Chapter 12 Optical Instruments Physics Beyond 2000

Geometric Optics In this chapter, the lenses and mirrors have dimension much longer than the wavelength of light. Effect of diffraction can be ignored. Light is regarded as ray.

Reflection Laws of reflection.

Plane mirror What are the properties of the image? Note that we cannot capture a virtual image on a screen.

Locate a virtual image Method of no parallax Use a long search pin to locate the image behind the mirror. In front of the mirror, view the image in the mirror and the search pin. search pin object plane mirror image

Locate a virtual image Method of no parallax If the search pin is at the exact position of the image, the image in the mirror and the search pin always coincide even if we change the angle of view. search pin image in the mirror eye

Locate a real image The method of no parallax can be applied to locate the position of real images. O I search pin real image

Rotation of a plane mirror Rotate the plane mirror from position 1 to position 2 by an angle θ. The reflected ray will turn through 2θ. Normal 2 Reflected ray 2 2θ θ Fixed incident ray Reflected ray 1 Normal 1 Mirror 1 Mirror 2

Rotating a plane mirror Light-beam galvanometer.

A moving mirror If the plane mirror moves at a speed v, the image moves at speed 2v. fixed object image 1 v 2v image 2 position 1position 2

Concave mirrors lens_e.htmlhttp:// lens_e.html

Spherical aberration If the aperture of the mirror is large, the reflected rays do not all passes through the focus. This is called spherical aberration.

Spherical aberration It can be corrected by using a parabolic mirror.

Focal length f and radius of curvature R Show that r = 2f for small angles. θ θ CF r f 2θ θ h paraxial ray

Images for concave mirror

Example 1 The rays from the sun are parallel and the image of the sun is on the focal plane. C F θ θ h f

Mirror formula for small angles θ θ CF r v O u α β γ I h

Mirror formula for small angles Nature of object/image Object distance u Image distance v Realpositive Virtualnegative Real-is-positive convention:

Mirror formula for small angles Nature of mirrorfocal length f concavepositive convexnegative Real-is-positive convention:

Linear magnification m CF v O u I

Example 2 Justify the nature of the image from the sign of image distance v.

Variation of image with object distance

Variation of magnification with object distance

Convex mirror for small angles Nature of mirror focal length f object distance u image distance v convexnegativepositivenegative (virtual)

Convex mirror

Example 3 F v O u I I

Measure the focal length of concave mirror Object at infinity. Image is at the focal plane. Measure the distance between the mirror and the screen. Method A C F θ θ h f

Measure the focal length of concave mirror Object at the radius of curvature. Image is at the radius of curvature. Use the method of no parallax to locate the image. Adjust the position of the object so that its image is coincide with the object. Measure the distance between the object and the mirror. Method B

Measure the focal length of concave mirror Object at different positions to produce real images. Images are captured by a screen. Method C 0

Measure the focal length of concave mirror Object at different positions to produce real images. Images are captured by a screen. Calculate the linear magnification m. Method D 0 m v slope =

Measure the focal length f m of a convex mirror It is not possible to capture a virtual image on a screen. Put a converging lens of focal length f lens in front of the convex mirror. Adjust the position of the object so that a real image is at the same position as the object. O I lens mirror C PQ f lens 2.f m s

Measure the focal length f m of a convex mirror O I lens mirror C PQ f lens 2.f m s Measure s, the separation between the lens and the convex lens. 2  focal length of the convex mirror is f lens – s.

Refraction Medium 1:  1, 1, c 1 and n 1 Medium 2:  2, 2, c 2 and n 2 11 22

Refraction Medium 1:  1, 1, c 1 and n 1 Medium 2:  2, 2, c 2 and n 2 11 22 Snell’s law: n 1.sin  1 = n 2.sin  2

Refraction Medium 1:  1, 1, c 1 and n 1 Medium 2:  2, 2, c 2 and n 2 11 22 If n 2 > n 1, then medium 2 is an optically denser medium and medium 1 is an optically less dense medium

Total internal reflection This occurs when light travels from an optically denser medium to an optically less dense medium and the angle of incidence > critical angle c. medium 1 medium 2 refraction with  1 < c. total internal reflection with  3 > c. critical case with  2 = c. 11 22 33

Total internal reflection medium 1 of n 1 medium 2 of n 2 refraction with  1 < c. total internal reflection with  3 > c. critical case with  2 = c. 11 22 33 The critical angle c is given by

Fish-eye view a/fishEye/fishEye.htmlhttp:// a/fishEye/fishEye.html

Example 4 The critical angle of glass with n = 1.5 is about 42 o in air. It depends also on the medium in which the glass is immersed.

Reflecting prism Angle of incidence = 45 o > Critical angle = 42 o. Total internal reflection occurs inside the glass prism. The glass prism can be used as a reflecting mirror. 45 o

Optical fibre There is total internal reflection inside the optical fibre. Light is guided to travel in the optical fibre.

Real depth and apparent depth The image I is displaced upwards relative to the object O. O I apparent depth real depth air medium with refractive index n BC

Real depth and apparent depth for small angles. O I apparent depth real depth air medium with refractive index n BC

Real depth and apparent depth air medium with refractive index n Where would be the image if we are inside the medium? Suppose that the angles are small. O I BC

Measure the refractive index of glass O1O1 O1O1 I O2O2 h1h1 h2h2 Find the real depth and apparent depth from h 1 and h 2. real depth = h 2 apparent depth = h 2 – h 2 glass block h h h travelling microscope eye

Rectangular glass block The incident ray and the emergent ray are parallel. The lateral displacement is i r r i d w incident ray emergent ray

Prism Find the angle of deviation D in terms of angles of incidence (  1 and  2 ) and angles of refraction (  1 and  2 ). D = (  1 -  1 )+(  2 -  2 ) A D 11 22 11 22

Prism Find the refracting angle A of the prism in terms of the angles of incidence and angles of refraction. A =  1 +  2 A D 11 22 11 22

Prism The angle of deviation is a minimum D min when the light ray is symmetrical. Find the refractive index n of the glass prism. A D 11 22 11 22

Small-angled prism For a prism with small refracting angle A. The angle of deviation D = (n - 1).A The angle of deviation is independent of the angle of incidence. D A

Convex lenses F’ F F The focal lengths on both sides are equal. f f

Aberration of lens Spherical aberration. Parallel incident rays far from the centre do not meet at the same focus as paraxial rays.

Aberration of lens Chromatic aberration. Violent light bends more than red light in glass.  f violet < f red white parallel light

Location of image for convex lens Ray diagrams.

Lens formula for thin lens. Nature of object/image Object distance u Image distance v Realpositive Virtualnegative Real-is-positive convention:

Lens formula for thin lens. Nature of lensfocal length f convexpositive concavenegative Real-is-positive convention:

Lens formula: proof D F f B P The angle of deviation D = (n – 1).A where A is the refracting angle Note that for small angle prism (thin lens), D is independent of the angle of incidence.

Lens formula: proof for thin lens. Suppose that there is a real image. Prove that O I D   u v B P F F’ f

Interchange of locations of object and real image O O I I u u v v

Object-image distance for real image To produce a real image, the object-image distance d must be longer than or equal to 2.f. Prove it. O I d

Thin lenses in contact Let f 1 be the focal length of the first thin lens and f 2 be the focal length of the second thin lens. When they are in contact, the combined focal length f is given by

Example 5 The image formed by the first lens is the object of the second lens.

The converging power of a lens Definition of converging power P = Unit: dipotre (D) For two thin lenses in contact, the combined power is D 1 + D 2.

Concave lens For a real object, virtual image is always formed.  v is negative. Focal length is negative.

Example 6 I 1 is the virtual object of the concave lens. u is negative  The concave lens produces a real image I 2. v is positive. I2I2 I1I1 convex lens concave lens v u

Measure the focal length of convex lens Method A. Object at infinity  Image is at the focal plane. f I focal plane parallel rays from distant object

Measure the focal length of convex lens Method B. Place the convex lens on a plane mirror. Adjust the position of the object so that it coincides with the image. This is method no parallax. eye object image convex lens plane mirror O

Measure the focal length of convex lens Method B. The distance between the object/image and the lens = f. O f I

Measure the focal length of convex lens Method C. Produce different real images. Measure the object distance u and the real image distance v.

Measure the focal length of convex lens Method D. Without changing the positions of the object and the image, find the two possible positions of the lens. Measure a and d.and find f from O Screen position of image a d object 1 st position of lens 2 nd position of lens

Measure the focal length of concave lens Use another convex lens to help producing a virtual object for the concave lens. Use the lens formula to calculate the focal length of the concave lens. I2I2 I1I1 O u v

Optical instruments Human eye: a convex lens with variable focal length. Far point of a normal eye = infinity. Near point of a normal eye = 25 cm. Least distance of distinct vision = 25 cm.

Short-sightedness A short-sighted eye can focus objects in the range from 25 cm to 200 cm. The far point of the short-sighted eye is 200 cm Find the focal length of the spectacle to correct the defect. Wearing a pair of spectacles, what is the new near point?

Long-sightedness A long-sighted eye can focus objects in the range from 200 cm to infinity. The near point of the long-sighted eye is 200 cm Find the focal length of the spectacle to correct the defect. Wearing a pair of spectacles, what is the new far point?

Visual angle Visual angle  of an object is the angle subtended by the object at the eye. The bigger the visual angle, the bigger the apparent size of the object. Most optical instruments are designed to magnify the visual angle. object eye 

Angular magnification M  is the visual angle of the final image  is the visual angle of the object Note that the visual angles are usually small so   tan   sin  and   tan   sin . It is used to measure the magnification of an optical instrument,

Normal adjustment An optical instrument is in normal adjustment when it forms the final image at a position which the user expects to see. Telescope: final image at infinity (far point). Magnifying glass: final image at 25 cm (near point). Microscope: final image at 25 cm (near point).

Magnifying glass A magnifying glass is a convex lens. It produces an enlarged virtual image.

Magnifying glass Without the magnifying glass, the largest visual angle of the object is  with the object at the least distance of distinct vision D = 25 cm. object eye  D h visual angle without optical instrument

Magnifying glass With the magnifying glass in normal adjustment, the final image is also at D. object eye v = D h image  visual angle with the optical instrument u

Magnifying glass Apply lens formula, (1) (2) (3) the angular magnification of a magnifying glass With the visual angles,

Compound microscope It is used to view small objects. It consists of two convex lenses. The objective lens and the eye-piece. Both are of short focal lengths. objective lens eye-piece object eye History of compound microscope:

Compound microscope The objective lens produces a magnified real image. The object is placed near the focus of the objective lens. This image is the object of the eye-piece. objective lens eye-piece object eye 1 st image

Compound microscope The eye-piece is a magnifying glass. It produces a magnified virtual image at D = 25 cm from the eye-piece. objective lens eye-piece object eye 1 st image final image D Note that the final image is an inverted image.

Compound microscope eye D h object  objective lens eye-piece object eye 1 st image final image D  h h1h1 h2h2

Compound microscope where m e is the linear magnification of the eye-piece and m o is the linear magnification of the objective The angular magnification M of a compound microscope is M can be increased by using lenses of short focal lengths.

Example 7 The angular magnification = 5.5

Refracting telescope It is used to view distant objects e.g. stars. Two convex lenses. The objective lens: Pointing to the object, with very long focal length. The eyepiece: A magnifying glass. Its focal length is short eye Objective lens Eyepiece

Refracting telescope The object is at infinity. The incident rays are parallel The objective lens produces a real image I 1 on its focal plane. eye Objective lens Eyepiece real image I1I1 fofo

Refracting telescope The first image I 1 is the object of the eyepiece. The eyepiece produces a virtual image at infinity. This is the normal adjustment. eye Objective lens Eyepiece real image I1I1 fofo fefe Image at infinity

Refracting telescope In normal adjustment, the angular magnification is

Refracting telescope The length of the refractive telescope is f o + f e The image is inverted. eye Objective lens Eyepiece real image I1I1 fofo fefe Image at infinity

Refracting telescope The aperture of the objective lens is large –to collect more light. –to reduce the diffraction effect.

Example 8 Note the focal lengths of the lenses.

Eye ring Eye ring is the position of the eye, at which most light enters the eye when using an optical instrument. The image is brightest when the eye is at the eye ring.

Eye ring All the light collected by the objective lens passes through the position of the eye ring. All the light enters your eye if you place your eye at the eye ring.

Locating the eye ring As the light comes from the objective lens, you may take the objective lens as the object. The position of the eye ring is the position of the image. Find the position of the eye ring from lens formula. where u is the distance from the objective lens to the eyepiece, f is the focal length of the eyepiece and v is the distance from the eye ring to the eyepiece.

Locating the eye ring The position of the eye ring is

Reflecting telescope It is similar to a refracting telescope. Light is collected by a concave mirror. F objective (concave mirror) plane mirror eye piece F’ eye

Reflecting telescope The angular magnification is F objective (concave mirror) plane mirror eye piece F’ eye

Reflecting telescope Advantages of using a reflecting telescope: The mirror reduces less light intensity than a lens. There is not any problem of chromatic aberration and spherical aberration. It is easier to produce a large mirror than a large lens.

Example 9 Locating the eye ring.

Spectrometer It is used for spectral analysis with the aids of a diffraction grating or a glass prism. It consists of three parts: collimator, diffraction grating/glass prism and a telescope.

Spectrometer Collimator: Place the source S near the slit. The collimator produces a parallel beam of light. S collimator

Spectrometer Diffraction grating: Use a diffraction grating to produce a spectrum of fringes. The diffraction grating is on a turntable to measure the angles of bright fringes. S collimator m = 0 m = 1 m = 2 diffraction grating turntable

Spectrometer Diffraction grating: Use a diffraction grating to produce a spectrum of fringes. The diffraction grating is on a turntable to measure the angles of diffraction. S collimator m = 0 m = 1 m = 2 diffraction grating turntable

Spectrometer Telescope: It can be rotated to any angle to view the diffraction spectrum. S collimator m = 0 m = 1 m = 2 diffraction grating turntable telescope eye

Spectrometer Telescope: the image is formed at the position of cross-wire. light from diffraction grating cross-wire objective lens eye piece eye

Spectrometer It is necessary to make a horizontal turntable. Adjust the levelling screws and use a spirit level to check the turntable. levelling screw turntable spirit level