School of Microelectronic Engineering CHAPTER 2 GEOMETRICAL OPTICS Prepared by NURJULIANA JUHARI School of Microelectronic Engineering

OBJECTIVES Studies about plane surfaces and prism Spherical surfaces Thin lenses Spherical Mirror

PLANE SURFACES AND PRISM (1) PARALLEL BEAM The parallel beam remains parallel after reflection or refraction at plane surface (Fig 2A(a)). Refraction causes a change in width of the beam which easily seen to be in the ratio (cosø’) and (cosø), whereas the reflected beam remains of the same width. Reflection at a surface where n increases (Fig 2A (a)) is called external reflection.

PLANE SURFACES AND PRISM Fig 2A(b) case of internal reflection or dense-to-rare reflection. In this particular case, the refracted beam is narrow because ø’ is close to 900 THE CRITICAL ANGLE & TOTAL REFLECTION Fig 2A(a) light passes from one medium like air into another medium like glass or water the angle of refraction is always less than the angle of incidence.

PLANE SURFACES AND PRISM Fig 2B several angles of refraction are shown from 00 to øc. In the limiting case, where the incident rays approach an angle of 900 with the normal, the refracted rays approach a fixes angle øc beyond which no refracted light possible. This particular angle øc for which ø=900, is called critical angle.

PLANE SURFACES AND PRISM Formula calculating critical angle is obtained by substituting ø=900 or sin ø=1, in Snell’s Law n x 1= n’ sin øc sin øc=n/n’ 2.1 So that, a quantity which is always less than unity.

PLANE SURFACES AND PRISM When apply principle of reversibility of light rays Fig 2B, all incident rays will lie within a cone subtending an angle 2øc while the corresponding refracted rays will lie within a cone 1800. Angles of incidence greater than øc there can be no refracted light and every ray undergoes total reflection as shown in Fig 2B(b)

PLANE SURFACES AND PRISM The critical angle for the boundary separating two optical media is defined as the smallest angle of incidence, in the medium of greater index, for which light is totally reflected Total reflection is that no energy is lost upon reflection. In any device intended to utilized this property there will, however be small losses due to absorption in the medium and to reflections at the surfaces where the light enters and leaves the medium.

PLANE SURFACES AND PRISM The commonest devices of this kind are called total reflection prisms, which are glass prisms with two angles of 450 and one of 900. The principle of reversibility states that light will follow exactly the same path if its direction of travel is reversed.

PLANE SURFACES AND PRISM 3) PLANE-PARALLEL PLATE When a single ray traverses a glass plate with plane surfaces that that are parallel to each other, it emerges parallel to its original direction but with a lateral displacement d which increases with an angle of incidence ø Apply the law of refraction and some simple trigonometry to find the displacement d.

PLANE SURFACES AND PRISM Starting with the right triangle ABE, we can write, Which by the trigonometric relation for the sine of the difference between two angles,

PLANE SURFACES AND PRISM From the right triangle ABC we can write Substituted in Eq (2.3) gives (2.5)

PLANE SURFACES AND PRISM From Snell’s Law we obtain Which upon substitution in Eq (2.5) gives

From 00 up to appreciably large angles, d is nearly proportional to ø (2.6) From 00 up to appreciably large angles, d is nearly proportional to ø The ratio of cosines becomes appreciably less than 1 causing the right-hand factor to increase. The sine factor drops below the angles itself in almost the same proportion

PLANE SURFACES AND PRISM 4. Reflection By A Prism In a prism the two surfaces are inclined at some angle α so that the deviation produced by the first surface is not annulled by the second but is further increased The geometrical optics of the prism for light of a single color i.e. for monochromatic light such as is obtained from a sodium arc.

PLANE SURFACES AND PRISM The solid ray in Fig 2F shows the path of a ray incident on the first surface at the angle ø1 Reflection at the second surface as well as at the first surface, obeys Snell’s Law, so that in terms of the angles shown (2.7)

PLANE SURFACES AND PRISM The angle of deviation produced by the first surface and that produced by the second surface is The total angle of deviation between the incident and emergent rays is given by Since NN’ and MN’ are perpendicular to the two prism faces, is also the angle at N’. From triangle ABN’ and the exterior angle we obtain (2.8)

PLANE SURFACES AND PRISM Combining the above equation, we obtain 5) Minimum Deviation When the total angle of deviation for any given prism is calculated by the use of the above equation, it is found to vary considerably with the angle of incidence. (2.9) (2.10)

PLANE SURFACES AND PRISM The angles thus calculated are in exact agreement with the experimental measurement. During the time a ray of light is refracted by a prism the prism is rotated continuously in one direction about an axis (A in Fig 2F) parallel to the refracting edge, the angle deviation will be observed to decrease, reach a minimum & increase again as shown in Fig 2G

PLANE SURFACES AND PRISM The smallest deviation angle, called the angle of minimum deviation occurs at that particular angle of incidence where the refracted ray inside the prism makes equal angles with the two prism faces (see Fig.2H). In this special case

PLANE SURFACES AND PRISM To prove these angle equal assume ø1 not equal to ø2 when minimum occurs. From principle reversibility of light rays there would be two different angles of incidence capable of giving minimum deviation. There must be symmetry and the above equalities must hold.

PLANE SURFACES AND PRISM Triangle ABC in Fig 2H the exterior angle equals the sum of the opposite interior angles . Similarly, for the triangle ABN’, the exterior angle α equals the sum consequently

PLANE SURFACES AND PRISM Solving these three equations for ø’1 and ø1 gives Since by Snell’s Law n’/n=(sin ø1/ sin ø’1 )

PLANE SURFACES AND PRISM 6) Graphical Method of Ray Tracing Consider first a 600 prism of index n’=1.5 surrounded by air of index n=1.00 Fig 2J, the prism drawn to scale The angle of incidence ø1has been selected and the construction begins as in Fig 1G

PLANE SURFACES AND PRISM Line OR is drawn parallel to JA Two circular arcs are drawn with radii proportional to n and n’. Line RP is drawn parallel to NN’ and OP is drawn to give the direction of the refracted ray AB Carrying on from the point P, a line is drawn parallel to MN’ to intersect the arc n at Q

PLANE SURFACES AND PRISM The line OQ then gives the correct direction of the final refracted ray BT In the construction diagram at the left the angle RPQ is equal to the prism angle α and the angle ROQ is equal to the total angle of deviation

PLANE SURFACES AND PRISM 7) Reflection of Divergent Ray When a divergence pencil of light is reflected at a plane surface, it remains divergent. All rays originating from point Q (Fig 2M) will after reflection appear to come from another point Q’ symmetrically placed behind the mirror.

PLANE SURFACES AND PRISM The proof of this proposition follows at once from the application of the law of reflection. Under these conditions the distances QA and AQ’ along the line QAQ’ drawn perpendicular to the surface must be equal i.e. s=s’ Object distance= image distance

PLANE SURFACES AND PRISM The point Q’ is said to be a virtual image of Q since when the eye receives the reflected rays They appear to come from a source at Q’ but do not actually pass through Q’ as would be case if it were real image. In order produced a real image a surface other than a plane one is required.

PLANE SURFACES AND PRISM 8) Refraction of Divergent Rays If an object embedded in clear glass or plastics or is immersed in a transparent liquid such as water, the image appears closer to the surface. Fig 2N has been drawn accurately for an object Q located in water of index 1.333 at a depth s below the surface.

PLANE SURFACES AND PRISM Light rays diverging from this object arrive at the surface at angles ø There are refracted at larger angles ø’, only to diverge more rapidly as shown. Extending these emergent rays backward, we locate their intersection in pairs. These are image points, or virtual images.

PLANE SURFACES AND PRISM 9) Images formed by Paraxial Rays Particular interest to many observers are the object and image distances s and s’ for rays making small angles ø and ø’. Rays for which angles are small enough to permit setting the cosines equal to unity and the sines and tangents equal to the angles are called paraxial rays.

PLANE SURFACES AND PRISM Consider the right triangles QAB and Q’AB in Fig 2N, redrawn in Fig 2P. Since there is a common side AB=h, we can write Therefore (2.13)

PLANE SURFACES AND PRISM Applying snell’s Law We obtain on substitution in Eq (2.13) For paraxial rays like the ones shown in the diagram, angles ø and ø’ are very small; Eq 2.13 can be written (2.14)

Together Eq 2.15 and 2.16 provide the simple relation And Eq 2.14 written Together Eq 2.15 and 2.16 provide the simple relation The ratio of the image to object distance for paraxial rays is just equal to the ratio of the indices of refraction. (2.15) (2.16) (2.17) Paraxial rays

SPHERICAL SURFACES a) equiconvex d) equiconcave b) plano-convex e) plano-concave c) positive meniscus f) negative meniscus

SPHERICAL SURFACES 1) FOCAL POINTS & FOCAL LENGTHS Point A where the axis crosses the surface Is called the vertex. Fig (a), rays are shown diverging from a point Source F on the axis in the first medium and Refracted into a beam everywhere parallel to the Axis in second medium

SPHERICAL SURFACES A beam converging in the first medium toward the point F and then refracted into a parallel beam in the second medium. F in each of these two cases is called the primary focal point, distance f is called the primary focal length

SPHERICAL SURFACES Parallel incident beam is refracted and brought to a focus at the point F’

SPHERICAL SURFACES A parallel incident beam is refracted to diverge as if it came from the point F’. F’ in each case is called secondary focal point and distance f’ is called secondary focal length

SPHERICAL SURFACES Diagram (a) and (b) for reference, we now state that the primary focal point F is an axial point having the property that any ray coming from it or proceeding toward it travel parallel to the axis after refraction. Diagram (c) & (d), is similar statement that the secondary focal point F’ is an axial point having the property that any incident ray traveling parallel to the axis will, after refraction, proceed toward or appear to come from F’.

SPHERICAL SURFACES A plane perpendicular to the axis and passing through either focal point is called focal plane. Focal length f for the convex surface diagram (a) is not equal to the secondary focal length f’ of the same surface diagram (c). The ratio of focal length f’/f is equal to the ratio n’/n of the corresponding refractive indices

SPHERICAL SURFACES 2) Image Formation Diagram illustration image formation by a single refracting surface is given in Fig 3D. 1st medium is air with n=1 and 2nd medium is glass with n=1.60. The focal length f and f’ therefore have ratio 1:1.60 (eq 2.18). Experimentally observed: if the object is moved closer to the primary focal plane, the image will be formed farther to the right away from F’ and will be larger, i.e magnified. If the object is moved to the left farther away from F, the image will be found closer to F’ and will be smaller in size.

All rays coming from the object point Q, brought to a focus at point Q’ Rays from any other object point like M brought to a focus at corresponding image point like M’

Departures from it give rise to slight defects of the image known as aberrations. Paraxial rays are defined as those rays, which make very small angles with the axis and lie close to the axis throughout the distance from object to image. Virtual Images The image M’Q’ Fig 3D is a real image. A sharply defined image of the object MQ will be formed on the screen. Fig 3E image is virtual

4) Conjugate Points & Planes The principle of reversibility of a light rays has the consequence that if Q’M’ in Fig 3D were in object, an image would be formed at QM. If any object is placed at the position previously occupied by its image, it will be imaged at the position previously occupied by the object. The object and image are thus interchangeable or conjugate. Any pair of object and image points such as M and M’ in Fig 3D called conjugate points

Planes through this points perpendicular to the axis are called conjugate planes. Three methods is used to measure size and position of an image a) graphical method b) experiments c) calculation using a formula s- object & s’- image. This equation called gaussian formula for a single spherical surface. (2.19)

Example 1 The end of solid glass rod of index 1.50 is ground and polished to a hemispherical surface of radius 1 cm. A small object is placed in air on the axis 4 cm to the left of the vertex. Find the position of the image. Assume n=1 for air.

As an object M is brought closer to the primary focal point Eq 2 As an object M is brought closer to the primary focal point Eq 2.19 shows that the distance AM’ of the image from the vertex become steadily greater and that in the limit when the object reaches F the refracted rays parallel and the iamge is form at infinity. Then Eq 2.19 become

Since this particular object distance is called the primary focal length f, can write If object distance is made larger and eventually approaches infinity, the image distance diminishes and become equal to f’ in the limit s=∞ then (2.20)

Or since this value of s’ represents the secondary focal length f’ Equating the left-hand members of Eq (2.20 & 2.21), obtain When (n’-n)/r in Eq 2.19 is replaced by n/f of by n’/f’ according to Eq 2.20 and 2.21, the results (2.21) (2.22) (2.23)

5) Convention of Signs All figures are drawn with the light traveling from left to right All object distance (s) are considered positive when they are measured to the left of the vertex and negative when they are measured to the right. All image distance (s’) are positive when they are measured to the right of the vertex and negative when to the left Both focal length are positive for a converging system and negative for a diverging system

Cont.. Object and image dimensions are positive when measured upward from the axis and negative when measured downward. All convex surfaces are taken as having a negative radius.

Example 2 A concave surface with radius of 4 cm separates two media of refractive index n=1.00 and n’=1.5. An object is located in the first medium at a distance of 10 cm from the vertex. Find a) primary focal length b) secondary focal length c) image distance

6) GRAPHICAL CONSTRUCTION (i) Parallel- Ray Method

(ii) Oblique-Ray Method

7) Magnification In any optical system, ratio between transverse dimension of the final image and corresponding dimension of the original object is called the lateral magnification. To determine the relative size of the image formed by a single spherical surface, ref Fig 3F

Here the undeviated ray 5 forms two similar right triangles QMC and Q’M’C. The theorem of the proportionality of corresponding sides requires that We now define y’/y as the lateral magnification m and obtain M is positive the image will virtual and erect, if negative image is real and inverted

THIN LENSES A thin lens may be defined as one whose thickness is considered small in comparison with the distance generally associated with optical properties. Such distances are example, radii of curvature of the two spherical surfaces primary and secondary focal length and object and image distances.

Focal Points & Focal Lengths The primary focal point F is an axial point having the property that any ray coming from it or proceeding towards it travels parallel to the axis after refraction. Every thin lens in air has 2 focal points. The secondary focal point F’ is an axial point having the property that any incident ray traveling parallel to the axis, after refraction, proceed toward or appear to come from F’

The distance between centre of a lens and either of its focal points is its focal length. +ve sign converging lens –ve sign diverging len. For lens with the same medium on both side, f=f’ , by the reversibility of light rays.

2) Image formation When an object is placed on side or the other of a converging lens and beyond the focal plane, an image is formed on the opposite side (Fig 4C). If object put closer to primary focal point, the image is magnified and farther away from secondary focal plane.

3) Conjugate Points and planes Principle of reversibility of light rays is applied to Fig 4C, Q’M’ become object and QM its image. The image and object are therefore conjugate Any pair of object and image points such as M and M’ in Fig 4C are called conjugate plane, and planes through these points perpendicular to axis are called conjugate planes. Use lens formula to determine the distance of object or image or focal length.

4) The power of thin lens The power of a thin lens in diopters is given by the reciprocal of the focal length in meters; P=1/f or Diopters=1/focal length,m Also can used from lens maker’s formula, P= (n-1)(1/r1-1/r2)

SPHERICAL MIRROR

SPHERICAL MIRROR Law of reflection ø’’= ø Focal length for concave mirror

Mirror Formula In order to apply the standard lens formula, we must adhere to follow sign conventions: Distances measured from left to right are positive while those measured from right to left are negative. Incident rays travel from left to right and reflected rays from right to left The focal length is measured from the focal point to the vertex. The gives f a positive sign for concave mirror and negative sign for convex mirror.

4) The radius is measured from the vertex to the centre of curvature 4) The radius is measured from the vertex to the centre of curvature. This makes r negative for concave mirrors and positive for 5) Object distances s and image distances s’ are measured from the object and from the image respectively to the vertex. This makes both s and s’ positive and the object and image real when they lie to the left of the vertex; they are negative and virtual when they lie to the right.

Using a well-known geometrical theorem, MC/MT=CM’/M’T’ Now for paraxial rays MT~MA=s and M’T~M’A=s’, where the symbol ~ means is approximately equal to. MC=MA-CA=s+r and CM’=CA-M’A=-r-s’=-(s’+r) Substituting in the above proportion gives s+r/s=-s+r/s’

Which can easily be put in the form 1/s+1/s’=-2/r (Mirror formula) (Eq 6b) The primary focal point is defined as that axial object point for which the image is formed at infinity, so substituting s=f and s’=∞ in Eq (6b), we have 1/f+1/ ∞=-2/r For which 1/f=-2/r or f=-r/2 (Eq 6c)

The secondary focal point is defined as the image point of an infinitely distant object point. This is s’=f’ and s= ∞, so that 1/ ∞+1/f’=-2/r From which 1/f’=-2/r or f’=-r/2 (Eq 6d) Therefore the primary and secondary focal point fall together, and the magnitude of the focal length is one-half the radius curvature. When -2/r is replaced by 1/f Eq 6b becomes 1/s+1/s’=1/f (Eq 6e) Just as for lenses

The lateral magnification of the image from a mirror can be evaluated from the geometry of Fig 6C. From the proportionality of sides in the similar triangles Q’AM’ and QAM we find that –y’/y=s’/s, giving m=y’/y=-s/s (Eq 6f)

POWER OF MIRROR Describe the image-forming properties of lenses can readily be extended to spherical mirror as follows. P=1/f V=1/s V’=1/s’ K=1/r Eq (6b), 6(e), 6(c) and 6(f), then take the forms V+V’=-2K (6h) P=-2K (6j) V+V’= P (6i) m=y’/y=-V/V’ (6k)