GEOMETRICAL OPTICS Laws of Reflection Laws of Refraction
- Slides: 67
GEOMETRICAL OPTICS
• Laws of Reflection • Laws of Refraction
Law of reflection
Law of refraction
Huygen’s Principle • All point on a given wavefront are taken as point sources for the production of spherical secondary waves (wavelets). These propagate outward. After some time the new position of the wavefront is the surface tangent tot the wavelets.
Huygen’s Principle
Fermat’s Principle – least time • When light travels between two points, its path is the one that requires the least time.
Imaging by optical system • O and I are conjugate points • If O and I are points – perfect imagery • Non ideal images due to (in practice) (i) light scattering (ii) aberrations (iii) diffraction • Some rays leaving O do not arrive at I because of (i) reflection losses at refracting surfaces (ii) diffuse reflection from reflecting surfaces (iii) scattering by inhomogeneities in transparent media - thus diminishing of brightness
• Cartesian surfaces form perfect images - reflecting and refracting
Sign Convention • Assume light is from left to right • Reflection (i) s> 0 if object is on same side as incoming light (ii) s′ > 0 if image is on same side as incoming light (iii) R > 0 if centre of curvature is away from incoming light • Refraction (i) and (iii) same (ii) s′ > 0 if image is on the other side of vertex V
Reflection at a spherical surface
• In paraxial approx
• Observe if R →∞, plane mirror • How do you describe the image (i) image distance (ii) magnification (iii) orientation
• From diagram
• + sign if O and I have same orientation • - sign if O and I have different orientation
Refraction at a spherical interface • Snell’s law of refraction • Paraxial approximation nθ ≈ n′θ 1 2
• For an object at infinity, Image distance = second focal length
• n′- n is called the power of the refraction R surface If R→∞ (plane surface) s′ apparent depth
Refraction by a thin lens
• If n′′ = n i. e. same medium on both sides of lens (ii) becomes …… (iii) Subs. s = -s ′ in ( iii ) and adding (i) and (iii) 2 1
• If object at infinity i. e. s = ∞ then s′ = f lensmaker’s equation NB (1) Convex lens – converging lens Rays refracted by lens bend towards axis (2) Concave lens – diverging Rays bend away from axis Magnification m = -s′ s
Sample ray diagram for converging (convex)
diverging (concave) lens
• A lens thicker in the middle than at the ends convex • A lens thicker at the ends than in the middle – concave • Magnification m = -s′ s The image is formed in the same medium as the object
Vergence and Refracting power
Newtonian equation for thin lens
Newtonian Equation for a Thin Lens • Object and image distances measured relative to F • Similar ∆’s and using + for height above and – for height below
Matrix methods for complex optical systems • Complex optical systems, involving a number of refracting and/or reflecting elements may be analyzed using system matrices. • A spherical thick lens is a lens whose thickness along the optical axis cannot be ignored without leading to serious errors when analyzed. • Just when a lens moves from a thin to thick depends on the accuracy required.
Matrix methods for complex optical systems • The thick lens can be treated by the methods discussed earlier where the glass medium is bounded by two spherical refracting surfaces. • The thick lens can also be described in a way that allows graphical determination corresponding to arbitrary objects. This description, in terms of the cardinal points of the lens, is useful because it can be applied to more complex optical systems.
Cardinal points and Cardinal planes • There are 6 cardinal points on the axis of a thick lens from which its imaging properties can be deduced. • F 1 and F 2 – First and Second focal points • P 1 and P 2 – First and Second principal points • N 1 and N 2 – First and Second nodal points • Planes normal to the axes at these six cardinal points are called Cardinal planes, focal planes, principal planes and Nodal planes.
Cardinal points and Cardinal planes • NB 1. Between principal planes – unit magnification 2. At nodal planes rays laterally displaced 3. Distances to right of reference planes – positive and to the left – negative
Cardinal points and Cardinal planes
Cardinal points and Cardinal planes
Translation matrix
Matrix Method • One relates output (y′, α′) to input (y, α) • 1. Translation matrix – translation through a medium
• Refraction Matrix • Refraction occurs at a point on the refracting surface
• In refraction matrix, if surface is concave, R is negative; if surface is plane R =∞ For a plane surface the refraction matrix is
Reflection Matrix • A concave mirror is used, so R < 0 • Sign convection is used • Angles positive if rays point upwards and negative if downward • y′ = y since reflection occurs • ie y′ = (1)y + (0)α …(i)
Thick Lens Matrix • Refraction at first surface, R 1
• For a thick lens, where there is a refraction, followed by a translation, then a refraction, the system matrix is
Thin Lens But for a thin lens, lensmaker’s formula is
• Element of the composite ray transfer matrix as • The input plane was at the left surface (first refraction) and output plane at the right surface (second refraction) • The determinant of the matrix given
Physical significances of system Matrix • If m = 0, then αf= m 21 y 0 (from (ii)) Now y 0 is fixed, so rays leaving a point on input plane emerge with same αf at the output plane (independent of α 0. input plane coincides with the First Focal Plane and parallel rays emerge. 22
Physical significances of system Matrix • If m = 0, then y = m α from (i) y is independent of y. All rays leaving input plane at same α (parallel ray) arrive at the same point in the output plane ie at the Second Focal Plane. • m = 0 then from (ii), α = m α So α independent of y. Parallel input rays produce parallel output rays in another direction. is the angular magnification. This is called a “telescopic system. ” 11 f f 12 0 0 0 21 f f 0 22 0
Physical significances of system Matrix • If m = 0 then from (i) y = m y • Independent of α. Case of conjugate planes. Input plane is object plane and output plane is image plane. • is the linear magnification 12 f 0 11 0
Location of Cardinal Points
Location of Cardinal Points • P is towards the left and using paraxial approx. • p gives the location of the first focus relative to the input plane
Location of Cardinal Points • From the figure r = - (f 1 – p) …(iv) the location of the first principal plane relative to input plane.
Location of Cardinal Points v is the location of the nodal plane relative to the input plane
Cardinal point locations in terms of system matrix elements
Useful Generalizations (1) r – s = v – w • Note : If then and so (1) r = v and s = w ie Principal and Nodal points coincide (2) are equal in magnitude
Two thin lenses in air • Let input and output planes be located at lenses’ centre • System matrix involves (1) refraction by f (2) translation through d (3) refraction by f 1 2
• i. e. 1 st principal point and 1 st nodal point coincide • Also 2 nd principal point and 2 nd nodal point coincide
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