A more detailed and exhaustive discussion of total internal reflection and the critical angle can be found at The Physics Classroom Tutorial.

where n1 represents the index of refraction of the incident medium and n2 represents the index of refraction of the refractive medium. In order for total internal reflection to occur, the value of n2 must be greater than the value of n1. This restriction is consistent with the criteria that light be in the more dense medium and heading towards the less dense medium.

The majority of photographic lenses have the lens elements made from glass although the use of high-quality plastics is becoming more common in high-quality lenses and has been common in inexpensive cameras for some time. The design of photographic lenses is very demanding as designers push the limits of existing materials to make more versatile, better-quality, and lighter lenses. As a consequence many exotic glasses have been used in modern lens manufacturing. Caesium[1] and lanthanum[2] glass lenses are now in use because of their high refractive index and very low dispersion properties. It is also likely that a number of other transition element glasses are in use but manufacturers often prefer to keep their material specification secret to retain a commercial or performance edge over competitors.

The lenses of the very earliest cameras were simple meniscus or simple bi convex lenses. It was not until 1840 that Chevalier in France introduced the achromatic lens formed by cementing a crown glass bi-convex lens to a flint glass plano-concave lens. By 1841 Voigtländer using the design of Joseph Petzval manufactured the first commercially successful two element lens.

In Physics, there are numerous physical situations in which a student would want to relate the angle in a right triangle to the length of one or more of its sides. In such instances, a trigonometric function is chosen and used to analyze the physical situation. For instance, the refraction of light at a boundary causes the angle that the light path makes with the normal (and also with the boundary) to change. It is different on one side of the boundary than on the other side of the boundary. If this light path is projected through space on either side of the boundary it will likely strike a physical object. There is subsequently a triangle (or even a pair of triangles) formed by the boundary, the light ray and physical objects present in the vicinity.

Abbe was instrumental in the development of the famous Jena optical glass. When he was trying to eliminate astigmatism from microscopes, he realised that the range of optical glasses available was insufficient. After some calculations, he realised that performance of optical instruments would dramatically improve, if optical glasses of appropriate properties were available. His challenge to glass manufacturers was finally answered by Dr Otto Schott, who established the famous glassworks at Jena from which new types of optical glass began to appear from 1888, and employed by Zeiss and other makers.

For the lens designer, achieving these objectives will also involve ensuring that internal flare, optical aberrations and weight are all reduced to the minimum whilst zoom, focus and aperture functions all operate smoothly and predictably.

Carl Zeiss was an entrepreneur who needed a competent designer to take his firm beyond just another optical workshop. In 1866, the service of Dr Ernst Abbe was enlisted. From then on novel products appeared in rapid succession which brought the Zeiss company to the forefront of optical technology.

The most important Zeiss lens by Rudolph was the Tessar, first sold in 1902 in its Series IIb f/6.3 form. It can be said as a combination of the front half of the Unar with the rear half of the Protar. This proved to be a most valuable and flexible design, with tremendous development potential. Its maximum aperture was increased to f/4.7 in 1917, and reached f/2.7 in 1930. It is probable that every lens manufacturer has produced lenses of the Tessar configuration.

Incorporating a commercial made Compur type shutter required lens designers to accommodate the width of the shutter mechanism in the lens mount and provide for the means of triggering the shutter on the lens barrel or transferring this to the camera body by a series of levers as in the Minolta twin-lens cameras.

There are 20 ready-to-use problem sets on the topic of Refraction and Lenses. The problems target your ability to mathematically relate the index of refraction and the light speed, to use Snell's Law and (at times) trigonometric principles to analyze physical situations regarding light refraction, to determine critical angles and to use the lens equation and magnification ratio to solve lens problems. Problems range in difficulty from the very easy and straight-forward to the very difficult and complex.

Refractive materials such as glass have physical limitations which limit the performance of lenses. In particular the range of refractive indices available in commercial glasses span a very narrow range. Since it is the refractive index that determines how much the rays of light are bent at each interface and since it is the differences in refractive indices in paired plus and minus lenses that constrains the ability to minimise chromatic aberrations, having only a narrow spectrum of indices is a major design constraint.

Lens makers formula

The table below summarizes the process of transforming a verbal statement into a mathematical equation which ultimately is used to substitute into the lens equation.

When reading a problem, give attention to cues within the problem in order to determine the sign on the given quantity. For instance, a problem may describe an object located to the left of a diverging lens and a virtual image located to the left of the diverging lens. Applying the above conventions, one can assign a positive value to the object distance and a negative value to the image distance. Failure to recognize such small nuances of problems will result in wrong answers despite the accuracy of the mathematical manipulations. A careful reading of the problem statement along with an understanding of the sign conventions (as stated in the table above) for the variables within the two equations allows one to make proper conceptual decisions. These types of decisions are critical to your success on these problems. Making the correct decisions has nothing to do with your mathematical skills. Rather, they are tests of your conceptual understandings and your willingness to read a problem carefully and to give attention to details which may be important.

These equations are used to assist in the solution of physics word problems. Values of two or more quantities are typically expressed in the problem; the goal of the word problem is to solve for an unknown quantity. Known quantities of the variables must be substituted into the above equations and proper algebraic manipulations must be performed in order to solve for the unknown variable.

However, because projected images are almost always viewed at some distance, lack of very fine focus and slight unevenness of illumination is often acceptable. Projector lenses have to be very tolerant of prolonged high temperatures from the projector lamp and frequently have a focal length much longer than the taking lens. This allows the lens to be positioned at a greater distance from the illuminated film and allows an acceptable sized image with the projector some distance from the screen. It also permits the lens to be mounted in a relatively coarsely threaded focusing mount so that the projectionist can quickly correct any focusing errors.

Knowing the index of refraction values of the two materials on opposite sides of the boundary and the angle of incidence allows one to make a calculation of the angle of refraction. In a similar manner, knowing the two angles that a light ray makes with the normal line for both sides of the boundary and knowing one of the index of refraction values of the two materials allows one to determine the second index of refraction value.

The critical angle is the angle of incidence that causes light to refract along the boundary at an angle of refraction of 90 degrees. Since 90 degrees is the largest possible angle of refraction, an incident ray with an angle of incidence greater than the critical angle cannot refract. Such a light ray will only reflect and stay within the original medium. Using Snell's law, one can show that the critical angle can be calculated from the index of refraction values of the two media on both sides of the boundary. The formula is

This is best illustrated in the diagram at the right. Laser light is sent from a submarine at an angle to the water surface. The light refracts at the boundary and strikes the top of a tall building on the water's edge. Refraction principles can be used to relate the angle of incidence (Θi) to the angle of refraction (Θr). These angles are in turn related to the distance measurements shown on the diagram.

The need to accommodate the shutter mechanism within the lens barrel limited the design of wide-angle lenses and it was not until the widespread use of focal-plane shutters that extreme wide-angle lenses were developed.

Lenses used in photographic enlargers are required to focus light passing through a relatively small film area on a larger area of photographic paper or film. Requirements for such lenses include

But the important thing about this Protarlinse is that two of these lens units can be mounted in the same lens barrel to form a compound lens of even greater performance and larger aperture, between f/6.3 and f/7.7. In this configuration it was called the Double Protar Series VIIa. An immense range of focal lengths can thus be obtained by the various combination of Protarlinse units.

There are several instances in these problems in which the lens equation must be used to solve for an unknown variable but only one of the other two variable values are known. Such problems usually have a statement of the effect: "the image is real and three times the size of the object." Such a statement reveals information about the magnification of the image. Since the ratio of the image to object height is equal to the (negative of the) ratio of the image distance to object distance, we can say that size and height can be treated synonymously. Stating that the image is three times the size of the object is stating that the ratio hi/ho is either +3 or -3. Determining whether hi/ho is +3 or -3 demands an understanding of the sign conventions (as discussed in the above table). The ho value is always positive (for our purposes). The hi value is positive for upright images and negative for inverted images. Since this statement asserts that the image is real (and thus inverted), a -3 value must be assigned to the hi/ho ratio. Since hi/ho is equal to -di/do, the -3 value can be equated with -di/do. This stream of logic allows one to write an expression for di in terms of do. This expression for di in terms of do can be substituted into the lens equation in order to transform it into a single equation with a single unknown. Customary algebraic manipulations can then be performed in order to solve for di or for do.

Phet geometricoptics

The speed of a light ray is dependent upon the medium through which it travels. Light travels relatively slowly in a more optically dense material and faster in less optically dense materials. The index of refraction value (n) provides a direct measure of the optical density of a material. It can be used to relate the speed of light in that material (v) to the speed of light in a vacuum (c = 3.0 x 108 m/s). The speed of light equation is stated above; it can be used to relate light speed to index of refraction values. A more detailed and exhaustive discussion of optical density, light speed and the index of refraction can be found at The Physics Classroom Tutorial.

The design of a fixed focal length lens (also known as prime lenses) presents fewer challenges than the design of a zoom lens. A high-quality prime lens whose focal length is about equal to the diameter of the film frame or sensor may be constructed from as few as four separate lens elements, often as pairs on either side of the aperture diaphragm. Good examples include the Zeiss Tessar or the Leitz Elmar.

Many modern cameras now use automatic focusing mechanisms which use ultrasonic motors to move internal elements in the lens to achieve optimum focus.

Rudolph also investigated the Double-Gauss concept of a symmetrical design with thin positive menisci enclosing negative elements. The result was the Planar Series Ia of 1896, with maximum apertures up to f/3.5, one of the fastest lenses of its time. Whilst it was very sharp, it suffered from coma which limited its popularity. However, further developments of this configuration made it the design of choice for high-speed lenses of standard coverage.

Probably inspired by the Stigmatic lenses designed by Hugh Aldis for Dallmeyer of London, Rudolph designed a new asymmetrical lens with four thin elements, the Unar Series Ib, with apertures up to f/4.5. Due to its high speed it was used extensively on hand cameras.

A light ray will undergo refraction (a change in direction of its path) at a boundary between two materials whenever it approaches the boundary at an angle of incidence other than zero degrees. This refraction occurs in a rather predictable manner as expressed by the Snell's law equation:

After the partitioning of Germany, a new Carl Zeiss optical company was established in Oberkochen, while the original Zeiss firm in Jena continued to operate. At first both firms produced very similar lines of products, and extensively cooperated in product-sharing, but they drifted apart as time progressed. Jena's new direction was to concentrate on developing lenses for the 35 mm single-lens reflex camera, and many achievements were made, especially in ultra-wide angle designs. In addition to that, Oberkochen also worked on designing lenses for large format cameras, interchangeable front element lenses such as for the 35 mm single-lens reflex Contaflex, and other types of cameras.

Diverging lens formula

Very-large-aperture lenses designed to be useful in very low light conditions with apertures ranging from f/1.2 to f/0.9 are generally restricted to lenses of standard focal length because of the size and weight problems that would be encountered in telephoto lenses and the difficulty of building a very wide aperture wide angle lens with the refractive materials currently available. Very-large-aperture lenses are commonly made for other types of optical instruments such as microscopes but in such cases the diameter of the lens is very small and weight is not an issue.

Many very early cameras had diaphragms external to the lens often consisting of a rotating circular plate with a number of holes of increasing size drilled through the plate.[3] Rotating the plate would bring an appropriate sized hole in front of the lens. All modern lenses use a multi-leaf diaphragm so that at the central intersection of the leaves a more or less circular aperture is formed. Either a manual ring, or an electronic motor controls the angle of the diaphragm leaves and thus the size of the opening.

Since the beginning of Zeiss as a photographic lens manufacturer, it has had a licensing programme which allows other manufacturers to produce its lenses. Over the years its licensees included Voigtländer, Bausch & Lomb, Ross, Koristka, Krauss, Kodak. etc. In the 1970s, the western operation of Zeiss-Ikon got together with Yashica to produce the new Contax cameras, and many of the Zeiss lenses for this camera, among others, were produced by Yashica's optical arm, Tomioka. Yashica's owner Kyocera ended camera production in 2006. Yashica lenses were then made by Cosina, who also manufactured most of the new Zeiss designs for the new Zeiss Ikon coupled rangefinder camera. Another licensee active today is Sony who uses the Zeiss name on lenses on its video and digital still cameras.

The angle of incidence value is the angle measure between the incident ray and the normal line. The normal line is an imaginary line drawn perpendicular to the boundary at the location where the incident ray strikes the boundary. The angle of refraction value is defined in a similar manner; it is the angle measure between the refracted ray and the normal line. Making these measurements simply involves the placement of a protractor along the boundary with its origin placed at the location where the light ray strikes the boundary. A more detailed and exhaustive discussion of the angles of incidence and refraction can be found at The Physics Classroom Tutorial.

As a discipline concerned about the relationships present in the physical environment, physics is clearly interested in questions concerning dimensions, directions, angle measures and the like. These three trigonometric functions, when combined with physics principles such as Snell's Law, allows one to make quantitative predictions and conclusions concerning the dimensions and angles associated with the origin and the destination of the path of a light ray.

Perhaps one of the most problematic areas of lens problems is dealing with the sign conventions associated with the image distance, focal length and image height. The table below summarizes the sign conventions associated with these quantities.

Rudolph left Zeiss after the First World War, but many other competent designers such as Merté, Wandersleb, etc. kept the firm at the leading edge of photographic lens innovations. One of the most significant designer was the ex-Ernemann man Dr Ludwig Bertele, famed for his Ernostar high-speed lens.

A positive image height (hi) corresponds to an upright image. A negative image height (hi) corresponds to an inverted image. All upright images (positive hi values) are virtual images located on the object's side of the lens; upright images will thus be virtual images with negative di values. Likewise, inverted images with their negative hi values are real images that have positive di values.

Most modern lenses for 35mm format rarely provide a stop smaller than f/22 because of the diffraction effects caused by light passing through a very small aperture. As diffraction is based on aperture width in absolute terms rather than the f-stop ratio, lenses for very small formats common in compact cameras rarely go above f/11 (1/1.8") or f/8 (1/2.5"), while lenses for medium- and large-format provide f/64 or f/128.

However, because photographic films and electronic sensors have a finite and measurable resolution, photographic lenses are not always designed for maximum possible resolution since the recording medium would not be able to record the level of detail that the lens could resolve. For this, and many other reasons, camera lenses are unsuited for use as projector or enlarger lenses.

Later developments adopted designs in which internal elements were moved to achieve focus without affecting the outer barrel of the lens or the orientation of the front element.

The design of the lens is required to work effectively with light passing from near focus to far focus - exactly the reverse of a camera lens. This demands that internal light baffling within the lens is designed differently and that the individual lens elements are designed to maximize performance for this change of direction of incident light.

At the time, single combination lenses, which occupy one side of the diaphragm only, were still popular. Rudolph designed one with three cemented elements in 1893, with the option of fitting two of them together in a lens barrel as a compound lens, but it was found to be the same as the Dagor by C.P. Goerz, designed by Emil von Höegh. Rudolph then came up with a single combination with four cemented elements, which can be considered as having all the elements of the Protar stuck together in one piece. Marketed in 1894, it was called the Protarlinse Series VII, the most highly corrected single combination lens with maximum apertures between f/11 and f/12.5, depending on its focal length.

Thin lens magnification

Many of the problems within this problem set pertain to converging and diverging lenses. For these problems, you will make frequent use of two equations - the lens equation and the magnification equations. These equations are shown below:

Zeiss' innovative photographic lens design was due to Dr Paul Rudolph. In 1890, Rudolph designed an asymmetrical lens with a cemented group at each side of the diaphragm, and appropriately named "Anastigmat". This lens was made in three series: Series III, IV and V, with maximum apertures of f/7.2, f/12.5, and f/18 respectively. In 1891, Series I, II and IIIa appeared with respective maximum apertures of f/4.5, f/6.3, and f/9 and in 1893 came Series IIa of f/8 maximum aperture. These lenses are now better known by the trademark "Protar" which was first used in 1900.

In the above equations, the variable do represents the object distance or the distance between the lens and the object. The variable di represents the image distance or the distance between the lens and the image. The variable ho represents the object height and the variable hi represents the image height. The variable f stands for the focal length of the lens. The variable M stands for the magnification of the image; it represents how many times bigger the image is than the object.

Upon reaching a boundary, a light ray will undergo partial reflection and partial transmission. The transmitted light will change its direction whenever it approaches the boundary at any angle of incidence other than 0 degrees. The reflected ray simply reflects according to the law of reflection; that is, the angle of reflection is equal to the angle of incidence. There are instances however when the incidence light does not follow the usual rule of partial reflection and partial transmission. In such instances, all the light which approaches the boundary will undergo reflection and stay within the original medium. This phenomenon is referred to as total internal reflection and occurs whenever the following two criteria are met:

An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. An effective problem-solver...

Thin lens

The last important Zeiss innovation before the Second World War was the technique of applying anti-reflective coating to lens surfaces invented by Olexander Smakula in 1935.[8] A lens so treated was marked with a red "T", short for "Transparent". The technique of applying multiple layers of coating was also described in the original patent writings in 1935.[9]

The design of photographic lenses for use in still or cine cameras is intended to produce a lens that yields the most acceptable rendition of the subject being photographed within a range of constraints that include cost, weight and materials. For many other optical devices such as telescopes, microscopes and theodolites where the visual image is observed but often not recorded the design can often be significantly simpler than is the case in a camera where every image is captured on film or image sensor and can be subject to detailed scrutiny at a later stage. Photographic lenses also include those used in enlargers and projectors.

The new Jena optical glass also opened up the possibility of increased performance of photographic lenses. The first use of Jena glass in a photographic lens was by Voigtländer, but as the lens was an old design its performance was not greatly improved. Subsequently, the new glasses would demonstrate their value in correcting astigmatism, and in the production of achromatic and apochromatic lenses. Abbé started the design of a photographic lens of symmetrical design with five elements, but went no further.

Except for the most simple and inexpensive lenses, each complete lens is made up from a number of separate lens elements arranged along a common axis. The use of many lens elements serves to minimise aberrations and to provide a sharp image free from visible distortions. To do this requires lens elements of different compositions and different shapes. To minimise chromatic aberrations, e.g., in which different wavelengths of light are refracted to different degrees, requires, at a minimum, a doublet of lens elements with a positive element having a high Abbe number matched with a negative element of lower Abbe number. With this design one can achieve a good degree of convergence of different wavelengths in the visible spectrum. Most lens designs do not attempt to bring infrared wavelengths to the same common focus and it is therefore necessary to manually alter the focus when photographing in infrared light. Other kinds of aberrations such as coma or astigmatism can also be minimized by careful choice of curvature of the lens faces for all the component elements. Complex photographic lenses can consist of more than 15 lens elements.

Until recent years, focusing of a camera lens to achieve a sharp image on the film plane was achieved by means of a very shallow helical thread in the lens mount through which the lens could be rotated, moving it closer or further from the film plane. This arrangement, while simple to design and construct, has some limitations, not least the rotation of the greater part of the lens assembly including the front element. This could be problematic if devices such as polarising filters are used that require accurate rotational orientation irrespective of focus distance.

Magnification values are positive whenever image heights (hi) are positive. Thus, positive M values correspond to upright, virtual images located on the object's side of the lens. Conversely, negative M values correspond to inverted, real images located on the side of the lens opposite of the object.

There is a unique relationship between the angles within a right triangle and the ratio of the length of the sides. Trigonometric functions are mathematical functions that relate the length of the sides of a right triangle to the angles within the triangle.

Lensequation

Most lens elements are made with curved surfaces with a spherical profile. That is, the curved shape would fit on the surface of a sphere. This is partly to do with the history of lens making but also because grinding and manufacturing of spherical surface lenses is relatively simple and cheap. However, spherical surfaces also give rise to lens aberrations and can lead to complicated lens designs of great size. Higher-quality lenses with fewer elements and lower size can be achieved by using aspheric lenses in which the curved surfaces are not spherical, giving more degrees of freedom to correct aberrations.

From the perspective of the photographer, the ability of a lens to capture sufficient light so that the camera can operate over a wide range of lighting conditions is important. Designing a lens that reproduces colour accurately is also important as is the production of an evenly lit and sharp image over the whole of the film or sensor plane.

Projector lenses share many of the design constraints as enlarger lenses but with some critical differences. Projector lenses are always used at full aperture and must produce an acceptably illuminated and acceptably sharp image at full aperture.

With the advent of the Contax by Zeiss-Ikon, the first serious challenge to the Leica in the field of professional 35 mm cameras, both Zeiss-Ikon and Carl Zeiss decided to beat the Leica in every possible way. Bertele's Sonnar series of lenses designed for the Contax were the match in every respect for the Leica for at least two decades. Other lenses for the Contax included the Biotar, Biogon, Orthometar, and various Tessars and Triotars.

A shutter controls the length of time light is allowed to pass through the lens onto the film plane. For any given light intensity, the more sensitive the film or detector or the wider the aperture the shorter the exposure time need to be to maintain the optimal exposure. In the earliest cameras, exposures were controlled by moving a rotating plate from in front of the lens and then replacing it. Such a mechanism only works effectively for exposures of several seconds or more and carries a considerable risk of inducing camera shake. By the end of the 19th century spring tensioned shutter mechanisms were in use operated by a lever or by a cable release. Some simple shutters continued to be placed in front of the lens but most were incorporated within the lens mount itself. Such lenses with integral shutter mechanisms developed in the current Compur shutter as used in many non-reflex cameras such as Linhof. These shutters have a number of metal leaves that spring open and then close after a pre-determined interval. The material and design constraints limit the shortest speed to about 0.002 second. Although such shutters cannot yield as short an exposure time as focal-plane shutter they are able to offer flash synchronisation at all speeds.

The placement of the diaphragm within the lens structure is constrained by the need to achieve even illumination over the whole film plane at all apertures and the requirement to not interfere with the movement of any movable lens element. Typically the diaphragm is situated at about the level of the optical centre of the lens.

To be useful in photography any lens must be able to fit the camera for which it is intended and this will physically limit the size where the bayonet mounting or screw mounting is to be located.

A positive image distance (di) corresponds to an image location on the opposite side of the lens as the object. A negative image distance (di) corresponds to an image located on the same side as the lens as the object. Thus, all real images have positive image distances and all virtual images have negative image distances.

The sine function relates the angle measure of one of the acute angles to the ratio of the lengths of the side opposite the angle and the length of the hypotenuse. The cosine function relates the angle measure of one of the acute angles to the ratio of the lengths of the side adjacent the angle and the length of the hypotenuse. And finally the tangent function relates the angle measure of one of the acute angles to the ratio of the lengths of the side opposite the angle and the length of the side adjacent the angle. The meaning of the functions can be easily remembered by the catchy mnemonic

The aperture control, usually a multi-leaf diaphragm, is critical to the performance of a lens. The role of the aperture is to control the amount of light passing through the lens to the film or sensor plane. An aperture placed outside of the lens, as in the case of some Victorian cameras, risks vignetting of the image in which the corners of the image are darker than the centre. A diaphragm too close to the image plane risks the diaphragm itself being recorded as a circular shape or at the very least causing diffraction patterns at small apertures. In most lens designs the aperture is positioned about midway between the front surface of the objective and the image plane. In some zoom lenses it is placed some distance away from the ideal location in order to accommodate the movement of floating lens elements needed to perform the zoom function.

Any triangle has three angles. A right triangle has two acute angles and one right angle. Trigonometric functions are typically used to express the relationship between the measure of one of the acute angles and the ratio of the length of the sides. Each of the acute angles is formed by the intersection of one of the short sides and the hypotenuse side of the triangle. The short sides are often referred to as the adjacent side and the opposite side. The adjacent side is the side that when combined with the hypotenuse side forms a specific angle. The side of the triangle that is located along the triangle opposite of this specified angle is the opposite side.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.