Chapter 12 Optical Instruments Physics Beyond 2000

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.

http://www.phy.ntnu.edu.tw/demolab/index.html

http://webphysics.davidson.edu/physletprob/ch18_v4_physlets/optics4/default.html

Reflection

• Laws of reflection.

http://www.netzmedien.de/software/download/java/brechung/

Plane mirror

• What are the properties of the image?

Note that we cannot capture a virtual image on a screen.

http://www.continental.clara.net/physics/lt31.htm

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 parallaxIf 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 mirroreye 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

• http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/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.

θθ

C Fr

f

2θθ

h

paraxial ray

Images for concave mirror

http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html

Example 1• The rays from the sun are parallel and the

image of the sun is on the focal plane.

C Fθθ

h

f

Mirror formula

fvu

111 for small angles

θ θ

C Fr

v

O

u

α β γI

h

Mirror formula

fvu

111 for small angles

Nature of object/image

Object distance

u

Image distance

v

Real positive positive

Virtual negative negative

Real-is-positive convention:

Mirror formula

fvu

111 for small angles

Nature of mirror focal length f

concave positive

convex negative

Real-is-positive convention:

Linear magnification m

u

v

objectofheight

imageofheightm

C F

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

http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html

Variation of magnification with object distance

fu

fm

Convex mirror

fvu

111 for small angles

Nature of mirror

focal length f

object distance u

image distance v

convex negative positive negative

(virtual)

Convex mirror

http://www.iln.net/html_p/c/453262/453270/453373/454123/56652_2079292.asp

http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html

Example 3

F

v

O

u

II

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 Fθθ

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

http://www.usafa.af.mil/dfp/physics/webphysics/Physlet_examples/concave_mirror_f.html

Measure the focal length of concave mirror

•Object at different positions to produce real images.•Images are captured by a screen.

Method C

v

1

u

1

f

1

f

10

fvu

111

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

-1

1.1

vf

m

slope = f

1

Measure the focal length fm of a convex mirror

• It is not possible to capture a virtual image on a screen.• Put a converging lens of focal length flens in front of the conv

ex 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

P Q

flens

2.fm

s

Measure the focal length fm of a convex mirror

O

I

lens mirror

C

P Q

flens

2.fm

s

• Measure s, the separation between the lens and the convex lens.

• 2 focal length of the convex mirror is flens – s.

Refraction

1

2

2

1

2

1

2

1

sin

sin

n

n

c

c

Medium 1:1, 1, c1 and n1

Medium 2:2, 2, c2 and n2

1

2

http://www.netzmedien.de/software/download/java/brechung/

http://www.fed.cuhk.edu.hk/sci_lab/download/project/Lightrefraction/LightRefract.html

Refraction

Medium 1:1, 1, c1 and n1

Medium 2:2, 2, c2 and n2

1

2

Snell’s law:n1.sin1 = n2.sin2

Refraction

Medium 1:1, 1, c1 and n1

Medium 2:2, 2, c2 and n2

1

2

If n2 > n1, then medium 2 is an optically denser medium and medium 1 is an optically less dense medium

Total internal reflectionThis occurs when light travels from an optically densermedium to an optically less dense medium and the angle of incidence > critical angle c.

medium 1

medium 2

refraction with 1 < c.

total internalreflection with 3 > c.

critical case with 2 = c.

12 3

http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/light/flashLight.html

Total internal reflection

medium 1 of n1

medium 2 of n2

refraction with 1 < c.

total internalreflection with 3 > c.

critical case with 2 = c.

12 3

The critical angle c is given by 1

2sinn

nc

http://www.continental.clara.net/physics/lt23.htm

Fish-eye view

• http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/fishEye/fishEye.html

Example 4

• The critical angle of glass with n = 1.5 is about 42o in air.

• It depends also on the medium in which the glass is immersed.

Reflecting prism• Angle of incidence = 45o > Critical angle = 42o.• Total internal reflection occurs inside the glass

prism.• The glass prism can be used as a reflecting mirror.

45o

45o

45o

45o

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

Iapparentdepth

real depth

air

medium withrefractive indexn

B C

Real depth and apparent depth

depthapparent

depthrealn for small angles.

O

Iapparentdepth

real depth

air

medium withrefractive indexn

B C

Real depth and apparent depth

air

medium withrefractive indexn

Where would be the imageif we are inside the medium?Suppose that the anglesare small.

O

I

B C

Measure the refractive index of glass

O1 O1

IO2

h1

h2

Find the real depth and apparent depth from h1 and h2.

real depth = h2

apparent depth = h2 – h2

glass blockhh h

travelling microscope

eye

Rectangular glass blockThe incident ray and the emergent ray are parallel.The lateral displacement is

r

riwd

cos

)sin(.

ir

ri

d

wincident ray

emergent ray

PrismFind the angle of deviation D in terms of angles ofincidence (1and 2) and angles of refraction (1 and 2).

D = (1 - 1)+(2 - 2)

A

D1

212

PrismFind the refracting angle A of the prism in terms of theangles of incidence and angles of refraction.

A = 1+ 2

A

D1

212

PrismThe angle of deviation is a minimum Dmin when thelight ray is symmetrical.Find the refractive index n of the glass prism.

2sin

)(21

sin min

A

DAn

A

D1

212

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 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.

http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/thickLens/thickLens.html

Aberration of lens• Chromatic aberration.• Violent light bends more than red light in glass. fviolet < fred

white parallel light

Location of image for convex lens

http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html

Ray diagrams.

Lens formula

fvu

111 for thin lens.

Nature of object/image

Object distance

u

Image distance

v

Real positive positive

Virtual negative negative

Real-is-positive convention:

Lens formula

fvu

111 for thin lens.

Nature of lens focal length f

convex positive

concave negative

Real-is-positive convention:

Lens formula: proof

DF

f

B

P

The angle of deviation D = (n – 1).Af

BP

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

fvu

111 for thin lens.

Suppose that there is a real image.

Prove that

O I

D

u v

B

PFF’

f

Interchange of locations of object and real image

O

O

I

I

u

u

v

v fvu

111

Object-image distancefor 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

http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/thinLens/thinLens.html

• Let f1 be the focal length of the first thin lens andf2 be the focal length of the second thin lens .• When they are in contact, the combined focal lengthf is given by

21

111

fff

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 = f

1Unit: dipotre (D)

• For two thin lenses in contact, the combined power isD1 + D2.

Concave lens

• For a real object, virtual image is always formed. v is negative.

• Focal length is negative.

Example 6I1 is the virtual object of the concave lens. u is negative

The concave lens produces a real image I2. v is positive.

I2I1

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 fromdistant 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 coincideswith the image. This is method no parallax.

eye

object image

convex lensplane 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.

v

1

u

1

f

1

f

1

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

22

41

ad

d

f

O

Screenpositionof image

ad

object

1st positionof lens

2nd positionof 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.

I2I1O

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.

objecteye

Angular magnification M

is the visual angle of the final image is the visual angle of the object

M

Note that the visual angles are usually small so tan sin and tan sin .

It is used to measure the magnification of an opticalinstrument,

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.

objecteye

D

h

D

h tan 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

u

h tan visual angle with the optical instrument

u

Magnifying glass

Apply lens formula,fDu

111

(1)

D

h (2)

u

h (3)

1f

DM

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

objecteye

History of compound microscope:http://www.utmem.edu/~thjones/hist/hist_mic.htm

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

objecteye

1st 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

objecteye

1st image

finalimage

D

Note that the final image is an inverted image.

Compound microscope

eyeD

h

object

objective lens eye-piece

objecteye

1st image

finalimage

D

h

h1

h2

D

h

D

h2

Compound microscope

oe mmh

h

h

h

h

hM .. 1

1

22

where me is the linear magnification of the eye-pieceand mo 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 I1 on its focal plane.

eye

Objective lens Eyepiecereal image

I1

fo

Refracting telescope• The first image I1 is the object of the

eyepiece.

• The eyepiece produces a virtual image at infinity. This is the normal adjustment.

eye

Objective lens Eyepiecereal image

I1

fo

fe

Image at infinity

Refracting telescopeIn normal adjustment, the angular magnification is

ef

fM 0

Refracting telescope• The length of the refractive telescope is fo +

fe

• The image is inverted.

eye

Objective lens Eyepiecereal image

I1

fo

fe

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 ringAll the light collected by the objective lens passes throughthe position of the eye ring.All the light enters your eye if you place your eye at theeye 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.

vuf

111

where u is the distance from theobjective lens to the eyepiece,f is the focal length of the eyepieceand v is the distance from the eye ringto the eyepiece.

Locating the eye ring

The position of the eye ring is

e

e

fL

fLd

.

Reflecting telescope• It is similar to a refracting telescope.

• Light is collected by a concave mirror.

Fobjective(concave mirror)

plane mirror

eye piece

F’

eye

Reflecting telescope

• The angular magnification is

Fobjective(concave mirror)

plane mirror

eye piece

F’

eye

e

o

f

fM

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 = 1

m = 2

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 = 1

m = 2

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 = 1

m = 2

diffraction grating

turntable

telescope

eye

SpectrometerTelescope: the image is formed at the position of cross-wire.

light fromdiffraction grating cross-wire

objective lenseye 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

turntablespirit level