Geometrical Optics And Its Applications

Optics is the cornerstone of photonics systems and applications. Geometrical optics, or ray optics, is to study the geometry of paths of lights and their imagery through optical systems. Light will be treated as a form of energy which travels in straight lines called rays. When light comes to be regarded as waves, it will be seen that shadows cast by objects are not as sharp as rectili (Ariel Lipson,Stephen G. Lipson,Henry Lipson) (Ariel Lipson,Stephen G. Lipson,Henry Lipson, 2010)near propagation suggests due to diffraction and interference effects of wave. Thus, there is a simple assumption for geometrical optics, which is rays of light propagate along straight lines until they get reflected, refracted, or absorbed at a surface.

2 Fermat’s Principle

2.1 Background

Fermat’s Principle, also known as the principle of the shortest optical path, was introduced by Pierre de Fermat in the early seventeenth century. This principle is used to state and explain the motion of light ray through different media, which helps to demonstrate laws of reflection and refraction later on.

2.2 Theory

“The path of a light ray connecting two points is the one for which the time of transit, but not the length, is a minimum.”

The time, T , for a light ray through space to travel from a point A to another point B can be calculated as:

It is known that is velocity, which can also be regarded as for light, where c is the speed of light and n is the refractive index of the medium. Thus, we have

Then the path taken by light should be the path that minimizes this integral, which would be:

Therefore, it is the fact that light will travel along paths of stationary optical path length, where the optical path length is a local maximum or minimum with respect to any small variation in the path. And many paths will take exactly the same time to travel from point A to point B. (Ariel Lipson,Stephen G. Lipson,Henry Lipson, 2010)

3. Reflection

3.1 Derivation for Law of Reflection

In general, law of reflection states that when a light ray incident upon a reflective surface, it will be reflected with an reflective angle that exactly equal to the incident angle with respect to the normal of the surface.

Law of reflection can be dervied from Fermat’s Principle. Assume that the medium of the light travel is homogeneous, we haveC:UsersGaryDesktop未命名.png

Total path length S from A to B is

Based on Fermat’s Principle we know that light would travel the path with minimum time. As in homogeneous medium, light travels with a constant speed and therefore the minimum time path is equilvant to the minimum distance path, which can be obtained by taking the first derivative of S with respect to x.

which is sinã„¥AOC = sin ã„¥BOC

Thus shows that ã„¥AOC = ã„¥BOC, the angle of incidence equals the angle of reflection

NB: It has to be reminded that the incident ray, the refracted ray and the normal are co-planar. (Philip D. Straffin, C. T. Benson, 1993)

3.2 Specular and Diffuse Reflection

In details, there are two types of reflection – Specular reflection and diffuse reflection.

http://titan.bloomfield.edu/facstaff/dnicolai/images/ImagesPhy106/lesson2.gif

Specular or regular reflection is said to occur when parallel rays reflect from a high smooth and flat surface. For instance, a flashlight beam is said to have specular reflection as the reflective surface is mirror which is highly smooth and it, hence, makes the reflected beams travel towards the same direction in parallel as in (a).

Most object, however, reflect light diffusely and the rays in an incident parallel beam are reflected in many direction as in (b) because of diffuse reflection. Diffuse reflection is due to the surface of the object not being perfectly smooth like a mirror. In fact, under microscopic scale, the surface of most of objects, if not all, is quite rough. Although at each point on the surface the law of reflection is observed, the angle of incidence and the angle of reflection varies from point to point. Each of the initially parallel rays, therefore, is reflected in a different direction.

3.3 Mirror Concepts and its Applications

3.3.1 Image Formation with Mirrors

Mirrors, undoubtedly, are ubiquitous in daily, especially used in mirror and optical instruments for gathering light and forming images. Law of reflection can be applied to use in locating the reflected image graphically as long as the size, location and orientation of the object is known.

3.3.1.1 Images in Plane Mirrors

Images with mirrors are formed when many non-parallel light beams are reflected from the mirror surface and converge to form a corresponding image point.

Image formed by plane mirror is erect, virtual, same size as the object and laterally inverted.C:UsersGaryDesktopreflection in mirror.png

For a point object, in Fig 3.3.1.1a, rays from the object at O are reflected in all directions based on the laws of reflection so that they appear to come from a point I behind the mirror and this is where the observer imagines the image to be.

For an extended object, in Fig 3.3.1.1b, the image of a point A on the object is at A’, and two points are being equidistant from the mirror. Similarly, the image of point B is at B’. The image size, therefore, is identical to the object seize, giving a magnification of unity. However, the right-hand side of the object becomes the left-hand side of the image and vice versa. The image is said to be laterally inverted.C:UsersGaryDesktopmirror extended object.png

To conclude, a planar mirror is strictly stigmatic in nature – Any incident rays issued from point A gives reflected rays passing through point B symmetrically to A with regard to the plane of the mirror. B, hence, is the image of A. (Katz, 2002)

3.3.1.2 Images in Curved Mirrors

There are mainly two types of spherical mirrors, concave and convex. Similar to planar mirror, image can be traced using the law of reflection.

Unlike image formed by planar mirror, which is always erect, virtual, same size as the object and laterally inverted, the image formed by a spherical mirror and its nature depend on the distance of the object from the mirror.

To construct the image, we assume that small objects on the principal axes of mirrors of small aperture are being considered so that all rays are paraxial. Point images will, thus, be formed of points on the object.

For concave mirrors, also known as converging mirrors because of their action on a parallel beam of light, is a mirror with a curved reflecting surface that bulges inward.

The formation of image using concave mirror can be in different locations of the object can be concluded by the following figure.

Concave Mirror

The nature of image using concave mirror and its applications are summarized as below:

Position of the object

Position of the image

Nature and size of the image

Use

At infinity

At the focus

Real, inverted and diminished

As collector of radiation in solar heating devices

Beyond the centre of curvature

Between the focus and the centre of curvature

Real, inverted and diminished

At the centre of curvature

At centre of curvature

Real, inverted and same size as object

As a reflecting mirror behind a projector lamp

Between the focus and centre of curvature

Beyond the centre of curvature

Real, inverted and magnified

In flood lights

At focus

At infinity

Real, inverted and magnified

As a reflecting mirror in car, head lights, search lights etc.

Between the pole of the mirror and the focus

Appears behind the mirror

Virtual, erect and magnified

As a shaving mirror or makeup mirror and dentist’s mirror

(Katz, 2002)

For convex mirrors, also known as diverging mirrors reflect the incoming parallel light beams to form divergent beams which appear to come from a point behind the mirror.Convex Mirror

Unlike that in concave mirrors, the nature of image formed by convex mirror is always virtual, diminished and erect. (Katz, 2002)

3.3.2 Derivation of the mirror formula

Mirror formula is useful when we have to calculate the image location.

Using triangles 2 and 3, we have,http://scienceworld.wolfram.com/physics/mimg232.gif

While using trangles 2 and 4, we have,

Where is the image distance, is the object distance, f is the focal length, is the image height and is the object height

It should be noticed that the sign convention is to be used in mirror formula. A real object or image distance is positive while a virtual object or image distance is negative.

Magnification of a mirror image, m, can also be calculated by

3.3.3 Mirror instruments

3.3.3.1 Catoptric Systems

Catoptric Systems is the system that merely consists of mirrors for the formation of image. (Board, 2012)

3.3.3.1.1 Newtonian telescopehttp://bdaugherty.tripod.com/gcseAstronomy/images/newtonian.jpg

Newtonian telescope is the first reflecting telescope invented by Isaac Newton in 1668. It consists of a concave primary mirror and a small flat diagonal secondary mirror.

In a bid to have a stigmatic axial image, a concave mirror has to be applied to act as a primary mirror and it reflects light back up the scope axis to the secondary mirror which is titled at 45o to the axis. The secondary mirror, which is a small plane mirror, is placed in the path of beams reflected by the primary mirror in order to divert the rays to one side of the tube. The reason for choosing a small mirror to act as secondary mirror is to prevent the influx of light beams is blocked from reaching the primary mirror.

Pros of the Newtonian telescope

Cons of the Newtonian telescope

Less expensive

Reduction of light intensity due to the blockage of central flat plane

Shorter focal ratio leads a wider field of view

Easily suffer from coma (i.e. an off-axis aberration)

Free of chromatic aberration

Simpler fabrication

(Kitchin, 2012)

3.3.3.1.2 Cassegrain Telescope

http://www.vikdhillon.staff.shef.ac.uk/teaching/phy217/telescopes/cassegrain.gif

Cassegrain Telescope is another type of reflecting telescope employing two principal mirrors, a concave parabolic primary mirror like the Newtonians’, but its secondary one is a convex hyperboloidal mirror.

It makes use of the special properties of parabolic and hyperbolic reflectors. A concave parabolic reflector will reflect all incoming light rays parallel to its axis of symmetry to a single point, the focus. A convex hyperbolic reflector has two foci and will reflect all light rays directed at one of its two foci towards its other focus. The mirrors in this type of telescope are designed and positioned so that they share one focus and so that the second focus of the hyperbolic mirror will be at the same point at which the image is to be observed, usually just outside the eyepiece. The parabolic mirror reflects parallel light rays entering the telescope to its focus, which is also the focus of the hyperbolic mirror. The hyperbolic mirror then reflects those light rays to its other focus, where the image is observed. (Waland, 1990)

3.3.3.1.3 Gregorian telescope

http://s4.hubimg.com/u/7341531_f520.jpg

Gregorian telescope employs concave parabolic mirror to act as both primary and secondary mirror in this reflecting telescope.

The light that first enters the tube is reflected by the primary concave mirrors and directed towards the secondary mirror, which is also a concave mirror. It will reflect the rays out to the telescope through the hole in its center. Observer can therefore view the image formed on the eyepiece. (Trümper, 2010)

3.3.3.2 Catadioptric Systems

Catadioptric system, on the other hand, is the system that consists of both mirrors and lenses for the formation of image. (Board, 2012)

Catadioptric Systems will not be included in this paper.

3.4 Other Applications

3.4.1 Optical lever and light-beam galvanometers

Galvanometer is an instrument that used to measure very small electric currents. Some sensitive galvanometers would use a beam of light in conjunction with a small mirror as a pointer. When there exists current flowing in the electric wire, a tiny mirror that is fixed to the part of the meter will rotate. And a light beam from a fixed lamp falls on the mirror and is reflected onto a translucent scale. For a given current, the longer the pointer (i.e. the reflected beam) the greater the deflection observed on the scale. This simply applies the reflective nature of light wave. (A.M.Badadhe, 2006)http://www.daviddarling.info/images/moving-coil_galvanometer.jpg

3.4.2 Red-eye effect

The Red-eye effect is a common phenomenon in photography. It occurs when the photographic flash unit place closely to the camera lens. Normally, the rationale of photo-taking is that light from the flash unit enters the eye and is reflected back to the camera lens. Nevertheless, under dark environment, pupil diameter would increase due to contraction of radial muscle fibers and relaxation of circular muscle fibers of iris under autonomic nervous system and therefore more light beams can enter into the retina of the eye. And the reflected light from the retina is red because of largely blood vessels in Choroid. It, therefore, gives the Red-eye effect in photograph.

3.4.3 Headlight of Carhttp://1.bp.blogspot.com/-dtZnF3MgZVc/T4_Rn6nSbaI/AAAAAAAAAtw/tzoIpGW9oFU/s1600/infiniti+headlight.jpg

It employs concave parabolic mirrors at the front of the car, which acts as reflectors in the head lights of cars, search lights etc.

3.4.4 Dentist’s Mirrorhttp://cdn7.fotosearch.com/bthumb/FSA/FSA132/x12292469.jpg

Dentist’s Mirror applies reflection to reflect and focus light on the tooth so that dentists can be examined in detail.

3.4.5 Optical Comparatorhttp://www.worldoftest.com/img/products/qv300_1.jpg

Optical Comparator is an instrument that projects a magnified image or profile of a part onto a screen for comparison to a standard overlay profile or scale based on the principles of optics.

The comparator, basically, is a non-contact device, which frequently used to measure, test, inspect, gauge or examine parts for compliance with specifications. (A.M.Badadhe, 2006)

3.4.6 Security Convex Mirrorhttp://dgmglass.com/images/mirrors/security/mirror-s-3.jpg

One of the distinguish feature of convex mirror is the widely view range. Therefore, convex mirror is applied to act as a security mirror in blind junctions of roads or at corners of walls of bug buildings. The person that is approaching from other side would be shown when the mirror is positioned properly.

3.4.7 Rear-view mirror in an automobilehttp://transport-futures.com/wp-content/uploads/2011/04/rear-view-mirror.jpg

Knowing that the size of image of an object would be smaller when the object comes closer to the convex mirror, this unique feature can be applied to use on automobiles as rear view mirrors so that the diver can clearly view an approaching vehicle.

3.4.8 Submarine’s PeriscopeC:UsersGaryDesktopreflecting Periscope.png

A periscope is a mirror instrument for observation from a concealed position. It works by employing two mirrors at 45o to each other, reflecting light rays from one place to another and finally out to the person’s eye.

4 Refraction

4.1 Law of Refraction

Qualitatively, when a light wave, which in fact composed of oscillating Electric fields and Magnetic field, crosses from a low optically density medium, say vacuum into a high optically density medium, say glass, E-Field and B-Fields are altered in terms of magnitude and direction of travel by the charges in the glass.

Law of refraction, however, can be proved mathematically as follows:

The first medium is supposed to be faster than the second medium and the speeds of propagation in 1st medium and 2nd medium are and respectively, where c is the speed of light in vacuum and n1, n2 ≥ 1. C:UsersLenovoDropboxU life2012-13, year2, 1st semesterCCST 9042 – The world of wavesSnellslaw_diagram2.png

Then, we have to evaluate the time taken by light ray from P to Q, which is

Based on Fermat’s Principle, light travels the path with the least time. Thus, in an attempt to minimize the transit time, we set the derivative of time is zero and we have:

By trigonometry, we have

As and, we have

This is Law of refraction, also known as Snell-Descartes law.

NB: Similar to law of reflection, the incident ray, the refracted ray and the normal are co-planar. (Philip D. Straffin, C. T. Benson, 1993)

4.2 Total internal reflection

According to Snell’s law, when a light travels from one a medium with a higher optical density, to a medium with a lower optical density, say from glass to air, it will be refracted away from the normal (i.e. Ray C in Fig 4.2a) and a weak internally reflected ray is, meanwhile, formed (i.e. Ray B in Fig 4.2a) C:UsersGaryDesktop未命名33.png

Increasing the angle of incidence i increases the angle of refraction r, and at a certain angle of incidence c, called the critical angle, the reflected ray beam just emerges along the surface of the glass and the angle of refraction is 90o (i.e. Ray D in Fig 4.2b).

In Fig 4.2 c, as the incident angle is increased continuously above critical angle (i.e. i > c), the refracted angle will be higher than 90o and sin r > 1, which is impossible and no refraction is possible. At this stage the internally reflected ray becomes bright and the refracted ray disappears. Hence, total internal reflection is now said to be occurring since all the incident light is reflected inside the optically denser medium.

Mathematically, the critical angle can be found based on Snell’s law, which is

Assume that n1 is refractive index of optically denser medium, n2 is refractive index of optically less dense medium, θ1 is the critical angle of denser medium and θ2 is the angle of refraction in less denser medium.

All in all, for Total internal reflection to happen:

Light must travel from denser medium to rarer medium.

Angle of incidence should be greater than critical angle (i.e. i > c.)

4.3 Thin Lenses

Thin lens is a lens that its axial thickness is small compared to the radii of curvature of its surfaces. (I.e. The thickness of the lens is negligible compared with the focal length of the lens)

There are mainly two types of thin lenses, Converging thin lenses and Diverging thin lenses. Converging thin lenses, also known as convex lenses, direct parallel light rays bending toward one another after passing through them Diverging thin lenses, or so-called concave lenses, cause parallel light beams to spread as the leave the lens.http://t1.gstatic.com/images?q=tbn:ANd9GcT6lXrPj04wfnztVlEvIlewlbzvIzhu_60Cup8oDYd840MY3w0SiOxnzxaVrQ

4.3.1 Image Formed by Thin Lenses

4.3.1.1 Convex Lens Ray DiagramFormation of Image by a Convex Lens

Object is located between focus and lens

The image is:

Virtual

Erect

Magnifiedobject at F1

2. Object is located at focus

The image is:

Real

Inverted

Magnified

3. Object is located between focus and F2

The image is:object between F2 F1

Real

Inverted

Magnified

4. Object is located at F2object between O F2

The image is:

Real

Inverted

Same size as the object

5. Object is located beyond F2object beyond F2

The image is:

Real

Inverted

Diminished

6. Object is located at infinityobject infinity

The image is:

Real

Inverted

Highly diminished

4.3.1.2 Concave Lens Ray Diagram

1. Object is located between focus and mirrorconcave lens object F O

The image is:

Erect

Virtual

Diminished

2. Object is located between mirror and infinityconcave lens object infinity O

The image is:

Erect

Virtual

Diminished

3. Object is located at infinityconcave lens object at infinity

The image is:

Erect

Virtual

Diminished

4.3.2 Derivation of the lens formula for thin lenses

Gaussian lens formula, also known as lens formula, can be derived as follow:

http://www.astarmathsandphysics.com/university_physics_notes/optics/university_physics_notes_proof_of_thin_lens_equation_html_5c3e1e20.gif

As,

————————— (1)

Where O is the size of the object and I is the size of the image.

Also,

Thus,

Combine with (1), we have

NB: The Cartesian Sign Convention for thin lens formulas are as follow:

Light travels initially from left to right toward the lens.

Object distance p is positive for real objects located to the left of the lens and negative for virtual objects located to the right of the lens.

Image distance q is positive for real images formed to the right of the lens and negative for virtual images formed to the left of the lens.

The focal length f is positive for a converging lens, negative for a diverging lens.

The radius of curvature r is positive for a convex surface, negative for a concave surface.

Transverse distances are positive above the optical axis, negative below

(Katz, 2002)

4.3.3 Power of a Lens

The optical power of the lens is used to describe the bending ability of lens in term of power. Basically the power of a lens of focal length f is,

Power, in the case, can be expressed in m-1as the unit of power is diopter

1 D = 1m-1

Therefore, if there exists a convex of power 1D, its focal length equals to 1 meter.

4.4 Optical instruments

4.4.1 Prismshttp://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Light_dispersion_conceptual_waves.gif/330px-Light_dispersion_conceptual_waves.gif

Optical prisms are components that commonly used in optical experimental arrangements and optical instruments, with the role of illustrating dispersion of light beams.

Rather than showing the complicated mathematical proof such as derivation of minimum deviation, application of prism will be illustrated below. (Thorington, 2009)

4.4.1.1 Prism BinocularsC:UsersGaryDesktopprism.png

Prism Binoculars consist of a pair of refracting astronomical telescopes with two totally reflecting prisms (angles 90o, 45o, and 45o) between each objective and eyepiece. Prism A causes lateral inversion and prism B inverts verticallt so that the final image is the same way round and the same way up as the object. (Thorington, 2009)

4.4.1.2 PeriscopeC:UsersGaryDesktoppp.png

Periscope, apart from using two reflecting mirror, employs two prisms. Under the occurring of total internal reflection on the hypotenuse, the incident rays and reflected rays are symmetric with regard to a plane orthogonal to the hypotenuse. In the case of normal incidence on one side of the right angle, the incident and reflected beams are orthogonal.

4.4.2 Magnifying Glasshttp://blog.timesunion.com/opinion/files/2011/02/0217_WVinternet.jpg

Magnifying glass, also called “Simple microscope”, consist of a converging lens forming a virtual, upright, magnified image of an object placed inside its principal focus.

For an object of height h is viewed at the near point by the unaided eye, the visual angle is, where D is the least distance of distinct visionhttp://www.astarmathsandphysics.com/ib_physics_notes/optics/ib_physics_notes_the_angular_magnification_of_a_magnifying_glass_when_the_image_is_formed_at_the_near_point_html_m367c081d.gif

Now the object is placed at distance u from the lens, the visual angle subtended by its image is given by http://www.astarmathsandphysics.com/ib_physics_notes/optics/ib_physics_notes_the_angular_magnification_of_a_magnifying_glass_when_the_image_is_formed_at_the_near_point_html_f9ca6a7.gif

The angular magnification is, therefore, given by

4.4.3 Compound Microscope

Compound Microscope is a kind of optical instrument that uses visible light and a lens system to magnify images of small samples. C:UsersGaryDesktopcompound micropound.png

The lens L1 nearer to the object, called the objective, forms a real, magnified, inverted image I1 of an object O placed just outside its principal focus Fo. I1 is just inside the principal focus Fo of the second lens L2, called the eyepiece, which act as a magnifying glass and produces a magnified, virtual image I2 of I1.

In normal adjustment, the final image I2 lies at the near point. Then the visual angle subtended by the final image to the eye is given by

When the object of height h is placed at the near point and viewed unaided, the visual angle subtended by the object is given by

Hence, the angular magnification is given by

And since, the linear magnification is equal to the linear magnification of the objective x linear magnification of the eyepiece. (Giordano, 2011)

4.4.4 Refracting Astronomical Telescope

Refracting Astronomical Telescope consists of two converging lenses; one is an objective with long focal length and the other an eyepiece with short focal length. The objective L1 forms a real, diminished, inverted image I1, of a distant object at its principal focus Fo since the rays incident on L1 from a point on such an object can be assumed parallel. The eyepiece L2 acts as a magnifying glass and forms a magnified, virtual image of I1 and, when the telescope is in normal adjustment, this image is at infinity. I1 must, therefore, be at the principal focus F2 of L2, hence F0 and F2 coincide.

http://images.tutorvista.com/content/optics/astronomical-telescope-magnifying-power.jpeg

From the above figure, we find that

The visual angle subtended by the object is given by and the visual angle subtended by the object is given by, where h is the image height.

Therefore, the angular magnification is

For telescope in normal adjustment, the separation of the two lenses is the sum of the focal lengths. And this, the foci of the objective lens and the eye-piece must be at the same place. (Giordano, 2011)

4.4.5 Galilean Telescopehttp://www.transtutors.com/userfiles/image/ARUN/IMAGES/Gal.JPG

Galilean Telescope consists of two lenses – a converging objective of large focal length and a diverging eyepiece of small focal length

The angular magnification is

5. Conclusion

In this term paper, the concepts of geometric optics are included comprehensively such as law of reflection, refraction, formation of images in mirrors and lens, etc. The notion of a light ray and the scientific study of light are involved in a bid to illustrating the working principle of different mirror and optical instruments, such as various telescope, efficiently.