Fermat's Last Theorem states that the sum of two integers each raised to the same integer power n cannot equal another integer raised to the power n if n > 2. This implies that no solutions to the optic equation have all three integers equal to perfect powers with the same power n > 2. For if 1 x n + 1 y n = 1 z n , {\displaystyle {\tfrac {1}{x^{n}}}+{\tfrac {1}{y^{n}}}={\tfrac {1}{z^{n}}},} then multiplying through by ( x y z ) n {\displaystyle (xyz)^{n}} would give ( y z ) n + ( x z ) n = ( x y ) n , {\displaystyle (yz)^{n}+(xz)^{n}=(xy)^{n},} which is impossible by Fermat's Last Theorem.

Aberrations in optics

Aberrations dredge

For a lens of negligible thickness, and focal length f, the distances from the lens to an object, S1, and from the lens to its image, S2, are related by the thin lens formula:

Multiplying both sides by abc shows that the optic equation is equivalent to a Diophantine equation (a polynomial equation in multiple integer variables).

Real lenses behave somewhat differently from how they are modeled using the thin lens equations, producing aberrations. An aberration is a distortion in an image. There are a variety of aberrations due to a lens size, material, thickness, and position of the object. One common type of aberration is chromatic aberration, which is related to color. Since the index of refraction of lenses depends on color or wavelength, images are produced at different places and with different magnifications for different colors. (The law of reflection is independent of wavelength, and so mirrors do not have this problem. This is another advantage for mirrors in optical systems such as telescopes.) Figure 1(a) shows chromatic aberration for a single convex lens and its partial correction with a two-lens system. Violet rays are bent more than red, since they have a higher index of refraction and are thus focused closer to the lens. The diverging lens partially corrects this, although it is usually not possible to do so completely. Lenses of different materials and having different dispersions may be used. For example an achromatic doublet consisting of a converging lens made of crown glass and a diverging lens made of flint glass in contact can dramatically reduce chromatic aberration (see Figure 1(b)).

Aberrations in lenses

The optic equation of the crossed ladders problem can be applied to folding rectangular paper into three equal parts. One side (the left one illustrated here) is partially folded in half and pinched to leave a mark. The intersection of a line from this mark to an opposite corner, with a diagonal is exactly one third from the bottom edge. The top edge can then be folded down to meet the intersection.[7]

The image produced by an optical system needs to be bright enough to be discerned. It is often a challenge to obtain a sufficiently bright image. The brightness is determined by the amount of light passing through the optical system. The optical components determining the brightness are the diameter of the lens and the diameter of pupils, diaphragms or aperture stops placed in front of lenses. Optical systems often have entrance and exit pupils to specifically reduce aberrations but they inevitably reduce brightness as well. Consequently, optical systems need to strike a balance between the various components used. The iris in the eye dilates and constricts, acting as an entrance pupil. You can see objects more clearly by looking through a small hole made with your hand in the shape of a fist. Squinting, or using a small hole in a piece of paper, also will make the object sharper.

(a) During laser vision correction, a brief burst of 193 nm ultraviolet light is projected onto the cornea of the patient. It makes a spot 1.00 mm in diameter and deposits 0.500 mJ of energy. Calculate the depth of the layer ablated, assuming the corneal tissue has the same properties as water and is initially at 34.0 oC. The tissue’s temperature is increased to 100 oC and evaporated without further temperature increase.

Spherical aberration

Components of an electrical circuit or electronic circuit can be connected in what is called a series or parallel configuration. For example, the total resistance value Rt of two resistors with resistances R1 and R2 connected in parallel follows the optic equation:

The harmonic mean of a and b is 2 1 a + 1 b {\displaystyle {\tfrac {2}{{\frac {1}{a}}+{\frac {1}{b}}}}} or 2c. In other words, c is half the harmonic mean of a and b.

Aberrations examples

Aberrations 5e

Quite often in an imaging system the object is off-center. Consequently, different parts of a lens or mirror do not refract or reflect the image to the same point. This type of aberration is called a coma and is shown in Figure 2. The image in this case often appears pear-shaped. Another common aberration is spherical aberration where rays converging from the outer edges of a lens converge to a focus closer to the lens and rays closer to the axis focus further (see Figure 3). Aberrations due to astigmatism in the lenses of the eyes are discussed in Chapter 26.2 Vision Correction, and a chart used to detect astigmatism is shown in Chapter 26.2 Figure 4. Such aberrations and can also be an issue with manufactured lenses.

In the crossed ladders problem,[2] two ladders braced at the bottoms of vertical walls cross at the height h and lean against the opposite walls at heights of A and B. We have 1 h = 1 A + 1 B . {\displaystyle {\tfrac {1}{h}}={\tfrac {1}{A}}+{\tfrac {1}{B}}.} Moreover, the formula continues to hold if the walls are slanted and all three measurements are made parallel to the walls.

So how are aberrations corrected? The lenses may also have specially shaped surfaces, as opposed to the simple spherical shape that is relatively easy to produce. Expensive camera lenses are large in diameter, so that they can gather more light, and need several elements to correct for various aberrations. Further, advances in materials science have resulted in lenses with a range of refractive indices—technically referred to as graded index (GRIN) lenses. Spectacles often have the ability to provide a range of focusing ability using similar techniques. GRIN lenses are particularly important at the end of optical fibers in endoscopes. Advanced computing techniques allow for a range of corrections on images after the image has been collected and certain characteristics of the optical system are known. Some of these techniques are sophisticated versions of what are available on commercial packages like Adobe Photoshop.

Aberrations bg3

Douglas College Physics 1207 Copyright © August 22, 2016 by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

Aberrations Physics

The special case in which the integers whose reciprocals are taken must be square numbers appears in two ways in the context of right triangles. First, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse. This holds whether or not the numbers are integers; there is a formula (see here) that generates all integer cases.[5][6] Second, also in a right triangle the sum of the squared reciprocal of the side of one of the two inscribed squares and the squared reciprocal of the hypotenuse equals the squared reciprocal of the side of the other inscribed square.

In a trapezoid, draw a segment parallel to the two parallel sides, passing through the intersection of the diagonals and having endpoints on the non-parallel sides. Then if we denote the lengths of the parallel sides as a and b and half the length of the segment through the diagonal intersection as c, the sum of the reciprocals of a and b equals the reciprocal of c.[4]

In a bicentric quadrilateral, the inradius r, the circumradius R, and the distance x between the incenter and the circumcenter are related by Fuss' theorem according to

In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c:[1]

Let P be a point on the circumcircle of an equilateral triangle △ABC, on the minor arc AB. Let a be the distance from P to A and b be the distance from P to B. On a line passing through P and the far vertex C, let c be the distance from P to the triangle side AB. Then[3]: p. 172  1 a + 1 b = 1 c . {\displaystyle {\tfrac {1}{a}}+{\tfrac {1}{b}}={\tfrac {1}{c}}.}

1: (a) 0.251 μm (b) Yes, this thickness implies that the shape of the cornea can be very finely controlled, producing normal distant vision in more than 90% of patients.