# Primary Aberrations, a lecture in optics. 1

## Lecture-II; delivered on 27-1-2017

In our Lecture-I  we discussed the phenomena of aberrations that arise because of a discrepancy of a first order theory and the 3rd order theory as depicted by the Maclaurin series; where we saw that first order theory represents the so called paraxial optical systems. Please have a look of the linked article to get a basic view of the ground on which we are discussing this topic. At-least going half-way through the lecture and stopping short of the derivation will do well.

We discussed that there are two kind of aberrations. Monochromatic and Chromatic. As the name suggests the monochromatic aberrations are a result of the discrepancy when we considered our incoming ray to be having a single wavelength of light. The chromatic rays on the other hand can have multiple colors or wavelength of light. The monochromatic aberrations are also called as Seidel or Primary aberrations and we will shed more light on them today. The chromatic aberrations were dealt in greater detail — eg the derivations pertained to the chromatic aberrations. We did so because the chromatic aberrations are simple to understand.

So lets discuss in detail the 5 types of primary aberrations now.

## Primary Aberrations.

### 1. Spherical Aberration.

When Paraxial Rays  refract after emerging from an object point they meet at a sharp focus.
But when non-paraxial or marginal rays emerge — or appear to emerge, from an axial object point they do not meet at a sharp focus.
Therefore different rays meet at different focal points. The resulting aberration is called as spherical aberration.

Photo-Credit; Spherical Aberration. Rays of different height have different focus. Creates a blurred image.

No spherical aberration would mean a clearer image.

For a refraction from spherical surface lets apply the Fermat’s Principle . The principle states that optical path length is stationary wrt any change in the position coordinates, of the ray. Time and again we will be applying the Fermat’s Principle, which is the backbone of geometrical or ray optics, as defined in our Lecture-I. Two articles (linked next) — art-1, art-2, were written outside of the nature of the lecturing styles of lecture-I and the concurrent lecture, which shed a good deal of conceptual light, on the meaning of Fermat’s Principle. You might wanna have a look first.

During the lecture I take some efforts to explain the Fermat’s Principle eg by drawing a reflection diagram and a refraction diagram and by explaining the ideas of optical path. If you read the linked articles, art-1, and art-2 you will soon get an article devoted towards Fermat’s Principle itself, in our concurrent lecturing style; in our Lecture-III for optics, by the name Geometrical Optics. Yes, its coming next.

The Fermat’s Principle applied for a refracting spherical surface, would give in the first order, the following.

$\inline&space;\fn_cs&space;\frac{n_1}{u}+\frac{n_2}{v}=\frac{n_2-n_1}{R};&space;\hspace{10pt}sin(\phi)=\phi.$

If we apply 3-rd order corrections as-well we would get the following.

$\inline&space;\fn_cs&space;\frac{n_1}{u}+\frac{n_2}{v}=\frac{n_2-n_1}{R}+h^2[\frac{n_1}{2v}(\frac{1}{v}+\frac{1}{R})^2+\frac{n_2}{2u}(\frac{1}{R}-\frac{1}{u})^2];&space;\\&space;\hspace{10pt}sin(\phi)=\phi-\frac{\phi^3}{3!}.&space;\hspace{10pt}SA&space;\propto&space;h^2$

Where h is the height at which the incident ray is refracted at the spherical surface. We see that h is larger for marginal or peripheral rays hence SA (henceforth: spherical aberration) is larger for marginal ray compared to rays near optical axis (known as paraxial ray) and grows as rapidly as the square of the height of ray. SA is proportional to square of h.

Photo-Credit: SA is proportional to the square of height of the corresponding ray incident upon a refracting surface, for spherical surfaces.

The distance between axial intersection of a ray and paraxial focus Fp is known as longitudinal spherical aberration. It is abbreviated as L.SA or Slong.
L.SA is positive for converging or convex lens which are considered positive lenses. Similarly L.SA is negative for diverging or concave lens which are considered negative lenses. The image blur is smallest on circle of least confusion.
The height from the paraxial focus where a marginal ray meets is called as transverse spherical aberration or T.SA.

Lets discuss some of the important results of spherical aberration without having to go through the strenuous exercise of deriving them by fundamental principles.

1. Plane refracting surface.

Spherical aberration due to refraction at plane surface.

$\fn_cs&space;L.SA=\Delta&space;z=-\frac{h^2}{2n&space;|z_0|}(n_2-1)$

2. For a single spherical refracting surface.

$\inline&space;\fn_cs&space;L.SA=\Delta&space;z=\\-\frac{n_2-n_1}{2n_2(\frac{1}{z_0}+\frac{n_2-n_1}{n_1R})^2}(\frac{1}{R}+\frac{1}{z_0})^2(\frac{-n_2+n_1}{n_1-z_0}+\frac{1}{R})h^2$

3. For a thin lens in air we define coefficient of spherical aberration A.

$\inline&space;\fn_cs&space;A=-\frac{f(n-1)}{2n^2}\times&space;[-(\frac{1}{R_2}-P)^2(\frac{1}{R_2}-P(n+1))+\frac{1}{R_1^3}]&space;\\&space;P=power=\frac{1}{f}=(n-1)(\frac{1}{R_1}-\frac{1}{R_2})$

Then we would have T.SA = Slat = Ah and L.SA = Slong = Ah2f.

Note: The only primary aberration that occurs when object points are axial is spherical aberration. All other primary aberrations are absent for axial object points.

### 2. Coma

We learned that for object points that are axial only one type of primary aberration occur, namely spherical aberration. What would happen if we move slightly off the optical axis? Both coma and SA would occur. Coma is a comet like — hence the name, image of a slightly off-axis object point.

Photo-Credit : Coma occurs due to off-axis rays, its a difference in magnification whereas SA was difference in focus.

Photo-Credit : Point like Stars appear like comets. Its more prominent in periphery.

If a parallel bundle of rays making an angle θ with optical axis are incident on a thin lens coma is given by:

$\inline&space;\fn_cs&space;coma=\frac{3(n-1)}{2}fh^2\tan^2\theta[\frac{(n-1)(2n+1)}{nR_1R_2}-\frac{n^2-n-1}{n^2R_1^2}-\frac{n}{R_2^2}]$

For a lens of shape factor q = +0.8 coma is zero, where:

$\fn_cs&space;q=\frac{R_2+R_1}{R_2-R_1}$

If Abbe sine condition is satisfied both SA and coma are zero:

$\fn_cs&space;\frac{n_1\sin\theta_1}{n_2\sin\theta_2}=\frac{y_2}{y_1}=linear&space;\hspace{3pt}&space;magnification\hspace{5pt}=\hspace{3pt}&space;constant.$

For Plano-Convex lens when light is incident on convex side q = 1, and SA and coma are minimum. Coma is more prominent when object point is off axis and rays are obliquely incident.

### 3. Astigmatism.

When the object point is considerably off-axis the incident rays of light strike the lens asymmetrically. There are two important planes defined for an optical system, which are mutually perpendicular. These two planes are called as meridional — or tangential, and sagittal plane. Due to asymmetry of incidence of rays in these two planes the focal lengths corresponding to these planes differ. This gives rise to astigmatism.

Astigmatism comes from Greek: for a = not and stigma = point or spot. Similarly sagitta is Latin for arrow.

Remember that meridional plane contains principal ray and optical axis and sagitta plane is perpendicular to meridional plane.

Photo-Credit : Astigmatism is due to asymmetry in amount of light incident in two mutually perpendicular planes, called sagittal and meridional planes, as shown here.

### 4. Curvature of field.

We saw that astigmatism occurs when sagittal and meridional planes have an asymmetric amount of incidence. This happens when object-point is highly off-axis and the resultant focal lengths of the two planes differ. On sagittal plane there is a line of focus perpendicular to the meridional plane and is called as sagittal focal-line. Similarly on meridional plane there is a focal line perpendicular to the sagittal plane and its called as meridional focal-line.

Photo-Credit : The image field is curved.

So, the focal points lie on two different surfaces. When they coincide astigmatism is zero. But even then, resulting image surface is curved. This is known as curvature of field.

### 5. Distortion.

Like SA, distortion can be an exclusive aberration. All other primary aberrations can be absent even when distortion is the only primary aberration that is present in a system. It is caused due to non-uniform magnification.

Photo-Credit : Distortion is caused due to different magnification.

Actual

Positive

Negative