Chapter 6

More on geometrical optics

Thick lensesAnalytical ray tracingAberrations

Phys 322Lecture 17

First focal point (f.f.l.)

Second focal point (b.f.l.)

Primary principal plane

Secondary principal plane

First principal point

Second principal point

Thick lens: terms

Nodal points

If media on both sides has the same n, then:N1=H1 and N2=H2

Fo Fi H1 H2 N1 N2 - cardinal points

Thick lens: principal planes

Principal planes can lay outside the lens:

Thick lens: equations

fss io

111

Note: in air (n=1)

2fxx io effective focal length:

2121

11111RRn

dnRR

nf l

lll

2

11

Rndnfh

l

ll

12

1Rn

dnfhl

ll Principal planes:

o

i

o

i

o

iT x

ffx

ss

yyM Magnification:

Thick lens: exampleFind the image distance for an object positioned 30 cm from the vertex of a double convex lens having radii 20 cm and 40 cm, a thickness of 1 cm and nl=1.5

cm 30.22cm22.0cm30 os

so si

30 cm

fss io

111

1 2 1 2

11 1 1 1 1 0.5 1 11 0.520 40 1.5 20 40 cm

l ll

l

n dn

f R R n R R

cm 8.26f

cm22.0cm5.14015.08.26

1

h

f

cm44.0cm5.120

15.08.262

h

f

cm8.2611

cm22.301

is

cm 238is

Compound thick lens

Can use two principal points (planes) and effective focal length fto describe propagation of rays through any compound systemNote: any ray passing through the first principal plane will emerge at the same height at the second principal plane

For 2 lenses (above):2121

111ff

dfff

1222

2111

ffdHH

ffdHH

Example: page 246

Analytical ray tracing

Refraction equation: 111111 ynn iitt D

Transfer equation: 12112 tdyy

note: paraxial approximation

Ray tracing of lens system: sequentially apply these equations

1

111 R

nn it D

Matrix treatment: refraction

note: paraxial approximation

In any point of space need 2 parameters to fully specify ray:distance from axis (y) and inclination angle () in respect to optical axis. Optical element changes these ray parameters.

111111 iiitt ynn D

1111 0 tiit yny

Refraction:

yt1=yi1

1

111

1

11

101

i

ii

t

tt

yn

yn D-Equivalent matrix

presentation:

Reminder:

DyCByA

yDCBA

rt1 - output ray

ri1 - input ray R1 - refraction matrix111 it rRr

Matrix: transfer through space

11122 0 tttii ynn

11212 tti ydy

1

11

1212

22

101

t

tt

ti

ii

yn

ndyn Equivalent matrix

presentation:

ri2 - output ray

rt1 - input ray T21 - transfer matrix

1212 ti rTr

Transfer:

yt1

yi2

112122 it rRTRr

System matrix

yt1yi1

yi2 yi2

ri1 rt1 ri2 rt2R1 T21

ri3 rt3R3T32

111 it rRr 11211212 iti rRTrTr

Thick lens ray transfer:

1212 RTRASystem matrix: 12 it rr ACan treat any system with single system matrix

1212 RTRA

101 D-

R

101

ndTReminder:

DdCbDcCaBdAbBcAa

dcba

DCBA

101

101

101 12 D-D-

ll ndA

yt1yi1

yi2 yi2

nl

dThick lens matrix

l

l

l

l

l

l

l

l

nd

nd

nd

nd

1

2121

2

1

1

D

DD-D-D- D

A

system matrix of thick lens

For thin lens dl=0

10/11 f-

A

Thick lens matrix and cardinal points

2221

1211

1

2121

2

1

1

aaaa

nd

nd

nd

nd

l

l

l

l

l

l

l

l

D

DD-D-D-D

A

i

t

o

i

fn

fna 21

12 in airf

a 112 effective focal length

12

11111

1a

anHV i

12

22222

1a

anHV i

Matrix treatment: example

rO

rI

OOlII rTTr 12A

O

OO

OOIII

II

yn

ndaaaa

ndyn

101

101

12221

1211

2

(Detailed example with thick lenses and numbers: page 250)

10/21 Rn

M

Mirror matrix

i

i

r

r

yn

yn

M

ir yy

Rnynn iir /2

note: R<0

Ryiir /2

AberrationsAberrations - deviations from Gaussian optics (paraxial approx.).

Chromatic aberrations - n depends on wavelengthMonochromatic aberrations - rays deviate from Gaussian optics

• spherical aberrations• coma• astigmatism• field curvature• distortion

Paraxial approximation: sin

Third order theory:!3

sin3

Departures from the first order theory observed in the third order leave to the following primary aberrations:

Taylor series: ...!7!5!3

sin753

“The five Seidel aberrations”

Philipp Ludwig Seidel(1821-1896)

Spherical aberrationsParaxial approximation:

Rnn

sn

sn

io

1221

Third order:

2

2

2

121221 112

112 iiooio sRs

nRss

nhR

nnsn

sn

deviation from the first-order theory

L.SA = longitudinal spherical aberrationsimage of an on-axis object is longitudinally stretched

T.SA = Transverse (lateral) spherical aberrationsimage of an on-axis object is blurred in image plane

positive L.SA - marginal rays intersect in front of Fparaxial

Spherical aberrations

LC - circle of least confusion, smallest image blur

Spherical aberration depends on object and lens arrangement:

Wavefront aberrations

John William Strutt(Lord Rayleigh)

1842-1919

Lord Rayleigh criterion: wavefront aberration of /4 produces noticeably degraded image (light intensity of a point object image drops by ~20%)

Zero SA

For points P and P’ SA is zero

Oil-immersion microscope objective

http://micro.magnet.fsu.edu/primer/java/aberrations/spherical/

Hubble telescope

COSTAR - corrective optics space telescope axial replacement module

Arecibo observatory

1000 ft radiotelescope (3cm - 3m)

Coma (comatic aberration)• Aberration associated with a point even slightly off the optical axisReason: principal planes are not flat but curved surfacesFocal length is different for off-axis points/rays

Negative coma: meridional rays focus closer to the principal axis

Coma (comatic aberration)

Vertical coma

Horizontal coma

Astigmatism

http://www.microscopyu.com/tutorials/java/aberrations/astigmatism/

• Aberration associated with a point considerably off the optical axis

Focal length for rays in Sagittal and Meridional planes differ for off-axis points

Field curvature

Focal plane is curved:Petzval field curvature aberration

http://www.microscopyu.com/tutorials/java/aberrations/curvatureoffield/index.html

Negative lens has field plane that curves away from the image plane:Can use a combination of positive and negative lenses to cancel the effect

DistortionTransverse magnification MT may be a function of off-axis image distance: distortions

http://www.microscopyu.com/tutorials/java/aberrations/distortion/index.html

Positive (pincushion) distortion

Negative (barrel) distortion

Correcting monochromatic aberrations

• Use combinations of lenses with mutually canceling aberration effects

• Use apertures• Use aspherical elements

Example:

Chromatic aberrations

Refraction index n depends on wavelength

21

1111RR

nf l

Chromatic aberrations

A.CA: axial chromatic aberration

L.CA: lateral chromatic aberration

21

1111RR

nf l

Achromatic Doublets

http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html

Combine positive and negative lenses so that red and blue rays focus at the same point

Achromatized for red and blue

For two thin lenses d apart:2121

111ff

dfff

21

1111RR

nf l

22112211 11111 nndnn

f

Achromat: fR=fB

22112211

22112211

11111111

BBBB

RRRR

nndnnnndnn

Achromatic Doublets 22112211

22112211

11111111

BBBB

RRRR

nndnnnndnn

Simple case: d = 0

RB

RB

nnnn

11

22

2

1

Focal length in yellow light

(between red and blue):

11 Y

Y

nf

Y

Y

Y

Y

ff

nn

1

2

1

2

2

1

11

1

1

111

222

1

2

YRB

YRB

Y

Y

nnnnnn

ff

Combine:

Achromatic Doublets

http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html

1

1

111

222

1

2

YRB

YRB

Y

Y

nnnnnn

ff

12

22

Y

RB

nnn

Dispersive powers: 11

11

Y

RB

nnn

Abbe numbers (dispersive indices, V-numbers):RB

Y

nnnV

22

21

1

RB

Y

nnnV

11

12

1

2

1

1

2

VV

ff

Y

Y 01122 VfVf YY

Typical BYR colors: B = 486.1327 nm (F-line of hydrogen)Y = 587.5618 nm (D3 line of helium)R = 656.2816 nm (C-line of hydrogen)Table of V numbers - page 270

Achromatic lenses CrownFlint

http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html

Achromatic triplet:focus match for 3 wavelengths

Cooke triplet(Denis Taylor, 1893)

Flint

Crown

Achromatic doublet: exampleDesign an achromatic doublet with f = 50 cm Use thin lens approximation.

Solution:21

111fff

01122 VfVfff

fff

1

12

01121

1

VfVff

ff

fV

VVf1

211

f

VVVf

2

122

Technically: want smaller R, i.e. longest possible f1 and f2Solution: use two materials with drastically different V

Use figure 6.39 (page 271)

Achromatic doublet: example

V

V1= 63.46

V2= 36.37

Achromatic doublet: exampleDesign an achromatic doublet with f = 50 cm

Solution:f

VVVf

1

211

f

VVVf

2

122

m 2134.050.046.63

37.3646.631

f m 3724.02 f

21

1111RR

nf l

f1 f2 n1=1.51009 (for yellow line!)n2 = 1.62004

Negative lens:

1111

12

2 Rn

f m 2309.01221 nfR

Positive lens:

m2309.0

1111

11

1 Rn

f m 2059.01 R