IMAGE FORMATION e.g. lens, mirors Spherical wavefronts expand, rays diverge Spherical wavefronts contract, rays converge The optical system may consist of reflecting and/or refracting surfaces that may change the direction of rays from an object point O. Fermat’s principle implies that as every ray starts at O and ends at I, each ray takes the same transit time  isochronous rays. Principle of reversibility states if I becomes the object point, each ray will reverse its direction but maintain its path and ends at its corresponding image point O  I and O are conjugate points of optical system. - ideal optical system: every ray from object point O intercepted by the system must pass through its conjugate image point I

In non-ideal optical system, some rays leaving O may not reach I because of: light scattering - reflection losses at refracting surfaces, diffuse reflections from reflecting surfaces, scattering by inhomogeneities in transparent media. These degrade image & cause loss of brightness. aberrations - departure from ideal, paraxial imaging. diffraction limited - effect due to using a limited portion of the wavefront being intercept by optical system.

Cartesian surfaces are surfaces that form perfect images. Examples of Cartesian reflecting surfaces Rays are isochronous.

Cartesian refracting surface: Arbitrary point Every ray from O that refracts at surface  is required to pass through image point I (e.g. isochronous rays OPI and OVI). OPI and OVI are not equal in length because of different refractive indices of media on either side of , but have equal transit times. Transit time of ray through medium of thickness x with refractive index n is: v = speed in medium of refractive index n; nx = optical path length (equivalent path of the ray if it were to travel in air at speed c)

constant (3.1) constant (3.2) However, optical path of ray OPI = optical path of ray OVI Thus, constant (3.1) Rewriting in terms of (x,y) coordinates: constant (3.2) Eq. (3.2) describes a Cartesian ovoid of revolution.

Hyperbolic surface images O at infinity when O is at one focus and ni > no Ellipsoid surface images O at infinity when O is at one focus and no > ni Cartesian refracting surfaces Cartesian ovoid images O at I by refraction.

A lens refracts light rays twice, once at each surface, and can produce a real image outside the lens (image may be in same medium as object) Double hyperbolic lens produces aberration-free image of point object O; only if point O is at correct distance from lens and on axis Shortcomings of hyperboloid surfaces: difficult to fabricate images of actual objects (extended) are not totally aberration-free Therefore, most optical surfaces are spherical (ease in fabrication outweighs its spherical aberrations)

REFLECTION AT SPHERICAL SURFACE (with radius of curvature R) Ray 1 Ray 2 R (1) Spherical mirrors Convex spherical mirror (Center of curvature C on right of vertex V) Rays 1 & 2 diverge after reflection. Image point I conjugate to O is at intersection of rays 1 (extended backward) & 2 after reflection. Image is VIRTUAL, located BEHIND the mirror surface.

To obtain a relationship between s and s’ in terms of R only: s & s’ are respective object and image distances measured from vertex V. From triangle POC: (3.3) (external angle = sum of two internal opposite angles) From triangle POI: (3.4) (3.5) Combining (3.3) & (3.4): Using small-angle approximation (leads to first-order, or Gaussian, optics) (3.6)  (3.7) 

As paraxial optics is considered,  and  are small, axial distance VQ can be neglected, and angles , ’ and  in Eq.(3.5) may be replaced by their tangents as: (3.8) Canceling h: (3.9) Employing an appropriate sign convention to represent convex as well as concave mirrors, the mirror equation becomes: (3.10)

Sign convention chosen is as follows (assume light ray to enter from the left):- Object Real + s O is at left of V Virtual  s O is at right of V Image + s’ I is at left of V  s’ I is at right of V Radius of curvature + R when C is at right of V (CONVEX)  R when C is at left of V (CONCAVE) For plane mirror, R   and s’ = s (negative means virtual image for real object)

 +f f Object focus is defined by: and image focus is defined by: Concave mirror Convex mirror +f f Object focus is defined by: and image focus is defined by: Therefore,  (3.11) Thus, (3.12) another form of mirror equation

To determine lateral magnification: As i, r and  are equal, (3.13) Lateral magnification defined by ratio of lateral image size to corresponding object size: (3.14) Again employing the appropriate sign convention, lateral magnification is given by: (3.15)

Object height + ho Above axis, erect object  ho Below axis, inverted object Image height + hi  hi Lateral magnification + m Image has same orientation as object  m Image is inverted relative to object

Concave mirror Convex mirror Virtual, magnified, erect Real, minified, inverted Real, magnified, inverted Virtual, magnified, erect Always virtual, minified, erect RO = real object RI = real image VI = virtual image C = center of curvature F = focal point Image formation by spherical mirrors of object placed at different distances from the mirror surface. Height of arrow from optical axis shows relative height of object/image.

Ray diagrams for spherical mirrors: (a) Real image, concave mirror (b) Virtual image, concave mirror Intersection of any 2 rays will locate the image: Ray-1: Line parallel to optical axis from tip of object to reflecting surface, then bends to pass through F after reflection Ray-2: Line from tip of object to/away from F to reflecting surface before bending to become parallel to optical axis after reflection Ray-3: Line from tip of object to/through reflecting surface to center of curvature. (c) Virtual image, convex mirror

Example: An object 4 cm high is placed 40 cm from (a) a convex, and (b) a concave spherical mirror, each of 20 cm focal length. Find the position and type of image in each case. (a) Convex mirror: f =  20 cm and s = +40 cm Using mirror equation: cm (virtual, right of V) (minified, erect)

(b) Concave mirror: f = +20 cm and s = +40 cm Using mirror equation again: cm (real, left of V) (same size, inverted)

REFRACTION AT SPHERICAL SURFACE (with radius of curvature R) Ray 2 refracted (Snell’s law) Axial ray 1 refracted with no change Triangle CPO: (3.16) Triangle CPI: (3.17) From Snell’s law: (3.18) For paraxial rays, we use approximation Eq.(3.18) becomes For small angles, distance QV can be neglected, and with approximation   (3.19) Employing the same sign convention as for mirrors, the refraction equation becomes: (3.20)

Sign convention for spherical refracting surface: Object Real + s O is at left of V Virtual  s O is at right of V Image + s’ I is at right of V  s’ I is at left of V Radius of curvature + R when C is at right of V (CONVEX)  R when C is at left of V (CONCAVE) For plane surface: R  , or which is the apparent depth.

Lateral magnification of refracting surface is determined as: From Snell’s law and small angle approximation:  Lateral magnification: (3.21) (negative sign inserted according the sign convention used)

From refraction equation: Example: A real object in air (na = 1) is placed 30 cm from a convex spherical surface of radius R = +5 cm. On the right of the interface, the medium is water (nw = 4/3). Find the image distance s’ and lateral magnification m of the image. From refraction equation:   cm (real image, on right of refracting surface in water) Magnification is (inverted, same size)

Suppose now there exists a 2nd concave refracting surface that reduces the length of the 2nd medium to only 10 cm thick, forming a thick lens. n3 = n1 = 1 The rays from object after refracting through 1st surface will still form a 1st image as calculated previously. But this image will now be the object for the 2nd surface, producing a different final image. Real image of 1st surface is on right of 2nd surface  virtual object for 2nd surface at distance s2 = 30 cm

For 2nd surface:    cm (real image, to right of 2nd surface) Lateral magnification: ( the size of 1st image or 2nd object, i.e. size of original object; same orientation as 1st image, i.e. inverted relative to original object)