py3101 optics - physics department ucc · 2014-03-11 · py3101 optics optical power. 1 f p= ......

$: PY3101 Optics - Physics Department UCC · 2014-03-11 · PY3101 Optics Optical power. 1 f P= ... deviation of a ray light, we can find the refractive index of the prism Prism –angular$
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M.P. Vaughan

PY3101 Optics

Optical instruments

Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland

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Department of PhysicsPY3101 Optics

• Lenses

• Apertures

• The prism

• The human eye

• The magnifying glass

• The refracting telescope

• Chromatic aberration

• The achromatic doublet

• The reflecting telescope

• Coma

Learning objectives

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Applications using lenses and mirrors

Photography Telescopes Microscopes

Ophthalmetry

Largely applied using geometrical optics

Lenses: a summary

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Convex lens


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Convex lens – transverse magnification

,f

xM i−= ,

ox

fM −= .2

ioxxf =

4


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Concave lens


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Concave lens – transverse magnification

,ox

fM −=,

f

xM i−=

as we had for the convex lens.

.2

ioxxf =

5


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Other types of lenses

plano-convex plano-convex doublet


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Optical power

.1

f=P

The optical power P of a lens is a measure of the degree to which it converges or diverges light.

P is defined as the reciprocal of the focal length.

The SI unit of P is called the dioptre (m-1).

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Apertures


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Apertures

Aperture StopImage Plane

Field Stop

The larger the aperture the more light collected.

DD.Flux 22 Dr ∝∝

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Camera Lenses

The larger the focal length, the larger the image size.

f

f

2Area Image f∝


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Flux density and f number

An exposure time will require a certain amount of optical energy.

Time ExposureDensityFlux Exposure ×=

( )2f/#Time Exposure ∝

f number

2

2

DensityFlux f

D∝

f

D=Aperture Relative

D

f=/#f

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Examples (Canon)

IXUS 70

“Maximum” f/#

f/2.8“Maximum” f/#

f/2.8 – f/4.9

EF-S 17-55mm

Prisms

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Prisms


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Prism v diffraction grating

Note that, for a prism, blue

light sees the greatest deviation...

... in contrast to a diffraction grating.

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2121 )90()90(180 ititA θθθθ +=−−−−=

Prisms – angular deviation

βα +=D

1iθ

1tθ 2iθ2tθ

D

A

n

190 tθ−α β

D is the angular deviation


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• From the angle of the minimum

deviation of a ray light, we can find the

refractive index of the prism

Prism – angular deviation

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( ) ( ))sin(sinsinsin 1

1

2

1

2 tit Ann θθθ −== −−

AD tiitti −+=−+−= 212211 )()( θθθθθθ

1iθ

1tθ 2iθ2tθ

D

A

n

190 tθ−α β


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( )11

1

2 sincoscossinsin ttt AnAn θθθ −= −

( )11

21 sincossin1sinsin tt AnAn θθ −−= −

( )11

221 sincossinsinsin ii AnA θθ −−= −

AD ti −+= 21 θθ

( )11

221

1 sincossinsinsin iii AnAA θθθ −−+−= −

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Prism – angular deviation


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The minimum deviation

Also:

Start with:

AD ti −+= 21 θθ

(A is a constant). Minimum where:

01

=id

dD

θ101

1

2

1

2 −=→=+→i

t

i

t

d

d

d

d

θθ

θθ

1102

1

2

1

2

−=→+==i

t

i

t

i d

d

d

d

d

dA

θθ

θθ

θ

21 itA θθ +=

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111111 coscossinsin ttiiti dndn θθθθθθ =→=

222222 coscossinsin iittit dndn θθθθθθ =→=

22

11

22

11

cos

cos

cos

cos

ii

tt

tt

ii

dn

dn

d

d

θθθθ

θθθθ

=

12

1 −=i

t

d

d

θθ

11

2 −=i

t

d

d

θθ

Use and

2

1

2

1

cos

cos

cos

cos

i

t

t

i

n

n

θθ

θθ

=2

2

1

2

2

2

1

2

sin1

sin1

sin1

sin1

i

t

t

i

n

n

θ

θ

θ

θ

−

−=

−

−→


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( )( )22

2

2

1

2

2

2

1

2

sin

sin

sin1

sin1

i

t

t

i

nn

nn

θ

θ

θ

θ

−

−=

−

−→

2

2

1

22

2

2

1

2

sin

sin

sin1

sin1

t

i

t

i

n

n

θθ

θθ

−−

=−−

→

With n > 1, this can only be true if:

12 ti θθ =So

21 ti θθ =

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,212 ntiA θθθ =+=

.22221 AAD anati −=−=−+= θθθθθ

21 ti θθ = and .12 ti θθ =

Using n for in the prism and a for in the air

Now

So

2

An =θ .

2

ADa

+=θand


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+

==→=

2sin

2sin

sin

sinsinsin

A

DA

nnn

ana θ

θθθ

Hence, from the minimum deviation D,

you can calculate n

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The human eye


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Alhazen

Alhazen

(c. 965 - c. 1040)

Kitab al-Manazir (Book of Optics) (1011 – 1021)

Gave early detailed description of the human eye.

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The anatomy of the eye


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• Cornea• Transparent part of front of the eye. Also refracts

light and accounts for about 2/3 of the optical

power of the eye. Typically the optical power of the

cornea is about 43 dioptres.

• Iris• Variable aperture controlling amount of light

entering the eye

• Sclera• The white part of the eye providing a protective

covering


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• Retina• The photosensitive region of the eye onto which

the image is projected

• Choroid• vascular layer of the eye, containing connective

tissue, lying between the retina and the sclera.

• Fovea• fovea centralis – pit near back of eye responsible

for sharp, central vision



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The lens

The curvature of the lens may be changed by muscle contractions in the eye.

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Near and far points

• The near point is the closest distance for

which the lens can focus light on the retina

• Typically at age 10, this is about 18 cm

• It increases with age, ~ 25 cm for an adult

• The far point of the eye represents the

largest distance for which the lens of the

relaxed eye can focus light on the retina

• Normal vision has a far point of infinity


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Farsightedness – hyperopia

Distant objects may be focussed but not nearby objects.

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Correcting farsightedness


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Nearsightedness – myopia

• axial myopia

• Lens too far from retina

• refractive myopia

• Lens-cornea system too powerful to focus properly onto the

retina

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Correcting nearsightedness

The magnifying glass

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Angular size of unaided image

The object at o subtends an angle of αu at the viewing point.


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Aided image

The image i of an object placed at

o within the focal length of a convex mirror subtends an angle of αa at the viewing point.

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Angular magnification

The angular magnification Mα is defined as the ratio of the aided and unaided viewing angles

.u

aMαα

α =

We shall employ the paraxial approximation to obtain an expression for this.


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.,i

i

o

oa

ou

d

h

d

h

D

h=== αα

From the diagrams

So

.ou

a

d

DM ==

αα

α

where D is the distance to the object in the unaided case.

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.fd

h

f

h

i

io

+=

From the diagram

,i

o

i

o

d

d

h

h=

.111

oi ddf=+

From the expression for αa,

Leading to


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From

,od

DM =α

.11

+=

idfDMα

We then have

Thus the shorter the focal length, the greater the angular magnification.

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The system matrix


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Analytical ray tracing

We can trace a ray through an optical system using

( ) ( ).ˆˆˆˆnttnii nn ukuk ×=×

Clearly, this just states Snell’s

Law.

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Trace of a light ray through a lens passing through points P1and P2.


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1111 ttii nn θθ =

( ) ( ).111111 αααα +=+ ttii nn

.1

11R

y≈α

Applying the paraxial approximation, Snell’s Law becomes

or, from the diagram

We also have, from the approximation of the tan function,

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.1

111

1

111

+=

+

R

yn

R

yn ttii αα

Putting these results together,

.111

211

11

−

−=

RRn

nn

f i

it

We shall make use of the lens maker’s formula


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For a single surface, this reduces to

,1

f=P

We also recall that the optical power of a lens is defined as

.11

11

11

Rn

nn

f i

it −=

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Hence, rearranging

we have

,1

111

1

111

+=

+

R

yn

R

yn ttii αα

.11

111111 y

R

nnnn it

iitt

−−= αα

This is the refraction equation.


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.1111

1

11 ynyR

nni

it P=

−

In terms of the optical power, for the first surface we may put

.1111111 ynnn iiitt P−= αα

Soa

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Matrix analysis of lenses

We can write out our result in the form

,0

,

11

1111111

it

iiiiii

yy

ynnn

+=

−= Pαα

where yi1 and yt1 are the heights of the point P1 immediately on either side of the first lens surface. Then, clearly

.111 yyy it ==


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This becomes useful when we re-write the simultaneous equations in matrix form

.,10

1

1

111

1

11

−=

i

ii

t

ii

y

n

y

n αα P

We can then write this in the form

,111 it xRx =

where R1 is called the refraction matrix.

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We now consider the path P1 to P2.


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.12112 tdyy α+≈

This is the transfer equation. From the diagram, the required simultaneous equations for the ray are

Using the paraxial approximation for tan, we have, for the vertical heights of P1 and P2,

.

,0

11212

1122

tti

tttt

ydy

nn

+=

+=

α

αα

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In matrix form, this is

.,1

01

1

11

1212

22

=

t

tt

ti

tt

y

n

ndy

n αα

This can be written in the form

,1212 ti xTx =

where T21 is the transfer matrix.


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Combining

we may put

,1212 ti xTx =

111 it xRx =and

.11212 ii xRTx =

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Now, for the second surface, the lens maker’s formula gives

.11

22

22

Rn

nn

f t

ti −−=

Hence, the optical power is

.1

22

222

Rn

nn

t

it −=P


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The system matrix

The refraction at the second interface my be encapsulated by

where the second refraction matrix is

,222 it xRx =

.10

1 2

2

−=

PR

The system matrix is then defined as

.1212 RTRA =

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The system matrix

More generally, the matrix for a system of lenses is

.. 12122111 RTRTRTRA K−−−−−= mmmmmmmm

The system matrix facilitates a ray

tracing analysis of a system of

lenses.

In summary,

Refracting telescopes

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• Terrestrial

• Produces upright image

• Employs a concave lens for the eyepiece

• Astronomical

• Inverts the image

• Employs a convex lens for the eyepiece

Types of refracting telescope


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Galileo

Improved Hans Lippershey’s design of refracting telescope

Observes phases of Venus, sunspots and Galilean moons.

Galileo (1564 - 1642)

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Galilean (terrestrial) telescope

Gives upright image. Note the focal points coincide.


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Johannes Kepler

Improved Galileo’s design of refracting telescope

However, the Keplerian (astronomical) telescope inverts the viewed image.

Kepler (1571 - 1630)

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Keplerian (astronomical) telescope

Gives inverted image. Note the focal points coincide.


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Galilean telescope Keplarian telescope

From the system matrix of each telescope, both are found to have an angular magnification of

.e

o

f

fM −=α

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Chromatic aberration


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• An inherent problem of refracting

telescopes is that they suffer from

chromatic aberration

• This is due to different frequencies of

light having different refracting indices

and therefore refracting at different

angles

Chromatic Aberration

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Blue refracts more than red (greater refractive index for normal dispersion


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• Chromatic aberration may be

compensated to some degree via the

use of an achromatic doublet


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Achromatic doublet

An achromatic doublet (achromat) is often used to compensate for the chromatic aberration.


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Achromatic doublet

First lens:

−−=

21

1

1

11)1(

1

RRn

fR

R

−−=

21

1

1

11)1(

1

RRn

fB

B

Second lens:

−−=

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2

2

11)1(

1

RRn

fR

R

−−=

32

2

2

11)1(

1

RRn

fB

B

(The R and Bsubscripts stand for red and blue

respectively).

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Achromatic doublet

Red:

RRR fff 21

111+=

Blue:

BBB fff 21

111+=

fff

111

21

=+

Using the result for adding thin lenses in close combination

we obtain


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Achromatic doublet

−−+

−−=

32

2

21

1

11)1(

11)1(

1

RRn

RRn

fRR

R

Red:

Blue:

−−+

−−=

32

2

21

1

11)1(

11)1(

1

RRn

RRn

fBB

B

Writing these expressions out in full

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Achromatic doublet

011

)(11

)(32

12

21

11 =

−−+

−−

RRnn

RRnn RBRB

BR

BRff

ff11

=→=

Now, we require

Thus, we need to choose parameters such that


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Example of the use of an achromatic doublet, using a doublet as the objective.

Achromatic doublet

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Reflecting telescopes


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Newton

Isaac Newton

(1642 - 1727)

Demonstrated decomposition of white light into the different colours of the rainbow via refraction through a prism.

Built the earliest known functional reflecting telescope in 1668 in a bid to escape achromatic aberration.

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• Newton’s telescope focussed the

incoming light via a curved mirror

• This was then reflected to the eyepiece

Newtonian telescope


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Newtonian telescope

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Newtonian telescope

.e

o

f

fM −=α

Essentially, we have the same magnification system as for the astronomical telescope.

Hence, the angular magnification is


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• An inherent problem of reflecting

telescopes is that they suffer from a

type of monochromatic aberration

known a coma

• This occurs in off-axis points due to the

transverse magnification being a

function of ray height

• This creates a pattern for a point that

looks like a comet

Coma

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Coma


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Coma in a parabolic mirror

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• Corrective lenses for Newtonian

telescopes with f numbers less than f/6

have been designed

• These employ a dual lens system of a

plano-convex and a plano-concave lens

fitted into an eyepiece adaptor

• An example of a correction strategy for

coma is Baader Rowe Coma Correction

Coma


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Baader Rowe Coma Correction

Comparison of the coma in an uncorrected f/3.9 Newtonian telescope vs the affects of coma with the Baader Rowe Coma Corrector.

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py3101 optics - physics department ucc · 2014-03-11 · py3101 optics optical power. 1 f p= ......

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