a$> denotes the distance from the original focus to the lens. This established relationship indicates that <$b \approx f$> if <$a \gg f$>, yet <$b > f$> otherwise. Intuitively, this means a focusing power of <$1 / a$> would be necessary to collimate the incident beam (i.e., to neutralize beam divergence), leading to a residual focusing power of <$1 / f - 1 / a$> for actual focusing.

If <$a \le f$>, the equation becomes insolvable, indicating that the lens cannot adequately focus the beam.

Remember that the lens equation applies only to rays under the assumption that the paraxial approximation is valid, meaning all angle deviations concerning the beam axis remain small.

Focusing a Collimated Beam

When a collimated Gaussian beam with a beam radius <$w_0$> encounters a focusing lens with a focal length <$f$>, the beam radius at the beam waist (the focus) post-lens can be calculated using the equation:

$${w_{\rm{f}}} = \frac{{\lambda f}}{{\pi {w_0}}}$$

Here, it is assumed that the beam radius at the focus is considerably smaller than the initial beam radius <$w_0$>. If the incident beam's radius is too small, this condition may fail, resulting in a focus larger than indicated by this equation. Additionally, it is presumed that the beam radius significantly exceeds the wavelength <$\lambda$> for the paraxial approximation to hold.

Calculator for Beam Waist Radius

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Numerical Aperture and f-Number of a Lens

The numerical aperture (NA) of either a focusing or collimation lens is defined as the sine of the angle of the marginal ray originating from the focal point, multiplied by the refractive index of the medium from which the input beam originates. The NA of a lens (as opposed to its focal length) constrains the size of the beam waist that may be formed. Higher NA lenses, typically ranging from 0.5 to 0.9, are essential for devices like CD, DVD, and Blu-ray players and recorders. In microscopy, the NA constrains the achievable image resolution.

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Keep in mind that the numerical aperture definition is not dependent on focal length for lenses or objectives designed to image from an object plane to an image plane. Here, the angular opening is considered from a point in the object plane. For more details, refer to the article on numerical aperture.

High-NA lenses are likewise critical for collimating laser beams originating from narrow apertures, exemplified by low-power single-mode laser diodes. Utilizing a lens with an insufficiently high NA may lead to distorted (aberrated) or truncated collimated beams.

It is evident that a high-NA lens must possess considerable size to maintain a large focal length.

For camera lenses (photographic objectives), a specific minimum f-number or range of f-numbers is frequently outlined. For instance, a f/4 lens indicates a maximum open aperture (specifically the maximum entrance pupil) that is a quarter of the focal length. (Note that <$f$> represents the focal length, not the f-number!) This correlation implies a numerical aperture of <$\sin(1 / 4)$> ' 0.247, although f-number specifications are more customary for photographic objectives.

Lens Types

The focusing lenses depicted in the preceding figures all represent biconvex designs, characterized by convex surfaces on both sides. Plano-convex lenses, in contrast, are flat on one side and convex on the other. Additionally, biconvex lenses may exhibit differing curvature radii between surfaces. Just as defocusing lenses may be biconcave or plano-concave.

Figure 4:

Varieties of Optical Lenses.

According to lensmaker's equation (referenced earlier), differing lens designs can achieve specific dioptric powers, although they may vary in aberration performance (i.e., imaging errors). For imaging a small spot to an equally sized spot, a symmetric biconvex lens proves most effective. Alternately, for asymmetrical applications, such as collimating a divergent beam or focusing a collimated beam, a plano-convex lens may be preferred. The optimal solution would ideally involve an asymmetric lens with curved radii optimized for both sides. It should be oriented such that the curved surface faces the collimated beam. Consequently, both lens surfaces contribute to the focusing effect. Figure 5 illustrates examples of commonly used configurations. The precise ideal configuration may deviate somewhat from a plano-convex lens in scenarios requiring beam focusing, yet the plano-convex design tends to be nearly optimal.

Figure 5:

Recommended Lens Types for Refocusing and Collimation. In terms of aberrations, the scenario for symmetric refocusing is superior to the first, although it may incur higher losses due to the additional optical surfaces.

Meniscus lenses feature a convex-concave design, which means one surface is convex while the other is concave. The effects of both sides on the dioptric power partially offset, allowing the lens to be overall positive (focusing) or negative (defocusing). Positive meniscus lenses are suitable for beam focusing, but are more commonly used as corrective elements in objectives for mitigating image aberrations, apart from serving in condenser applications for illumination systems.

Ball lenses take on a spherical shape, while rod lenses are cylindrical.

Doublet lenses consist of two lenses bonded together, made from varying optical materials. The most frequent examples are achromatic doublets (refer to details below).

Fresnel lenses can be constructed to be thinner but may compromise optical performance.

Centration of Lenses

For many technical applications, an accurate centering of a lens is critical; that is, ensuring the lens's center aligns precisely with the intended beam path. Any deviation from this alignment could result in a beam trajectory offset from the center, resulting in minor beam deflections. Centration errors are typically described based on the angular deviation of such a beam.

Achieving precise centering is particularly vital for lenses exhibiting short focal lengths, as minor positioning errors can yield significant beam deviations.

Types of Aberrations from Lenses

Lenses can introduce a range of aberrations (degradation of images):

• While many lenses possess spherical surfaces because they are easier to manufacture, such spherical surfaces often deviate from the ideal shape. This discrepancy can result in image aberrations—especially in peripheral regions—or a decline in laser beam quality, known as spherical aberrations. Aspheric lenses can notably reduce these aberrations. Specialized lenses may have surface contours tailored to specific applications; check the article on spherical aberrations for further information.
• When light enters at an angle, astigmatism and coma may develop. Furthermore, lenses exhibit field curvature wherein focal points for incoming rays from various angles align along a curve, rather than a planar surface.
• Distortions in images can also occur, manifesting as barrel, pincushion, or mustache effects.
• Chromatic aberrations arise due to the chromatic dispersion characteristics of the lens material. Consequently, the focal length can become dependent on wavelength, leading to white light not being perfectly focused; various wavelength components will focus at distinct points. Achromatic lenses notably diminish these chromatic discrepancies.
• If a laser beam with an excessively large radius strikes a lens, its peripheral beam profile may become truncated, resulting in substantial beam distortions. This aperture diffraction phenomenon is also observed in imaging scenarios, where a lens's finite dimensions may limit an optical system's imaging resolution. However, diffraction does not always impose a quality limitation if the optical components and design maintain high standards.

Such imaging errors can frequently be mitigated either by optimizing individual lenses or through effective combinations of multiple lenses. This is a key reason why photographic objectives generally feature a significant number of lenses.

Aspheric Lenses

While spherical aberrations can often be effectively mitigated through the strategic combination of multiple lenses, it is sometimes preferable to utilize aspheric lenses, characterized by surfaces that deviate from a spherical form. Such designs facilitate high imaging quality (minimal spherical aberrations) with either a single lens or fewer lenses in an objective. However, aspheric lenses are more challenging to manufacture and hence tend to be more expensive.

Achromatic Lenses

The standard method for producing achromatic lenses—characterized by significantly reduced chromatic aberrations—involves combining two lenses made of distinct materials (as shown on the right side of Figure 4). A typical combination includes low-index crown glass shaped biconvex and high-index flint glass formed as plano-concave to create such an achromatic doublet. The radii of curvature at the bonding site are meticulously determined to minimize chromatic dispersion and must be precisely matched.

Cylindrical and Astigmatic Lenses

A lens exhibiting curvature solely in the horizontal dimension, without vertical curvature, is identified as a cylindrical lens. This type will either focus or defocus light only in the horizontal plane while maintaining unaffected wavefront curvature in the vertical direction.

Cylindrical lenses are advantageous when generating elliptical beam focuses or for cultivating or compensating astigmatism in a beam or optical system.

In the case of a curvature existing in both directions, albeit at unequal strengths, the result is an astigmatic lens. Such lenses can be beneficial for vision correction, for instance.

Infrared Lenses

Standard dielectric materials suffice for lens creation in the near-infrared spectrum. However, as wavelengths increase into mid-infrared ranges, specialized infrared optics materials must be employed. Semiconductors such as silicon and germanium are examples that can be utilized.

Multiple Element Lenses

In various scenarios, achieving multiple objectives concerning properties like minimized aberrations with a single lens may prove challenging. As such, using multiple-element lenses—combinations of lenses—may provide better outcomes. Lens doublets (comprising two lenses) and triplets (three lenses) are frequently employed; some systems may incorporate even more components. Many of these can be classified as objectives, including photographic objectives or projection objectives. (As previously mentioned, achromatic doublets fall into this category.)

These singular lenses may either be cemented together or arranged with air gaps in between (air-spaced lens systems). Regardless, a multiple-element lens functions similarly to a single optical element, such as a photographic objective.

Lens Fabrication

Lenses are traditionally crafted from prefabricated lens blanks that tend to be slightly larger. By employing suitable grinding and polishing techniques, unnecessary material is removed to create the desired shape and surface quality. An anti-reflection coating is commonly applied following this process, and frequently throughout the fabrication process, advanced optical characterization techniques (like interferometry) are employed for monitoring fabrication defects, which may interfere with wavefront precision.

Lens Surface Coatings

Numerous lenses include anti-reflection coatings on their surfaces, significantly diminishing reflection losses associated with refractive index changes at the surface. However, this function only operates within limited wavelength ranges. Achieving a balance between substantial reflection suppression and broad operational bandwidth can be challenging.

Coatings designed for abrasion resistance also enhance lens durability.

Lens Types by Application and Optical Function

Optical lenses and systems are often categorized per their optical functions within instruments. Examples of various terminologies include:

Compared with curved mirrors, lenses possess some disadvantages, such as reflection losses and chromatic aberrations. Conversely, lenses facilitate precise focusing without inducing astigmatism, a factor that is particularly relevant when using lenses (or mirrors) for focusing ultrashort light pulses.

Figure 6:

A macro lens designed for integration with a standard objective.

Standard Lenses Versus Custom Lenses

Given that lenses have limited essential parameters (such as focal length, numerical aperture, and operational wavelength range), standard lenses are often utilized, many of which are readily available in stock. However, customized lenses can also be produced according to specific requirements. This approach allows for lenses with unique optical performance metrics (e.g., aspheric lenses) as well as customized geometrical features and optical materials.

Specialty Lenses

Some specialty lenses exhibit unusual characteristics. For example, axicons, which feature a conical surface, are categorized as specialty lenses.

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