Coherent Interference Intensity

22

21 )()( IIIT

Huygens’ Principle

Section 25.4

Double Slit Analysis

Section 25.5

Constructive interferenced sin θ = m λ

Destructive interferenced sin θ = (m + ½) λ

Single-Slit AnalysisDestructive interference

w sin θ = ±m λ

Section 25.6

Diffraction Grating

ΔL = d sin θ = m λ

Section 25.7

Rayleigh Criterion

D1.22

Section 25.8

Applications of Optics

Chapter 26

Applications of OpticsMany devices are based on the principles of opticsEyeglasses around 1200s

Perhaps the oldest optical instrumentMicroscopes and telescopes around 1600CDs and DVDs around 1980sAlso improvements to devices have been made

Applications of a Single LensThe eye can be modeled as a single lens with a

focal length ƒeye

Eyeglasses and contact lenses add a lens in front of the eye

A magnifying glass is also a single lens

Section 26.1

Normal EyeLight emanating from a

point on the object is focused to a corresponding point on the retina

The near-point distance, sN, is the closest distance an object can be that you can focus (~25 cm)Objects nearer than the

near-point cannot be focused on the retina

Section 26.1

Normal Eye, cont.The normal eye can also focus on objects that are

very far aways ~ ∞

The eye must adjust its focal length to values between sN and ∞Does this by using muscles that deform and change

the shape of the eye’s lensNeeds to change from about 2.3 cm to 2.5 cm

Section 26.1

Glasses and Contact LensesGlasses or contact lenses are lenses placed in front

of the eyeAlong with the eye, these form a system of lenses

One lens from the eye and one from the glasses or contact

Systems of lenses contain two or more lensesThe same analysis idea will be applied to

telescopes, microscopes and other optical instruments

Section 26.1

Analysis for a System with Two or More LensesDraw a picture showing the object of interest and the

lenses in the problemUse the rules for ray tracing along with the thin-lens

equations to find the location and magnification produced by the first lens in the system

The image produced by the first lens then acts as the object of the second lens in the system

Use the rules for ray tracing and the thin-lens equations a second time to find the location and magnification produced by the second lens in the system

Section 26.1

Far-Sighted VisionThe near-point distance

is greater than for a normal eye

Objects located closer than the near-point distance cannot be focused

To compensate, a lens can be placed in front of the eye

Section 26.1

Far-Sighted CorrectionThe contact (or glasses)

lens is the first lens in the system

For example, if a person’s near-point distance is 75 cm, the corrective lens needs to be a converging lens with ƒlens = 38 cm

If the person’s near-point distance is greater than 75 cm, the focal length of the corrective lens needs to be shorter

Section 26.1

DioptersThe strength of a lens is sometimes measured in

terms of its refractive power

Units are m-1 which is called a diopterFor example, the lens with ƒ = 38 cm will have a

refractive power of 2.7 diopters

lens

refractive power1

ƒ

Section 26.1

Near-Sighted VisionA nearsighted person is

unable to focus light from distant objects on the retina

The incoming rays from an object very far away are approximately parallel to the axis (at infinity)

A nearsighted eye produces an image in front of the retina

Near-Sighted CorrectionThe object at ∞ needs

to focus on the retinaFor example, if the

person can focus objects within 2.0 m, the corrective lens needs to be a diverging lens with ƒlens = -2.0 m

GlassesThe eyeglass lens is a

short distance in front of the eyeInstead of touching it as

with the contact lensThe distance must be

taken into account This generally makes the

focal length of the eyeglasses about 10% shorter than a contact lens

Section 26.1

Magnifying GlassThe simplest magnifying glass is a single lensAgain it can be considered a system of two lenses

The magnifying lens and the eyeThe goal is to produce a greatly magnified image at

the retinaWant the image on the retina to be as large as

possibleAnalysis is similar to that for contact lenses or

eyeglasses

Section 26.1

Magnifying Glass, cont.The largest clearly

focused image for the unaided eye results when the object is at the near point

The object’s apparent size when it is located at the near point can be measured using the angle θ

Section 26.1

Image Properties with a Magnifying GlassThe object is positioned inside the focal length of

this lensThis position of the lens produces an upright virtual

image at a point farther from the eyeThe eye perceives the light as emanating from this

virtual imageThe image angle with the magnifying glass is greater

than the image angle for the eye aloneThe image on the retina is enlarged by the

magnifying glassSection 26.1

Angular MagnificationThe enlargement of the image on the retina is given

by the angular magnification mθ

From geometry and the small angle approximations,

The angular magnification of a typical magnifying glass is usually 10 or 20

Mm

i N N N

o o

h s s sm

h s1

ƒ ƒ

Section 26.1

MicroscopesLenses with focal lengths less than a few mm are difficult

to makeThere is a practical limit to the magnification of a single

lensA more useful way to achieve higher magnification is

using two lenses arranged as a compound microscopeThe image produced by one lens is used as the object of

the second lensThe image produced by the second lens is then viewed

by the eyeThe total magnification is the product of the

magnifications of the two lenses

Section 26.2

Compound Microscope

The two lenses are called the objective and the eyepieceTo analyze the image produced first apply ray tracing and the

thin-lens equation to find the image produced by the objective lens

This image acts as the object for the eyepieceThe image produced by the eyepiece is viewed by the eye

Section 26.2

Compound Microscope, cont.The distance between the objective lens and the

original object is adjusted so that the image produced by the objective falls at the focal point of the eyepiece

This gives a final virtual image for the observerThe linear magnification of the objective lens is

i iobj

o obj

h sm

h ƒ

Section 26.2

Compound Microscope, MagnificationThe total magnification of the microscope is the

product of the linear magnification of the objective and the angular magnification of the eyepiece

The negative sign indicates that the image is inverted

i Ntotal obj eyepiece

obj eyepiece

s sm m m , ƒ ƒ

Section 26.2

Advances in Microscope DesignThe index of refraction of the glass used to make the

lenses is slightly different for light of different colorsThis makes the focal length slightly different for

different colorsThis affects the focusing properties of a microscopeCalled chromatic aberration

Chromatic aberration can be corrected by using an achromatic lensThis is a lens composed of different types of glass with

different indices of refraction which approximately cancels the aberrations

Section 26.2

Resolution of a MicroscopeThere is a fundamental limit to the resolution that

can be achieved with any microscope that relies on focusing

This limit is due to the diffraction of light passing through the aperture of the microscopeDiffraction prevents the size of the focused spot from

being less than a value approximately equal to the wavelength of the light

Section 26.2

Resolution, cont.It is possible to resolve

the outgoing light from two features only if they are separated by a distance approximately equal to the wavelength of the light that is used

If they are closer, it is not possible to tell that there are two separate features

Section 26.2

Resolution, finalOptical resolution is set by diffraction

It is approximately equal to the wavelength of the light used

Applications requiring the best possible resolution use blue or ultraviolet light

These color have the shortest wavelength compared with other colors of visible light

Section 26.2

Confocal MicroscopeA confocal microscope

is designed so that features at only one particular depth form the final image

This is done by placing a pinhole in front of the observer

The depth of resolution is again limited by diffraction effectsDepths must be greater

than λ to be separated

Section 26.2

TelescopesWhen using a telescope, the light rays from the

object are nearly parallelThe object is approximately at infinity

One purpose of a telescope is to increase the angular separation between two starsThis allows your eye to distinguish one star from the

other

Section 26.3

Refracting Telescope

A refracting telescope use lensesObjective lens and eyepiece

Was invented around 1600 and was the type used by Galileo

The objective lens forms an image of the objectThis image then acts as the object for the second lens

Section 26.3

Refracting Telescope – Image For the objective lens

The object is at infinity (approximately)The image forms at the focal point of the lens

EyepieceThe eyepiece is located such that the image formed by

the objective is very close to the focal point of the eyepiece

The rays from the first image form a bundle of nearly parallel rays that are perceived by the observer

Section 26.3

Refracting Telescope – MagnificationThe magnification is determined by the angles the

incident ray (θ) and ray refracted by the eyepiece (θT) make with the axis

Actually, this is the angular magnificationFrom geometry and the small angle approximation

Tm

obj

eyepiece

mƒ

ƒ

Section 26.3

Reflecting Telescope – Newtonian DesignNewton designed a

reflecting telescopeUses mirrors

AdvantagesThe mirrors will not have

any chromatic aberrationEasier to make a high-

quality mirror than a lensFor a given diameter, a

mirror is lighter and easier to support

Section 26.3

Reflecting Telescope – Cassegrain DesignIn the Cassegrain design,

light reflects from the primary mirror, then from a secondary mirror and travels through a small hole in the primary mirror

The light then travels through an eyepiece to the observer

The Hubble Space Telescope is an example of a Cassegrain design

Section 26.3

Magnification – Reflecting TelescopeThe concave mirror forms a real image of a distant

object very close to the focal point of the mirrorA second mirror is positioned in front of the focal

point and reflects the light to an eyepieceThe magnification is similar to that for the refracting

telescope, with ƒM being the focal length of the primary mirror

M

eyepiece

mƒ

ƒ

Section 26.3

Resolution of a TelescopeResolution determines how close together in angle

two stars can be and yet still be seen as two separate stars

The resolution is limited by two factorsDiffraction at the telescope’s apertureAtmospheric turbulence

The aperture is generally the same diameter as the primary mirror

From the Rayleigh criterion, the limiting angular resolution set by diffraction is Dmin

1.22

Section 26.3

Resolution, cont.Most telescopes do not attain the resolution limitStarlight must pass through many kilometers of air

before reaching an observer on EarthThe turbulent motion of the air causes fluctuations in

the refractive index from place to placeThe fluctuations act like lenses and refract the

incoming light from the starThe “lenses” are constantly changing, so the direction

of the starlight changes as well This makes the star “twinkle”

Section 26.3

Atmospheric EffectsFor a location on the Earth’s surface, the angular

spread caused by atmospheric turbulence is typically 1" (one arc second)1° = 60 arc minutes1 arc minute (1') = 60 arc seconds

The value for this angular spread is smaller at higher altitudes

Telescopes in space eliminate atmospheric effects and the resolution is determined by the diffraction limit of the primary mirror

Section 26.3

Adaptive Optics

The technology of building telescopes with adjustable mirrors to compensate for atmospheric distortion is called adaptive optics

A reference star is an object known to appear as a point sourceAs the atmosphere causes the image of the reference star to be

smeared out, the telescope’s mirror is adjusted to make the image as perfect as possible

Computers allow for rapid and accurate control of the mirror shape

Section 26.3

CamerasCameras are common

optical devicesA simple camera consists

of a single lens positioned in front of a light-sensitive material

The lens forms an image on the detector

An aperture is opened for a short time to allow sufficient light energy to enter

Section 26.4

Film CameraThe distance between

the camera’s lens and the film determines which objects are in focus

The standard lens for a 35 mm camera is 40 mmThe “35 mm” is from

the size of the film24 mm x 35 mm

Section 26.4

Film Camera, cont.Other lenses can be purchased with different focal

lengthsSince the object is far away from the camera, a good

approximation is that the image forms at the focal point

The linear magnification of the image is

The image is real and inverted

i

o o

hm

h sƒ

Section 26.4

Digital CameraA digital camera replaces film with a CCDA CCD is a charge-coupled deviceA CCD uses a type of capacitor to detect light and

record its intensityThe optical system of a digital camera is basically

the same as that of a film cameraThere are important differences

Section 26.4

CCD

A CCD is fabricated in an integrated circuit chipThe chip contains many capacitors arranged in a gridWhen light strikes the chip, it is absorbed in the dielectric

layer and ejects some electrons from their normal chemical bonds

Section 26.4

CCD, cont.The ejected electrons move to the capacitor plateThis leads to a voltage across the capacitor that is

closest to where the light was absorbedThis voltage is detected by additional circuitry and its

value is stored in a computer memory in the cameraThe magnitude of the voltage depends on the light

intensityThe greater the intensity, the higher the voltage

The pattern of voltages on the capacitors gives the light intensity as a function of position

Section 26.4

CCD, finalOne way to measure

the color is to combine the information from four adjacent capacitors

Filters allow different colors to pass through to the capacitor

The camera’s computer can estimate the average color over the region

Section 26.4

PixelsEach region forms a pixel

From picture elementThe image produced by the CCD is stored by the

camera as a set of intensity and color values for each pixel

An important specification is the number of pixels in each photograph

A larger number of pixels indicates a finer level of detail in the photograph

Section 26.4

Optics of a Digital CameraThe size of the CCD is much smaller than the area

of the filmTypically about 6 mm x 8 mm

The magnification is still

Since the detector is smaller, the image height must be smaller

The focal length must be smaller for a digital cameraAbout 4 times smaller

Section 26.4

i

o o

hm

h sƒ

Optics, cont.The lens in the digital camera must be closer to the

detectorThe distance between the lens and the CCD is

approximately the focal length of the lensThis allows the digital camera to be much thinner than a

film cameraThe optical zoom function changes the magnification of a

digital camera by moving the lens relative to the CCD detector

A digital zoom process constructs the entire photo using just the image data from near the center of the CCD gridThis uses fewer pixels and has poorer resolution than

without the digital zoom

Section 26.4

ƒ-NumberSettings for both film and digital cameras include

shutter speed and the ƒ-numberShutter speed is the amount of time the film or CCD

is exposed to light from the objectThe ƒ-number is associated with the camera’s

apertureThe aperture is an opening that controls the open area

of the lens

Section 26.4

ƒ-Number, cont.The ƒ-number is the

ratio of the focal length to the aperture diameter

A large aperture gives a small ƒ-numberThis allows more light

to reach the film or the CCD

numberDƒ

ƒ

Shutter Speed and ƒ-NumberThere is a trade-off between

shutter speed and ƒ-numberIf you reduce shutter speed,

you need to compensate by increasing the ƒ-number

Same Exposure Value (Camera settings) can have different f-number and time

Halving f-number reduces EV by sqrt(2)

Section 26.4

time

fEV n

2

Depth of Focus and ƒ-Number

With a small aperture (large ƒ-number) the blurring of images away from the best focus is small

With a large aperture (small ƒ-number) some rays make a large angle with the central ray They diverge more quickly as one moves away from the image point

The ƒ-number is also related to the depth of focusHaving a large depth of focus means that objects that are not at the

best focusing point will produce images that are still close to idealSection 26.4

Pinhole CameraThe pinhole camera

makes the aperture very tinyNo lens is neededA sharp image can result

The intensity is very low and so you need long exposure times

Allows safe viewing of intense light sources such as the Sun

Section 26.4

CDCDs and DVDs are

applications of optics that can only be understood in terms of the ideas of wave opticsCDs and DVDs operate

through similar principlesStructure is a plastic layer

that is smooth on the bottom and contains a pattern of pits on the top

Section 26.5

CD StructureThe pattern of pits on the top surface is used to encode

information on the CDThe top surface is coated with a thin layer of aluminum to

make it reflectingIt is then covered with a protective layer of lacquerThe label is placed over the lacquerThe pits are arranged in a long spiral trackInformation encoded in the pits is read by reflecting a

laser beam from the aluminum surfaceLaser light passes in and out through the bottom surface

of the plastic, so the surface must be kept clean

Section 26.5

Reading a CDThe layer of aluminum

acts as a mirrorIt reflects the laser light

The pits influence this reflection through thin-film interference effects

The pit depth is designed to produce destructive interference

Reading a CD, cont.There is no reflected light when the laser beam is

over a pit edgeThe intensity is large when the laser beam is over

the center of a pit or is outside a pitAs the laser beam travels along a track, the reflected

light intensity varies between zero and a large valueThese high and low values of the intensity

correspond to ones and zeros in a binary encoding of information on the CD

Section 26.5

Reading a CD, finalTo store as much information as possible on the CD,

the pits must be as small as possibleThe minimum size is approximately equal to the

wavelengthThe limit is set by wave optics

Differences in DVDsShorter wavelength lasers allow pits to be closer

togetherMultiple layers of aluminumPits probed on both sides

Section 26.5

Optical FibersOptical fibers are

flexible strands of glass that conduct and transmit light by using total internal reflection

Remember whenever the angle of incidence is greater than the critical angle, all the light is reflected at the surface

Section 26.6

Optical Fiber DesignThe central cos is

surrounded by an outer layer called the cladding

The core and cladding are both made of glassDifferent compositions and

different indices of refractionWith ncladding < ncore, total

internal reflection of the light in the core can occur

Modern fibers have smaller core diameters that require wave optics to analyze

Section 26.6

NSOMThe wavelength of light

sets a limit on the resolution of a compound microscopeLight cannot be focused

to a spot size smaller than its wavelength

Details smaller than λ can be perceived using near-field scanning optical microscopy (NSOM)

Section 26.7

NSOM, cont.The technique uses an optical fiber to illuminate a very

tiny region at the end of the fiberThe light is most intense at the opening of the tip

This can be much smaller than the wavelengthThe tip is positioned very close to the object to be studiedIt is then scanned over one of its surfacesBecause the tip is so close to the object, only a very

small area near the tip is strongly illuminatedResolution is then determined by the spacing from the

surface and the tip diameterBoth can be much smaller than the wavelength

Section 26.7

NSOM, finalBy measuring the scattered or reflected light, an

image of the surface can be constructedBased on the scattered intensity as a function of the

tip’s positionCurrent NSOM fibers have openings much smaller

than visible light wavelengthsAbout 45 nm openings

The technique is being used to create images of objects as small as individual molecules

Section 26.7