aberrations

Geometric aberrations

For optical systems which are not rigorously stigmatic or when Gauss conditions are not respected (which means that the angle with the axis of light rays coming from an object is large and/or the object is far from the axis), geometrical aberrations may appear. In this case, the image of a point is no more a point but a spot and therefore the image of a scene is blurred. Geometric aberrations are different from chromatic aberration which only appears in optical systems containing refractive components (like lenses for example) but can occure also under Gauss conditions. Chromatic aberration may blurr the image by generating images of the same object at different locations depending on the wavelength. Therefore, the chromatic aberration appears for objects illuminated with polychromatic sources as geometric aberration happens also with monochromatic sources. Note that only systems with a cylindrical symetry are considered in the following sections.

The pictures on the left illustrate the aberrations of a single spherical lens for an off-axis object on the Y axis (vertical). The furthest to the left picture represents the ray tracing in the YZ plane, that one in the middle is a ray tracing in perspective view and the most to the right one is the spot diagram obtained in the paraxial image plane.

Before describing aberrations, one must define the aperture of an optical system.

Aperture of an optical system

The aperture is supposed to be the only surface of the system stopping light rays. This surface can be in the object space, in the image space or in an intermediate space.

In any case, it is possible to define the aperture in the object space: it is the conjugate of the aperture by the sub-system formed with the preceding components and used in the reverse way. Also, it is possible to define the aperture in the image space which is the conjugate of the aperture by the sub-system formed with the following components. The aperture in the object space enables to defines the solid angle inside which the rays are launched from the field(s). The aperture in the image space enables to eventually calculate the Wave Front Error and the associated parameters as well as the diffraction limit of the system.

Note that in some cases, vignetting can appear. Indeed, some rays coming from off-xis objects may be stopped by another surface than the said aperture surface. This can affect the aberrations, limit the system resolution and create some shadows on the image. In general, one try to avoid vignetting and this case is not considered in the rest of the tutorial.

An optical system is represented in the pictures below with different aperture positions. Theoretically, the chief rays (chief rays are defined further in this tutorial) coming from different objects (fields) intersect the aperture surface at the same point. This point is the aperture center.

In the left picture, the aperture is defined by the first surface. In the right picture, the aperture is defined by a surface in the middle of the optical system.

In the left picture, the system is telecentric (aperture at infinity) in the object space as the chief rays in the object space are all parallel. In the right picture, the system is telecentric in the image space as the chief rays in the image space are all parallel.

The aperture can be defined in several ways.

It can first be specified by the diameter D of the aperture itself.

It can also be defined by the half aperture angle θ_M in the object space. θ_M is the angle with the axis of a ray coming from the on-axis object and passing at the edge of the aperture in the object space (called marginal ray). θ_M is often used to specify the aperture of illumination optics. For these systems, θ_M may be larger than 90°.

The aperture is commonly specified with the numerical aperture in the object space NA.

where n is the refraction index in the object space.

The aperture is also often defined with the F number (common notation: F/N, F#, F_number). It is currently the case for many systems as for instance camera lenses.

.

. is the numerical aperture in the image space (n' is the refraction index in the image space). The larger the aperture, the larger the numerical aperture and the lower the F number. For optical systems with small aperture and working for objects located at infinity, the F number is approximately the ratio of the focal length f_i to the diameter D of the stop surface : .

Geometric aberration formula

At this stage, it is necessary to give the definition of the expressions commonly used to describe aberrations.

The tangential plane linked to the object A is the plane containing A and the optical axis of the system. Thus, it necessarily contains the aperture center (the aperture being supposed to be centered on the optical axis).

The sagittal plane linked to the object A is perpendicular to the tangential plane. It also contains A and the aperture center.

The chief ray linked to the object A is the ray coming from A and passing through the aperture center.

A marginal ray linked to the object A is a ray coming from A and passing at the edge of the aperture.

The coordinate x' and y' of the impact B in the observation plane of a ray coming from an objet A and propagating through an optical system can be calculated according to the formula below :

. .

The object A is considered on the Y axis. h is its coordinate along the Y axis. P is the impact of the ray on the aperture in the object space. s and α are the cylindrical coordinate of P referred to the aperture (see the picture on the left). This formula relates all the aberrations which are 5 in number : the spherical aberration, the coma, the astigmatism together with the field curvature and the distorsion.

This formula is a Taylor expansion. The first order terms (s.cos(α) and s.sin(α)) correspond to the terms of the gauss approximation. A₂ coefficient is related to the magnification. A₁ coefficient expresses a defocus and thus would be equal to "0" if the observation plane was the paraxial image plane. The higher order terms are related to the transversal geometric aberrations. The third order terms are generally preponderant in optical system with a relatively small aperture. They are the "B" coefficients in the formula. They are also called Seidel coefficients. B₁ is related to the spherical aberration, B₂ to the coma, B₃ to the astigmatism, B₄ to the field curvature and B₅ to the distorsion. Note that this aberrations classification is exhaustive and therefore is also suitable for higher orders (5th order aberrations expressed through the "C" coefficients, 7th order aberrations not expressed in the above formula, etc.). For simplicity, the following sections are sometimes based on third order aberrations interpretation which are existing in any case and preponderant for slightly opened systems.

Spherical aberration

Let consider an optical system with spherical aberration (coefficients B₁ for the third order spherical aberration, C₁ for the fifth order spherical aberration,...) Rays from an on-axis object and making the same angle θ with the optical axis are focusing at the same point (image) because of the cylindrical symetry of the system. This appears in the Taylor expansion by the fact that the spherical aberration terms are not depending on α (see section related to the "geometric aberrations"). However, the position of this image depends on θ (or s in the Taylor expansion). At the limit angle θ = 0, the image coincides with the paraxial image. The spherical aberration represent the variation of image position depending on θ. The maximum distance between the image and the paraxial image is the longitudinal spherical aberration. The transverse spherical aberration is equal to the aberration spot diameter in the paraxial image plane. Note that the spot is circular because of the cylindrical symetry of the system. The spherical aberration depends on the position of the object plane, on the lens shape and obviously on the aperture size. It does not depend on the aperture position and on the distance of the object to the optical axis as the related coefficients in the Taylor expansion are proportionnal to s^2.n+1. It means that the spherical aberration is the same everywhere in a given object plane.

The two figures on the left show the ray tracing of a single spherical lens with spherical aberration (leftmost picture) and the associated spot diagram in the paraxial image plane (rightmost picture).

The left side scheme illustrates the case of a converging spherical lens. The larger the angle θ, the more the image moves away backwards from the paraxial image. The longitunal spherical aberration noted LAS on the picture is then negative, which means that the lens is undercorrected. In this particular case, the transverse spherical aberration is .

The longitunal spherical aberration is positive for a diverging lens which is then overcorrected.

It is possible to minimize the spherical aberration of a single lens in certain conditions. That is for instance the case with an object at infinity and when the radius of curvature of the first facet is around 6 time smaller than the radius of curvature of the second facet (see left figure on the left side). Note that if the lens is reversed (right figure), the spherical aberration is no more minimized. Note also that the lenses displayed on the two picture seems different just because the pictures have different ratio vertical scale/horizontal scale. In general, when considering a lens with convex facets, the configuration with less spherical aberration is the following one : the facet with the smallest radius is oriented towards the "more parallel" rays.

In the general case, an optical system with spherical aberration can be either undercorrected or overcorrected. LAS is not necessary the distance d from the paraxial image to the marginal rays image : in many cases, LAS > d and it is the distance from the paraxial image to the image corresponding to an intermedate angle .

The spherical aberration depends on the position of the object plane and on the aperture size. It does not depend on the aperture position and on the distance of the object to the optical axis as the related terms in the Taylor expansion are proportionnal to s^2.n+1. It means that the spherical aberration is the same everywhere in a given object plane.

Most components as spherical lenses, spherical mirrors, plates with flat and parallel faces,...have spherical aberration whatever the object position. Some others like ellipsoidal or parabolic mirrors and dioptres have in general spherical aberration but may be rigorously stigmatic for particular object positions (see rigorous stigmatism tutorial).

An optical system can be corrected for the spherical aberration (generally partially up to a certain order) by using aberrated components that combined together finally minimize or cancel its effects. Also, a certain type of lenses with aspherial shapes can almost be totally corrected for spherical aberration, even for large apertures. These kind of lenses are for instance used for on-axis objects, for example to inject the beam from a VCSEL (Vertical Cavity Surface Emitting Laser) which emitting area is small (in the range of the micron²) into a single mode fiber optic. A group of two lenses, the first one collimating the beam and the second one focusing the beam at the fiber entrance can be used for an optimal coupling. There are other solutions for minimizing spherical aberration like using a Schmidt plate before a spherical mirror when the object is at infinity...

The leftmost picture shows the ray tracing of a lens which is flat on one side and aspherical on the other side. The aspherical side is optimized to correct almost perfectly the spherical aberration for an object at infinity. Its spot diagram (right picture) shows a spot much smaller than the Airy disk (black circle) defining the diffraction limit of the system (see the "diffraction" tutorial).

Chromatic aberration

Another type of aberration than geometric aberrations exists for optical systems including refractive components. It is the chromatic aberration which is depending on the refractive materials used. Unlike geometric aberrations, chromatic aberration, occurs also in Gauss conditions for objects close to the axis and rays with direction close to the optical axis direction. Unlike reflective components, refractive components generally contain a dispersive material (which refraction index depends on the wavelength). The same incident ray is refracted differently at the component interface depending on the wavelength. Consequently, the paraxial conjugation is different depending on the wavelength. Practically, if the object is illuminated by a polychromatic light source, it has as many images as wavelengths contained in the light source spectrum. Therefore, whatever the position of the observation plane, the image of a point is an iridescent spot and the image of a scene can be blurred with coloured outlines.

Considering an optical system and an axial object A illuminated with a polychromatic light source, the axial chromatism (or longitudinal chromatism) is defined as the algebric distance dx_i from Air to A_ib, A_ir and A_ib being the conjugated points respectively for the largest and the smallest wavelengths of the light source spectrum.

The principal axial chromatism corresponds to the case where the object is at infinity. Therefore, it is the algebric distance df_i from F_ir to F_ib, F_ir and F_ib being the back focal points respectively for the largest and the smallest wavelengths of the spectrum.

Obviously, the chromatism affects also the off-axis conjugation. As the position of the image plane changes with the wavelength, the conjugated of an off-axis object is at a different height depending on the wavelength and thus the magnification also changes depending on the wavelength. The height chromatism dy_i for an off-axis object B is defined by the algebric height difference between B_r and B_b, B_r and B_b being the paraxial images of B respectively for the largest and the smallest wavelengths of the spectrum. The height chromatism is currently called lateral colour.

The constringence (or Abbe number) characterizes the dispersion of a material. The dispersion is the variation of refractive index regarding the wavelength. The constringence is given by where n_B and n_R are the refraction indexes of the lens material respectively for the lowest and the highest wavelength of the spectrum, n_Y being the refraction index at an intermediate wavelength. As the refraction index decreases with the wavelength, the constringence is positive. Its values is in general between 20 and 100 in the visible spectral range.

In the visible, the constringence ν is commonly calculated using 486.13 nm (blue hydrogen line) and 656.27 nm (red hydrogen line) as the lowest and highest wavelength as well as 589,3 nm (yellow sodium line) as the intermediate wavelength. It is also calculated with other sets of wavelengths. Indeed, ν_d uses the same extreme wavelengths than ν but the intermediate wavelength is 587,56 nm ( yellow helium line ). ν_e is calculated using 480.0 nm (blue cadnium line) and 643.8 nm (red cadnium line) as the lowest and highest wavelength as well as 587,56 nm (mercury line) as the intermediate wavelength. One can divide the materials in two categories. The higher is the constringence, the less dispersive is the material. A material is usually considered to have a low dispersion when its constringence is higher than 50 and a high dispersion when its constringence is lower than 50.

The figure on the left (Source WIKIPEDIA), called Abbe Diagram, represents the constringence (or Abbe number ) regarding the wavelength for glasses labelled with the SCHOTT manufacturer code. It is very useful when designing an optical system supposed to work at different wavelengths as it gives an idea of the refractive index values and of the constringence.

For a thin lens, the principal axial chromatism is where f_i is the back focal length and ν is the constringence. Therefore, it is negative for a positive lens (converging lens) and positive for a negative lens (diverging lens). This is easily explained by the fact that as the refraction index decreases when the wavelength increases, the rays at short wavelength are more deviated than the rays at long wavelengths. Accordingly, focal lengths are shorter at short wavelengths than at long wavelengths.

As positive and negative lenses have opposite axial chromatism, one can obtain a system corrected from the chromatism by using two thin lenses ( L1 and L2 ) placed side by side. If L1 is converging, its chromatism is negative and can be compensated by the positive chromatism of the diverging lens L2. Given the focal length of the system, the respective focal lengths of L1 and L2 are calculated as follows : .

The system is called an achromat. According to the formula, the constringences of L1 and L2 must be different and thus the two lenses have to be manufactured with different materials. L1 has to be less dispersive than L2.

The left side curve on the left shows the principal axial chromatism (also called focal shift) of an achromat with a focal length of 100 mm. It is made with a converging lens L1 (f_i1=50 mm, ν₁=60) and a diverging one L2 (f_i2=-100 mm, ν₂=30). This curve has a minimum which means that the focal length is the same for two different wavelengths. It is obviously different from the right side curve representing the principal axial chromatism of a single lens with the same focal length and a constringence of 60. Even if it is not null, The achromat focal shift in the visible spectrum is significantly smaller than for the single lens ( about 8 times smaller in the present case).

MTF of an optical system

The modulation transfer function (MTF) is a mean to evaluate the spatial resolution of an optical system on an image surface supposed to be "conjugated" with the object surface. It gives the local contrast produced by the system on the image surface as a function of the spatial frequency. Therefore, it provides an indication on the aberrations of the system (geometrical and chromatic aberrations) by comparison with its maximum value obtained with a perfect system and allows to check if its resolution is suitable.

The MTF of an optical system is represented by the blue curve on the left side graphic. The spatial frequencies are on the abscissa and the contrast on the ordinate. The black curve is the ideal MTF of the system (without aberration) thus only limited by diffraction.

Let consider a point M of the object surface and its "conjugated" point M' on the image surface. "Conjugated" means here that M' is the intersection with the focusing surface of the chief ray from M after propagation through the system. The MTF at the vicinity of M' is computed with the wave front error (WFE - see the "WFE" tutorial in the "calculate"->"real propagation" menu of the "Advanced calculations" tool) calculated from the point M. Except if the system has a cylindrical symetry and if M is on-axis, the WFE has no cylindrical symetry and the contrast depends on the direction considered in the focusing plane. Therefore, the MTF is calculated in two directions : the tangential and the sagittal directions. It is computed for a pattern made of bands perpendicular to the considered direction and which intensity modulation is sinusoidal (the intensity modulation is then along the considered direction).The MTF in the tangential direction for the spatial frequency F_t at the wavelength λ is given by the formula:

,

.

t and s are the normalized coordinates on the aperture respectively along the tangential axis and the sagittal axis.

p(t, s) is the transmission function of the aperture (it is in general a complex function as the aperture can induce losses dans phase shifts. For a simple diaphragm, p(t, s)="1" inside the diaphragm boundary and p(t, s)="0" outside). WFE(t, s) is the wave front error function calculated from the point M. f_t is the normalized spatial frequency along the tangential axis T.

.

F_C is the cut-off frequency (above which the contrast is null). It is given by the formula :

.

F_num is the F-number of the system.

For sinusoidal bands perpendicular to the sagittal direction (the modulation is along the sagittal direction) and with a spatial frequency F_s, the MTF is given by the formula:

.

is the normalized spatial frequency along the sagittal axis S.

When several wavelengths are involved, the global MTF is the weighted sum of the MTF for each single wavelength. The weight is the proportionnal to the ratio of intensity contained at the wavelength over the total intensity

References

"Optique Fondements et applications" - 2004 - author : José-Philippe Perez.

"Optique géométrique Imagerie et instruments" - 2007 - author : Bernard Balland.

"Optique géométrique paraxiale" - Institut doptique théorique et appliquée - 1985 - author : Michel Cagnet.

"Formation des images Aberrations" - Institut doptique théorique et appliquée - 1985 - author : Michel Cagnet.