OPTICS
Geometric optics, or ray optics, refers to a model of optics that in terms of rays describes light propagation. Furthermore, the ray in geometric optics is an abstraction useful for approximating the paths along which the propagation of light takes place under certain circumstances. The concept of geometric optics should not be confused with the concept of physical optics, which studies the phenomena for which geometric optics is not valid.
Introduction to Geometric Optics
When it comes to geometric optics, there is no account for some optical effects like interference and diffraction. Furthermore, this simplification will turn out to be useful in practice. Moreover, it is a very good approximation when the wavelength is small in comparison to the size of structures with which the interaction of the light takes place.
Two important geometric optics examples are a reflection of light and refraction of light. Furthermore, when the reflection of a ray of light takes place by some angle by a barrier in its pathway, the rebounding of the light beam takes place and this procedure is called the reflection of light. Also, refraction of light is the procedure in which the beam of light diverges slightly from its route when it crosses from one medium to the other
The Laws of Reflection in Geometric Optics
The conduct of the ray of light, whose rebounding take place by a smooth mirror, must be judged on the basis of the laws of reflection. However, one must need to understand the basic terms.
Incident ray refers to the light that is approaching a mirror. Furthermore, reflected ray refers to a beam of light whose reflection takes place by the mirror. The normal refers to the spot of reflection, the drawn perpendicular.
The angle of incidence is the angle amidst the incident ray and the normal. Furthermore, the angle of reflection refers to the angle that is amidst the reflected ray and the normal. When one confers to the laws of reflection, the angle of reflection is always equivalent to the angle of incidence.
Formulae For Geometric Optics
- Laws of reflection of light
- Lying on the same plane are the incident ray, the refracted ray, and normal
- Snells law
sin i/sin r = constant
- Relative refractive index 1n2= v1/v2
v1 = velocity of light in first medium
v2 = velocity of light in second medium
- Absolute Refractive indexn = c/v
c= velocity of light in air
v = velocity of light in given medium
- Lateral Shift
T sin(i-r)/cos r
5. Relative refractive index 1n2 = v1/v2
v1 = velocity of light in first medium
v2 = velocity of light in second medium
6. Absolute Refractive indexn = c/v
c= velocity of light in air
v = velocity of light in given medium
- Refraction through a prism
Refractive index of the prism n = [sin(A+δ/2)]/sin A/2
- Deviation whose production happens by a thin film d = (n-1) A
(n-1) is called the refractivity of the material
- Angular dispersion between two coloursAngular dispersion = (nv-nr)A
nv, nr is the refractive index of violet light and red light
- Dispersive powerω = nv– nr/n-1
- Lens makers formula for thin lenses1/f = (n-1) { 1/R1 – 1/R2}
- Power of lensP = 1/f
- Equivalent focal length of combination of two thin lenses1/f = 1/f1+ 1/f2
FAQs For Geometric Optics
Question 1: What is meant by geometric optics?
Answer 1: Geometric optics is simply a model of optics that in terms of rays tells us about light propagation. Furthermore, geometric optics helps in the approximation of the paths along which the propagation of light takes place under certain circumstances. Moreover, the concept of geometric optics should not be confused with the concept of physical optics, the latter involves the study of the phenomena for which geometric optics is not valid.
Question 2: Explain the refractive index of the medium in geometric optics?
Answer 2: The refractive index is denoted by the letter ‘n’. Furthermore, it is a dimensionless number that shows the radiation or light that travels via a medium. Mathematically, its expression is as follows:
n = c/v
Where,
‘v’ represents the phase velocity of light that is present in a medium, and
‘c’ indicates the speed of light that is in a vacuum.
Refractive Index of the Medium in Geometric Optics:
The refractive index is denoted by ‘n’ and described as a dimensionless number which shows that the light or any radiation travels through a medium. It is mathematically expressed as:
n = c/v
Where,
‘v’ indicates the phase velocity of light in a medium, and
‘c’ indicates the speed of light in a vacuum.
Refraction through a Glass Prism:
The ray of light comprises different colors and also called white light. When the white light crosses through a prism, it is disintegrated into the colors it comprises, which is violet, indigo, blue, green, yellow, orange, and red. This procedure is also termed as dispersion that occurs sue to a different wavelength of each color. Because of these discrete wavelengths, the dispersed ray from the prism refracts into diverse angles and therefore displaying all the colors that are present in the white light. This color configuration is also termed as VIBGYOR titled with the first letter in the order of diminishing deviations.
Stay tuned with BYJU’S to learn more about Geometric Optics.
What is Geometric Optics?
When a ray of light is reflected by some angle by a barrier in its pathway, it is said that the light beam is rebounded and this procedure is known as a reflection of light.
When the beam of light crosses from one medium to the other, it diverges slightly from its route. This deviation hinges over the medium where the light is crossing through. This procedure is termed as the refraction of light.
Snell’s Law (Laws of Refraction and Reflection):
When light is refracted or reflected, one can check the basis of procedure for refraction and reflection along with the things linked to them.
The Laws of Reflection in Geometric Optics:
The conduct of the ray of light that is rebounded by a smooth mirror is judged based on the laws of reflection. But primarily, one needs to know the basic terms.
Light approaching a mirror is termed an incident ray and the beam of light that is reflected by the mirror is termed as a reflected ray. At the spot of reflection, the perpendicular that is drawn is termed as normal. The angle amidst the incident ray and normal is the angle of incidence, while the angle amidst the reflected ray and normal is the angle of reflection.
Conferring to the laws of reflection, the angle of incidence is always equivalent to the angle of reflection.
| ||
Far from the source, the curvature of the wave front is small, so the wave front appears to be a plane. Then, the light rays will be nearly parallel. Rays from the sun are considered to be parallel when reaching the earth.
The law of reflection
Most visible objects are seen by reflected light. There are few natural sources of light, such as the sun, stars, and a flame; other sources are man‐made, such as electric lights. For an object to be visible, light from a source is reflected off the object into our eyes (except in the special case of phosphors). In Figure 2, the light is coming from the sun, parallel due to the distance of the source. The light reflects off the object and travels in straight lines to the viewer. Through experience, the viewer has learned to extend the reflected rays entering the eye back to locate the object.
Figure 2
Vision is the result of light reflected from the object.
As shown in Figure 3, light strikes a mirror and is reflected. The original ray is called the incident ray, and after reflection, it is called the reflected ray. The angles of the incident and reflected rays are always measured from the normal. The normal is a line perpendicular to the surface at the point where the incident ray reflects. The incident ray, reflected ray, and normal all lie in the same plane perpendicular to the reflecting surface, known as the plane of incidence. The angle measured from the incoming ray to the normal is termed the incident angle. The angle measured from the outgoing ray to the normal is called the reflected angle. The law of reflection states that the angle of incidence equals the angle of reflection. This law applies to all reflecting surfaces.
| ||
Light undergoes either diffuse or regular reflection. The two are illustrated in Figure 4.
| ||
Diffuse reflection occurs when light reflects from a rough surface. Regular reflection is reflection from a smooth surface, such as a mirror. The reflected rays are scattered in diffuse reflection. This scattering is because the local direction of the normal to the surface is different for the different rays. By contrast, in regular reflection, the reflected light rays are orderly because each local region of the surface has a normal in the same direction.
Plane mirrors
Figure 5 illustrates the formation of an image by a plane mirror. Light rays are coming from a source and reflecting off each point of the object ( AB) in all directions. For simplicity, only a few of the rays are drawn. The rays spread upon leaving the object, and then each ray reflects from the mirror according to the law of reflection. The eye extends back the diverging reflected rays to see an image behind the mirror. An image formed in this manner by extending back the reflected diverging rays is called a virtual image. A virtual image cannot be projected on a screen. The light does not physically come together, but rather, the eye (or camera) interprets the diverging rays as originating from an image behind the mirror. Due to the law of reflection, the image formed by a plane mirror is the same distance behind the mirror as the object is in front of the mirror.
| ||
How tall does a mirror need to be so you can see your entire height? Assume the top of the mirror is in line with the top of your head. Does it matter where you stand? The ray diagram in Figure illustrates this situation. From the law of reflection and basic geometry, it can be proven that the marked angles are all equal; therefore, the necessary height of the mirror is approximately half your height. Draw a figure at a different distance to show that the distance from the person to the mirror does not change the result.
| ||
Concave mirrors
Regular reflection occurs not only for plane (flat) mirrors but also for curved mirrors. Picture a series of plane mirrors arranged in a semicircle as shown in Figure 7. The incoming light is from a distant source and, therefore, is nearly parallel, as Figure 7(a) shows. After reflection, the light converges on a region. As the number of mirrors increases—Figure 7(b)—the converging region of the light beams decreases.
Figure 7
A semicircular arrangement of mirrors focuses light in the region F.
A concave mirror reflects its light from the inner curved surface. The mirror can be a portion of a sphere, a cylinder, or shaped as a rotated parabolic curve. The light rays intersect after reflection at a common focus called the focal point ( F). The focal point is on the optical axis, the symmetry axis of the mirror. The distance f from focal point to the mirror is called the focal length. For a spherical mirror, the focal length is one‐half the radius of the sphere that defines the mirror. This distance c is called the radius of curvature, and the center of the sphere is denoted as C( c = 2 f). Figure 8 illustrates these definitions.
| ||
It is helpful to have a geometric system for locating an image formed by rays reflected from a curved mirror. Any reflected ray follows the law of reflection; however, certain rays have easily defined paths so that measuring angles and finding the normals are not necessary. Four of these rays are
- The ray directed parallel to the optical axis will reflect through F.
- The ray directed through F will reflect parallel to the optical axis.
- The ray directed to the center of the mirror will reflect at the same angle to the optical axis.
- The ray directed along the radius of the sphere will reflect back on itself.
Light rays are drawn for four different positions in Figure 9 : (a) far from F, (b) at nearly 2 F, (c) between F and 2 F, and (d) at F. It is only necessary to find the intersection of two reflected rays from a point on the object to define the corresponding point on the image. A third one can be used as a check. Sometimes one or another of the rays may be difficult to draw, and so choices can be made.
| ||
Notice that images are formed for the first three cases but not for the last one. No image is formed when the object is at the focal point or, alternatively, the image is formed at infinity and cannot be seen. The three images are all real images. Real images can be shown on a screen because the light physically comes together at a point in space. Note that real images are formed by light that converges after reflection. Also, real images are always inverted—upside down—with regard to the original object. In Figure , the light rays from the bottom of the object are not drawn. Light traveling along the optical axis will reflect back along the axis, and so if a point of the object is on the optical axis, the corresponding image point will also be on the optical axis.
The images formed can be characterized by size and placement. Let the distance from the object to the mirror be given by O. Then the image characteristics can be summarized as follows:
- If O > 2 F, the image is inverted, smaller, and located between F and 2 F.
- If O = 2 F (at C), the image is inverted, the same size as the object, and located at 2 F; that is, the distances of both the object and image to the mirror are equal.
- If 2 F < O < F, the image is inverted, larger than the object, and located >2 F. Light paths are reversible. If the object is placed in the position of its former image, the image will then be located where the object was originally; that is, the two will exchange positions.
Figure 10 shows the diagram for the case when the object is between the focal point ( F) and the mirror. In this case, a virtual image is formed because the reflected rays diverge from the surface of the mirror. The virtual image is upright, enlarged, and behind the mirror. Virtual images are never inverted.
| ||
The following approximate mirror equation relates the distances from the object to the mirror ( O), the distance from the image to the mirror ( I), and the focal length ( f):
The sign of f is positive if it is on the same side as the mirror (a concave mirror) and negative otherwise (convex mirror). Both O and I are positive in sign if they lie on the same side of the mirror as the incident light and negative if they lie on the opposite side.
The magnification is defined as the ratio of the image size to the object size. This ratio is the same as the ratio of the distances:
Thus, a magnification of 10 × means the image seen is 10 times the size of the object when viewed without a magnifying device.
Convex mirrors
The graphical technique for locating the image of a convex mirror is shown in Figure . For convex mirrors, the image on the opposite side of the mirror is virtual, and the images on the same side of the mirror are real. Figure shows a virtual, upright, and smaller image. In comparison to the virtual image of the concave mirror, the virtual image of the convex mirror is still upright, but it is diminished (smaller) instead of enlarged and on the opposite side of the mirror instead of the same side. Again, the virtual image is formed by extending back the reflected diverging rays.
| ||
The law of refraction
Refraction is the bending of light when the beam passes from one transparent medium into another. A transparent object allows the transmission of light, in contrast to an opaque object, which does not. Some of the light will also be reflected. The incident ray, reflected ray, normal, and refracted ray are shown in Figure 12.
| ||
When Willebrod Snell (1580–1626) observed light traveling from air into another transparent material, he found a constant ratio of the sines of the angles measured from the normal to the light ray in the material:
The constant ( n) is called the index of refraction and depends only upon the optical properties of the material. The index of refraction gives a measure of the amount of bending occurring when light travels from air into the material. It is a dimensionless number and can be located in tables of properties of materials. For example, the index of refraction of water is 1.33, and the index of refraction of crown glass varies from 1.50 to 1.62, depending upon the composition of the glass.
For the more general case of light traveling from medium 1 to medium 2, Snell's law can be written n 1 sin θ 1 = n 2 sin θ 2, where the subscripts 1 and 2 refer respectively to the angles and indices of the refraction for material 1 and material 2 respectively. A light ray traveling along the normal, with an incident angle of zero, will not be bent.
The index of refraction is also the ratio of the speed of light in a vacuum ( c) and the speed of light in that medium ( v); thus,
Consider the following problem involving both reflection and refraction. Imagine light entering an aquarium and reflecting off a mirror at the bottom. First, what will be the angle of refraction in the water if the angle in air is 30 degrees? Second, at what angle will the beam leave the water? See the setup in Figure 13.
| ||
Angle θ 2 is determined from θ 1, using Snell's law of refraction. Angle θ 2 = θ 2 by geometry, θ 3 = θ 4 by law of reflection, and θ 4 = θ 5 by geometry. θ 6 is related to θ 5 by Snell's law of refraction, in the same ration as θ 1 to θ 2. Therefore, θ 6—the angle of the ray leaving the water—must be 30 degrees. The problem is symmetrical.
A light ray passing through a rectangular block of transparent material will simply be displaced from its original path. For example, in passing from air to glass, the ray will bend toward the normal. Upon leaving the glass block, the ray will bend away from the normal so that the measured angles in the air on each side of the block are the same (Figure 14).
| Figure 14 | A light ray is displaced after passing through a refracting medium. |
Brewster's angle
Light reflected from the surface of a material is partially polarized. A ray incident on a transparent surface at a certain angle will be partly refracted and partly reflected in a plane polarized ray. This angle of maximum plane polarization is called Brewster's angle, named for Sir David Brewster (1781–1868). The equation is tan θ = n, where n is the index of refraction of the reflecting surface.
Total internal reflection
When light travels from a material with a higher n to one with a lower n, at certain angles all of the light is reflected. This effect is called total internal reflection.
Example 1: Figure 15 illustrates ray 1 along the normal (no bending), rays 2 and 3 are refracted, and rays 5 and 6 are reflected. Ray 4 is intermediate between reflection and refraction with an angle of refraction of 90 degrees. The incident angle for this case is called the critical angle (θ). If the angle of incidence is less than θ, the light will refract, and if it is greater, the light will reflect.
Figure 15
Total internal reflection at the interface of two different media.
The equation is
where n 1 > n 2. Find the critical angle from glass to air.
Therefore, if the incident ray on a glass to air interface is greater than 42 degrees, total internal reflection will occur. Figure 16 shows the light rays entering and leaving a 45‐45‐90 glass prism. This phenomenon has broad applications where a mirror is needed, but a silvered surface might corrode after a period of time.
| ||
Optical lenses
An optical lens functions by refracting light at its interfaces. In these examples, the lens will be assumed to be thin, in which case the thickness of the lens is negligible compared with it focal length. Lenses are basically of two types. A converging lens causes parallel rays to converge, and a diverging lens causes parallel rays to diverge. Figure 17 illustrates the paths of the rays through the lens and the focal point for each case. The definitions for optical axis, focal point, and focal length given for curved mirrors hold true for lenses, with the addition that lenses have focal points on each side of the lens.
| ||
Ray diagrams can be made for lenses similar to those drawn for curved mirrors. These three rays can be drawn to locate the image formed by the lens.
- The ray directed parallel to the optical axis refracts through F on the far side.
- The ray directed to the near F refracts parallel to the optical axis.
- The ray directed to the center of the lens is undeviated (in the thin lens approximation).
The ray diagrams for two cases of a converging lens are shown in Figure 18.
| ||
In Figure 18(a), a real image is formed, and in Figure 18(b), a virtual image is formed. The lens setup in Figure 18(b) is called a simple magnifier. With lenses as with mirrors, virtual images are right side up, and real images are inverted. (This is why slides inserted into a projector are inverted; the projector lens reinverts the image on the screen.)
The lens equation is the same relationship used for curved mirrors:
as is the equation for magnification:
The focal length is positive for a converging lens and negative for a diverging lens. The object and image distances are positive if they are on opposite sides of the lens and negative if they are on the same side. The relative sizes and positions of the object and image for a converging lens are similar to the four cases reviewed for the concave mirror.
- If O > 2 F, the image is inverted, smaller, and located between F and 2 F, on the opposite side.
- If O = 2 F, the image is inverted, the same size as the object, and located at 2 F; that is, the distances of both the object and image to the lens are equal but on opposite sides of the lens.
- If 2 F < O < F, the image is inverted, larger than the object, and located > 2 F.
- If O < F, the image is virtual, enlarged, and located on the same side of the lens where I > F.
Figure 19 shows the ray diagram for a diverging lens. The image formed by this lens is always virtual, upright, and diminished.
| ||
The compound microscope
When lenses are used in combinations, the image given by one lens becomes the object for a second lens. The compound microscope is an example of the use of several lenses to magnify an object. An objective lens near the object forms an enlarged image. This image is then further magnified by the second lens, called the eyepiece. Both are converging lenses.
In Figure 20, the object ( AB) is placed just below the focal point of the objective lens. The objective lens forms an enlarged, real, and inverted image at a distance greater than 2 F from the first lens. This image ( A′ B′) falls inside the focal point of the eyepiece lens; therefore, an enlarged, virtual image is formed by the eyepiece ( A″ B″). The total magnification is the product of the magnifications of each lens.
| ||
Dispersion and prisms
An important property of the index of refraction is that it is slightly dependent upon wavelength. For a given material—for example, glass— n decreases with increasing wavelength; thus, blue light bends more than red light. This effect is called dispersion.
Light is refracted twice as it enters and leaves the prism, as shown in Figure 21.
| ||
A given ray is bent from its original direction of travel by an angle (δ), called the angle of deviation. The angle of deviation for the red light is less than that for the blue light; therefore, the prism spreads the light into the colors of the spectrum. These colors are commonly called red, orange, yellow, green, blue, indigo, and violet (often abbreviated with the mnemonic Roy G Biv).
Rainbows are formed by dispersion and total internal reflection of sunlight in raindrops. The critical angle for water to air is approximately 40 degrees. The sunlight enters the drop and is reflected off the side of the drop away from the viewer. Due to dispersion, the violet ray emerges above the red ray. Figure 22 shows the refraction of sunlight on one idealized raindrop.
Reflection of Light Rays
On the surface of a flat mirror, a light ray is assumed to be reflected such that the output angle equals the input angle (both measured against the normal direction). For a curved mirror, one does that calculation based on a tangential flat plane. Figure 1 shows an example with reflection on a curved mirror.
Modified laws can be applied in case of diffraction gratings, where additional diffracted rays emerge at different angles.
Refraction of Light Rays
When a ray hits an interface between two different transparent media, a portion is reflected, and another portion is transmitted; for the latter one, which is refracted, the propagation direction is generally modified according to the Snellius law of refraction. Figure 2 shows an example case for a ball lens, where only the refracted rays (which are usually stronger) have been drawn.
Curved Rays
In optically inhomogeneous media, light beams may propagate along curves instead of straight lines. In geometrical optics, one may correspondingly assume curved ray paths. An example is shown in Figure 3, showing the focusing of light in a gradient-index lens. The rays get deflected in the lens and may exactly meet in a focal point if the lens is optimized.
Figure 4 shows how the beam radius, calculated with wave optics, evolves. Here, one can see that the beam radius in the focal point has a finite value related to diffraction.
Quantities Related to Light Rays
In geometrical optics, and particularly in Gaussian optics, certain quantities are often related to light rays:
- Points to which some bundle of rays converges are called focal points. For variable directions of input beams, for example, the focal points may lie in a focal plane, or actually often on a curved surface.
- Principal planes are defined based on the intersections of extrapolated rays in an optical system.
- A focal length is often defined based on a distance from a focal point to a principal plane, or sometimes to a geometrical position of an optical device.
Special Rays
Certain light rays have a special importance in geometrical optics:
- The chief ray of a ray bundle, which is limited by some aperture, is the one going through the center of that aperture.
- Marginal rays of those passing at the edges of apertures.
Ray Tracing
The propagation of light rays, as shown in the figures above, is calculated based on purely geometrical considerations. The used technique is called ray tracing and is usually applied with specialized optics software. The calculations can be geometrically exact, i.e., valid even for large incidence angles. Curved surface may have any geometrical shapes. Depending on the initial direction of a light ray, it may or may not hit a certain optical component.
Rays may be split up into multiple rays, e.g. due to partial reflection and transmission that interfaces, or due to multiple diffraction orders at gratings. In the context of diffuse optical scattering, one may employ stochastic methods for representing the scattered light with some limited number of rays. For multiple diffuse reflections, this may of course result in a very large number of rays to be treated.
One may not only need to know the paths of multiple rays, but also derive various results from them. For example, ray tracing software may locate focal planes, calculate image magnification or estimates resulting optical intensities and colors.
Ray tracing can be used for many purposes, for example for studying the detailed properties of imaging systems including their optical aberrations and effects of misalignment and imperfections from optical fabrication, or for the design of illumination systems.
Paraxial Optics
In many situations, one can use simplified equations which describe the approximate propagation of rays which stay close to the optical axis in terms of lateral offset and direction. Any terms of second or higher order are ignored; for example, one may consider the deflection of a ray at a curved lens surface as occurring in the plane touching the surface, ignoring a longitudinal position error of second order in the lateral offset.
Many useful relations can be derived based on such paraxial optics, which would otherwise be far more complicated or not analytically solvable at all.
The paraxial approximation is widely used in Gaussian optics.
The paraxial case is also often used as the reference case, for example for the study of optical aberrations.
Limitations of Geometrical Optics
The main limitation of geometrical optics is that it ignores the wave properties of light, as described in wave optics. In particular, that means that the phenomena of diffraction, interference and polarization are not taken into account. This is not a substantial problem in many practical cases, where such effects may be negligible or can be taken into account separately. For example, one can study the optical aberrations of an imaging system with geometrical optics, being aware that even for perfect compensation of aberrations one will not obtain perfectly sharp images due to the diffraction limit. Anyway, aberrations often remain a more severe limitation than diffraction, which can thus often be safely ignored.
Ignoring diffraction becomes a serious problem when treating the propagation of light under conditions where it experiences tight confinement. For example, light propagation in single-mode fibers can not be realistically described at all with geometrical optics. One may still define the numerical aperture of a fiber, for example, based on geometrical optics, but such a quantity then has only a limited meaning for the actual propagation of light in the fiber. Even for multimode fibers with many modes, geometrical optics is only of quite limited utility. It can be completely misleading, for example, concerning optical phase delays of fiber modes [1].
Similarly, light propagation in multilayer coatings cannot be realistically analyzed with ray optics, because interference effects are essential.
To some extent, the deficits of geometrical optics can be amended by adding additional properties to rays. For example, one may attribute some optical power to each ray in a ray tracing simulation, taking into account power losses by absorption, incomplete reflection, etc. Similarly, one may add polarization properties and optical phases, for example for calculations on an interferometer setup. A simpler example is the calculation of different ray paths for different polarization directions, for example when analyzing a polarizing prism.
In some areas, one finds an interesting correspondence between quantities from geometrical optics and wave optics:
- The ABCD matrix algorithm has been developed in geometrical optics, but the components of that matrix can also be used for Gaussian beam calculations.
- Optical resonators can be analyzed in terms of geometrical stability, and it turns out that the properties of resonator modes as calculated with wave optics have substantially different properties depending on whether or not a resonator is geometrically stable.
No comments:
Post a Comment