Nomenclature

Principal axisOptical centreNomenclature

Sign convention

All distances on the principal axis are measured from the optical centre. All distances measured in the direction of incident rays are positive All the distances measured in the direction opposite to that of the incident rays are negative. All distances measured above the principal axis are positive. Thus, height of an object and that of an erect image are positiveall distances measured below the principal axis are negative

All distances measured in the direction of incident rays are positive Incident ray directionDistances measured in the apposite direction of incident rays are negative+ ve direction+ ve direction- ve directionAll distances are measured from the optical centre & principal axis

All distances are measured from the optical centre & principal axis

- ve directiondistances measured above the principal axis are positive.

Let AB be a object placed at right angles to the principal axis at a distance > focal length .The image A1B1 is formed between O and F1 on the same side as the object is kept The image is erect and virtual. fuPrincipal axis

OF1 = Focal length = - f OA = CB = Object distance = - uOA1 = Image distance = -v vfu(-ve)(- ve)(-ve)

v

vFrom diagram at OC = AB ----- 2

----- 2AndFrom

Now

Dividing throughout by uvf

Convex lens formula

Let AB be an object placed at right angles to the principal axis at a distance greater than the focal length f of the convex lens. The image A1B1 is formed beyond 2F2 and is real and inverted. B1

OA = Object distance = - uOA1 = Image distance = + v OF2 = Focal length = + f

B1

Triangle OAB andOA1B1 are similar B1

Triangle OAB andOA1B1 are similar B1But OC = AB

Triangle OAB andOA1B1 are similar B1And

Triangle OAB andOA1B1 are similar B1Dividing equation throughout by uvf

Mirror formula

Since all distances are measured from P CB = PB PC And CB = PC PBNow

Using the sign convention,PB = - U (object distance) PC = - RPB = - V (Image distance) On substitutionUV - VR = UR- UVUR + VR = 2UV

UR + VR = 2UV Dividing both sides by UVR, we have

For ALL LensesFor ALL mirrorsU= object distance , V = image distance

Linear magnification is ratio of the size of the image to the size of the object.

Note:If m>1, image is enlarged.If m

Refraction

Principle of reversibility of lightThe path of light is reversible.

According to Snell's law : Ration of sin i / sin r is constant for a frequency & two mediums

Refractive index of medium 2 with respect to medium 1Light travels from medium 1 to medium 2

It can be also proved that=

According to the principle of reversibility&or

From the above relation it is clear that the refractive index of the second medium with respect to the first medium is the reciprocal of the refractive index of the first medium with respect to the second.

If the medium 1 is air or vacuum, the refractive index of medium 2 is referred to as the absolute refractive index. airglass = Refractive index of glass W.R.T. air/vacuum = Vc / VgVacuumGlassglass = Vc / Vg

Similarly, when light travels from air /vacuum to medium 1 & 2, we can writeDividing equation (ii) by equation (i) we geti

Thus the relative refractive index between a pair of media is the ratio of their absolute refractive indices. While the absolute refractive index of any material medium is always greater than unity, its relative refractive index may be greater or lesser than unity.

In general, if a ray of light is passing from medium 1 to medium 2, thenThe refractive index depends on the following factors:the nature of the mediums the color or wave length of the incident light

Different colors have different wavelength,Their 's are also different.Red light has a longer wavelength than violet So red < violet. This means violet deviates by a larger angle than red light. This is one of the reasons to have red light signals

Cause of DispersionThe refractive index of prism depends on wavelength according to cauchys relation namely.

Relation of to is complex depending on factors like crystalline structure , moleculer structure etc

Fundamental flaw in lens images:They are seen predominantly when we enlarge image as in microscope when Chromatic aberrationSpherical aberrationField Curvature Astigmatism aberrationsGeometrically distorted image etc

Chromatic aberration : Splitting of colors in lens

Splitting of colors in lens

Chromatic aberration : Colours in photo

Chromatic aberration during refraction

Spherical aberration

Spherical aberration

Multiple focusIdial focus for ideal thin lens

Ideal image without spherical abrasion

Distorted image with spherical abrasion

Ideal image without spherical abrasionGood imageDistorted image

Aberration correctedAberration present

Field Curvature

object on principle axisoffset object positionMicroscope imageClear imageDistortedimageAstigmatism aberrations

Correct imagedistorted imageGeometrical distortions in image

Human eye has a substantial amount of chromatic aberration. Fortunately, we are able to compensate for this effect when the brain processes images,but it is possible to demonstrate the aberration using a small purple dot on a piece of paper. When held close to the eye, the purple dot will appear blue at the centre surrounded by a red halo. As the paper is moved farther away, the dot will appear red surrounded by a blue halo.

Combination of lenses

Combination of lenses

lens errors are corrected byCombination of different type of lensLenses of different materials

Combination of crown glass and flint glassThey have different refractive index the blue rays and the red rays can get a common focus.This combination is termed a lens doublet or achromatic lenses or achromats This is the most widely used lens and is commonly found on laboratory microscopes.But all colours are not corrected

Chromatic aberration effect can be minimized by using combination of lenses of different materials (Called Achromatic doublet)

Aromatic doublet

Aromatic doublet

Chromatic aberration correcting triplet:

Combination of lens thickness, curvature, refractive index, reduce chromatic aberration by bringing two colours at common focus. If fluorspar + glass lens or a fluorspar lens is used, then 3 colours red, green, and blue can be brought into a single focus minimising chromatic aberrationFluorspar (halide mineral composed of calcium fluoride, CaF2).Fluorspar due to their strong hexagonal crystal structure show even refraction for all colours light.

Apochromatic lenses They contains two lens doublets and a lens triplet for correction of both chromatic and spherical aberrationsThey are used to build high-quality chromatic aberration-free microscope

Calculation of focal length for combination lens

VIOUvITwo thin lenses L1 and L2 of focal lengths f1 and f2 be placed in contact so as to have a common principal axis. Let O be a point object on the principal axis. The refractions by two thin lenses L1 and L2 of focal lengths f1 and f2 be placed in contact so as to have a common principal axis.

Converging lens

Lens formula derivation

Concave lens formula

VIOUvIPoint of contact of two lensesSince the lenses are thin, as approximation all distances can measured from the centre of the lens system (point of contact of two lenses

VThe image I' due to the first lens L1 acts on the virtual object for the second lens.L2 Let the final image be formed at I, at a distance v from the centre of the lens system. Writing the lens equation separately, IOUvI

For L1, O = object, U = Object distance (-ve) I = Image V = Image distance (+ve) For L2, I = object, V = Object distance I = Image V = Image distance Finalimage

For L1: U is - ve, V is +ve , f1 is +ve Putting these values in above equation we get

For L2: Object distance U = V (+ve) Image distance = V (+ve) Focal length = f1 (+ve)Virtual object for L2Ray direction +ve Putting these values in above lens equation ------- (II)

Adding I & II&'

Let us replace two lenses by a single lens with focus f giving the same effect i.e., for an object at O forms image at V. Such a lens is called an equivalent lens and its focal length is called the equivalent focal length. Then

Power of lens is defined as P = 1/ f. where f is expressed in metersIt is called diopter & lts units are m-1Ex A lens with a focal length of -20 cm has a power of-5.0 diopters. Powers are often used to describe eye glass lenses Power of combination of two thin lens is P = P1+P2 , where P1 & P2 are power of individual lens

Other refracting surfaces

Convex Spherical Refracting Surface Parts of Sphere of transparent refracting materialConvex refractingsurfaceP = poleC = center of curvature= Absolute refractive index

Concave Spherical Refracting Surface Parts of Sphere of refracting materialConcave refractingsurfaceP = poleC = center of curvatureAbsolute refractive index

Assumption: In dealing with refraction at spherical refracting surfaceThe object to be a point lying on the principal axis of the spherical refracting surface. The aperture (0pening for light) of the spherical refracting surface is small compared to radius of curvature. The incident and refracted rays make small angles with the principal axis of the surfacesign convention remain same

Formula for Refraction : Rarer to Denser Medium (Convex Spherical Surface21uvR

Formula for Refraction : Rarer to Denser Medium (Concave Spherical Surface

To make a lens of given focal length we need relation betweenFocal length, Radii of curvaturesRefractive index of the lens materialRefractive index of surroundings.Such a relation is called lens maker formula Lens maker formula

Flowing assumptions made for the derivation of these formula:The lens is thin, so that distances measured from the poles of its surfaces are almost equal as equal to the distances from the optical centre of the lens. The aperture of the lens is small. Point object is considered.Lens maker formula

Lens maker formula

Lens maker formula

END

XY=spherical surface.i = incident angleOA incident rayObjectImageir = angle of refractionrAI refracted ray

irSum of exterior angles = interior anglesRemember lim of sin / =1 as tends to 0 For small angles sin & Tan Sin

irSum of exterior angles = interior angles

ir

As the aperture is close, M is close to P. Using the sign convention, we putPO = -u , PI = +v, PC = Ror

For the virtual image, the point lies close to the pole of refracting surface. In this case the refracted rays PC and AB do not meet actually at any point but appear to come from a point I as shown below.

Refraction from denser to rarer medium at a concave spherical refracting surface

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