Prof. Jose Sasian

Chromatic Aberrations

Lens Design OPTI 517

Prof. Jose Sasian

Second-order chromatic aberrations

• Change of image location with λ (axial or longitudinal chromatic aberration)

• Change of magnification with λ (transverse or lateral chromatic aberration

2 2000 200 111 020, cosW H W W H W H W

Prof. Jose Sasian

Chromatic Aberrations

• Variation of lens aberrations as a function of wavelength

• Chromatic change of focus: W020

• Chromatic change of magnification W111

• Fourth-order:W040 and other• Spherochromatism

Prof. Jose Sasian

2020W 111 cosW H

Chromatic Aberrations

Prof. Jose Sasian

Topics• Chromatic coefficients• Optical glass and selection• Index interpolation• Achromats: crown and flint:

different solutions• Achromats: dialyte; single

glass• Mangin lens• Third-order behavior• Spherochromatism• Secondary spectrum• Tertiary spectrum

• Apochromats• Super-apochromats• Buried surface• Monochromatic design: one

task at a time• Lateral color correction as an

odd aberration• Color correction in the

presence of axial color• Field lens to control lateral

color: field lenses in general• Conrady’s D-d sum

Prof. Jose Sasian

Chromatic aberration coefficients

0201

1 /2

j

j j ji

W A n n y

1111

/j

j j ji

W A n n y

For a system of j surfaces

1111

j

i i

W yy

2020

1

12

j

i i

W y

For a system of thin lenses

Prof. Jose Sasian

With stop shift

020 0W

111 0202 yW Wy

Prof. Jose Sasian

Review of paraxial quantitiesy

ru’

s’

1/ nn nn

Prof. Jose Sasian

Chromatic coefficients

rsnn

rsnnyW 111

'1''

2

2

020

nnAy

rsn

rsnyW

2111

'1'

2

2

020

2

0201 1 1 1'

2 'yW n n

s r s r

Prof. Jose Sasian

Stop shifting

Prof. Jose Sasian

Prof. Jose Sasian

Stop shifting

Ey

Ey

Old stop plane at CCNew stop

Marginal ray height at old pupil

New chiefray

Hyyy EEshiftE

HAAH

yy

E

Eshift

New chief ray height at old pupil

Prof. Jose Sasian

Chromatic coefficients

HAAH

yy

E

Eshift

2

2shift shiftA AH H HA A

020 2y nW A

n

111nW A yn

Prof. Jose Sasian

Can show equality using the Lagrange invariant

Prof. Jose Sasian

The ratio E

E

y y Ay y A

Prof. Jose Sasian

The ratio

1 1

2 2

2 1 2 1

2 1 2 1

Ж A y AyЖ A y Ay

A y y y A A

y y A Ay A

/A A

1

2 1

0

cc

cc

Ayy y A

y y A

When the stop is shifted at the cc

Old chief ray

New chief rayMarginal ray

stopParameters are at the surface

Prof. Jose Sasian

The ratio

2 1 2 1q q p py y y yq p

q py yq p

Marginal ray

stop

Old and new chief rays

Does not depend on plane where it is calculated given similar triangles

2 1 cc

cc

yy yy y

pq

Prof. Jose Sasian

For a system of thin lenses

1111

j

i i

W yy

2020

1

12

j

i i

W y

V is the glass V-numberΦ is the optical powery is the marginal ray heighty-bar is the chief ray height

Prof. Jose Sasian

Glass• Schott, Hoya, Ohara glass catalogues (A wealth of

information; must peruse glass catalogue)• Crowns and flints are divided at V=50• Normal glasses: • Soda-lime, silica, lead (older glasses)• Crowns, light flints, flints, dense flints• Barium glasses (~1938)• Lanthanum or rare-earth glasses• Titanium• Fluorites and phosphate• Environmental and health issues in the production of

glass. Lead replaced with Titanium and Zirconium.

Prof. Jose Sasian

Other materials

• For the UV• For the IR• Plastics• Advances come usually with new

materials that extend or have new properties.

• The design is limited by the material

Prof. Jose Sasian

Prof. Jose Sasian

Glass properties

nd-1nF-nC

486.1 587.6 656.3589.8

F (H) d (He) C (H)D (Na)

nd -1 RefractivitynF-nC Mean dispersionnd -nC Partial dispersion

v=(nd-1)/(nF-nC) v-value, reciprocal dispersive power, Abbe numberP=(nd-nC)/(nF-nC) Partial dispersion ratio

λ

n

1

Prof. Jose Sasian

Glass properties

• Homogeneity• Transmission• Stria• Bubbles• Ease of fabrication; soft glasses• Coefficient of thermal expansion• Opto-thermal coefficient• Birefringence

Prof. Jose Sasian

Index of refraction variation

Rate of slope change in the blue makes it more difficult to correct for color

Prof. Jose Sasian

Index interpolation

2 22

2 2 ...b dn ac e

2 2 2 4 6 81 2 4 6 81 ...n A A A A A A

Schott

Sellmeier

HartmannConradyKettler-Drude

Must verify index of refraction

Prof. Jose Sasian

The optical wedge

δα

1n

The deviation is independent of the angle of incidence for small θ(First order approximation)

'1 1

'2 1'2 2

'1 2

1n

n

θ

Prof. Jose Sasian

Wedge

1

1 1

1 1

1

d

F C F C

d C d C

d

F C

d C

F C

n

n n n n

n n n n

nv

n n

n nP

n n

v

Pv

δ Deviation∆ Dispersionε Secondary

dispersion

Prof. Jose Sasian

Achromatic wedge pair

1 21 2

1 2

22 1

1

2 1 21 2 1 1 1 2 1 2

1 1 2

1 1

1 2 1

2 2

1 2 2

1 21 2

0

11

11

1

d

d

v vvv

v v v v vv v v

vv v n

vv v n

P Pv v

Deviation without dispersion

Prof. Jose Sasian

•There is deviation•There is no dispersion•Red and blue rays are parallel•Independent of theta to first order

Achromatic wedge

Schematic drawing

Prof. Jose Sasian

Achromatic doublet(Treated as two wedges)α

2

' '1 2 1

1 2

1 2

1 2

1 2

1 2

; '2

1 11

0

0

d

Y YZ sag Zr r

YZ Z Yr r n

v vY Yv v

Z’

Z

Prof. Jose Sasian

Achromatic doublet1 2

1 2

1 2

1 1

1 2

2 2

1 2

0Y Yv v

vv v

vv v

Independent of conjugateRequires finite difference

1 2v v

Can lead to strong optical powers

Prof. Jose Sasian

Relative sag(for 100 mm focal length)

Zonal spherical aberrationCritical airspace

Prof. Jose Sasian

Achromatic doublet

•Must have opposite power (Glass)•Strong positive and weaker negative lens•Cemented doublet•Crown in front•Flint in front•Corrected for spherical aberration•Degrees of freedom•Large achromats and cementing•Conrady D-d sum•Zonal spherical aberration

Prof. Jose Sasian

Conrady’s D-d sum

• In the presence of sphero-chromatism thebest state of correction is achieved when:

0D d n D

d

Is the difference of optical path between the marginal F and C rays.

Prof. Jose Sasian

Conrady’s D-d sum

_ _ f f c cOptical path difference Dn dn Dn dn D d n

D

d

Minimizes the rms OPD difference by joining the opd curves at the edge ofThe aperture. Valid for fourth order sphero-chromatism.

Prof. Jose Sasian

Cemented doublet solutions

• Correction for chromatic change of focus• Correction for spherical aberration• Degrees of freedom: relative powers for a

set of glasses; shapes• Crown in front: two solutions• Flint in front: two solutions• Note multiple solutions

Prof. Jose Sasian

Crown in front and flint in frontdoublet solutions (BK7 and F2)

Prof. Jose Sasian

Contact options for doublets

Full contact (cemented)Air spaced

Edge contactedCenter contacted

Prof. Jose Sasian

LimitationsSecondary spectrum, spherochromatism and

zonal spherical aberration set limits

F=100 mm, f/4, 0.5 wave scale

Prof. Jose Sasian

Achromatic doublet

20 inch diameter

F/12

BK7

F4

Prof. Jose Sasian

In this lecture

• Chromatic coefficients• Basic glass properties• Achromatic wedge-pair and lens doublets• Examples• D-d method• Achromatic doublet• Diversity of solutions