Thick lenses, lens system and cardinal points in paraxial optics

Thick lenses, lens system and 

cardinal points in paraxial optics 

Textbook (Hecht), Chapter 6

Image Formation Summary Table

Complex optical system: Thick lens 

Thick lenses, combinations of lenses etc.. 

Consider case where t is NOT 

negligible. 

We would like to maintain our 

Gaussian imaging relation 

n n’ 

n 

s 

n' 

+ 

s ' 

= 

P 

t 

n L 

But where do we measure s, s’ s ; f, f’ f 

from? How do we determine P? 

We try to develop a formalism that 

can be used with any system!!

Cardinal points and planes: 

The cardinal points and the associated cardinal planes 

are a set of special points and planes in an optical system, 

which help in the analysis of its paraxial properties. 

The analysis of an optical system using cardinal points is 

known as gaussian optics, named after Carl Friedrich 

Gauss. 

The cardinal points and planes of an optical system include: 

- The focal points and focal planes 

- The principal planes and principal points 

- The surface vertices (or vertexes) 

- The nodal points


Focal points and planes 

The front focal point of an optical system, by definition, has the 

property that any ray that passes through it will emerge from 

the system parallel to the optical axis. 

The rear (or back) focal point of the system has the reverse 

property: rays that enter the system parallel to the optical axis 

are focused such that they pass through the rear focal point. 

The front and rear (or back) focal planes are defined as the 

planes, perpendicular to the optic axis, which pass through the 

front and rear focal points. An object in infinite distance away 

from the optical system forms an image at the rear focal plane. 

For objects a finite distance away, the image is formed at a 

different location, but rays that leave the object parallel to one 

another cross at the rear focal plane.

Thin Lens 

AIR 

In paraxial region 

h1≈h2 

Focal Point 

F 

h2 

R 

R 

h1 

n 

(Glass) 

Collimated Beam 

(Parallel to the OA) 

Optical Axis 

(OA) 



How do we 

measure the focal length? 

Thick Lens 

AIR 

h1>h2 


F = ? 

h2 

R 

n 

(Glass) 

R 

h1 



Optical Axis 

(OA) 


(Parallel to the OA)

Thick Lens: F measured from the center of the lens? 

F1 

AIR 

h 2 

h 1 

= h’ 1 

; h 2 

= h’ 2 

R 

R 

n 

h 1 

(Glass) 


L1 



Optical Axis 

(OA) 



AIR 

F2 




It seems F2 > F1. 

Is this correct? 

h’ 2 

R 

n 

(Glass) 

L2 

h’ 1 

R 

Optical Axis 

(OA) 


(Parallel to the OA)

Thick Lens: Define Principal Planes 

AIR 

First 


F1 

R 

First Principal Plane (FPP) 

(Primary Principal Plane) 

n 

(Glass) 

R 



Optical Axis 

(OA) 



First Principal Point 

(FPP ⊥ OA)


Second Principal Plane (SPP) 

(Secondary Principal Plane) 



F2 

AIR 

Optical Axis 

(OA) 



R 

n 

(Glass) 

R 

Second Focal Point 

Second Principal Point 

(SPP ⊥ OA)


F1 = F2 

F1 

F2 

AIR 

Optical Axis 

(OA) 

First 


R 

n 

(Glass) 

R 

Second 



(Primary Principal Plane) 

Second Principal Plane (FPP) 

(Secondary Principal Plane)

Thick Lens: Front and Back Focal Length 


F1 

Second Principal Plane (FPP) 

F2 

AIR 

Optical Axis 

(OA) 

R 

n 

(Glass) 

R 

First 


Second 


Front Focal Length 

(Front Working Distance) 

Back Focal Length 

(Back Working Distance)


Principal planes and points 

• The two principal planes have the property that a ray emerging from the lens 

appears to have crossed the rear principal plane at the same distance from the 

axis that that ray appeared to cross the front principal plane, as viewed from the 

front of the lens. In this model, rays travel parallel to the optical axis between 

principal planes. The principal planes are crucial in defining the optical properties 

of the system, since it is the distance of the object and image from the front and 

rear principal planes that determines the magnification of the system. The 

principal points are the points where the principal planes cross the optical axis. 

• If the medium surrounding the optical system has a refractive index of 1 (e.g. 

air), then the distance from the principal planes to their corresponding focal 

points is just the focal length of the system. If the medium is not air or vacuum, 

the distance to the foci is multiplied by the index of refraction of the medium. 

• For a thin lens in air, the principal planes both lie at the location of the lens. The 

point where they cross the optical axis is sometimes misleadingly called the 

optical centre of the lens. Note, however, that for a real lens the principal 

planes do not necessarily pass through the centre of the lens, and in general may 

not lie inside the lens at all.

Cardinal points and planes 

Surface vertices 

The surface vertices are the 

points where each surface 

crosses the optical axis. 

They are important primarily 

because they are the physically 

measurable parameters for the 

position of the optical elements, 

and so the positions of the other 

cardinal points must be known 

with respect to the vertices to 

describe the physical system.


1. Nodal (N) points and planes 

Nodal points 

The front and rear nodal points have the property that a ray that 

passes through one of them will also pass through the other, and 

with the same angle with respect to the optical axis. The nodal 

points therefore do for angles what the principal planes do for 

transverse distance. If medium on both sides of the optical system 

is the same (e.g. air), then the front and rear nodal points coincide 

with the front and rear principal planes, respectively.


1. Nodal (N) points and planes 

n 

n’ 

N 1 

N 2 

n L 

NP 1 

NP 2

Cardinal planes of simple systems 

1. Thin lens 

V’ and V coincide and 

V’ 

V 

n 

n' 

+ ' 

= 

P 

H, H’ H 

s 

s 

is obeyed. 

Principal planes, nodal planes, 

coincide at center

Cardinal planes of simple systems 

1. Spherical refracting surface 

n 

V 

n’ 

Gaussian imaging formula 

obeyed, with all distances 

measured from V 

n 

n' 

+ ' 

= 

P 

s 

s


1. Focal (F) points & Principal planes (PP) and points 

R 1 

≠ R 2 

n n L 

n’ 

F 2 

H 2 

ƒ’ 

PP 2 

Keep definition of focal point ƒ’



R 1 

≠ R 2 

n n L 

n’ 

F 1 

H 1 

ƒ 

PP 1 

Keep definition of focal point ƒ

Utility of principal planes 

Suppose s, s’, s , f, f’ f all measured from H 1 and H 2 … 

n n L 

n’ 

h 

F 1 

F 2 

H 1 

H 2 

h’ 

ƒ 

ƒ’ 

s s’ 

PP 1 

PP 2 

Show that we recover the Gaussian Imaging relation: 

1 1 1 

+ = 

s s′ 

f

Conjugate Planes – where y’=y 

n n L 

n’ 

y 

F 1 

F 2 

H 1 

H 2 

y’ 

ƒ 

ƒ’ 

s s’ 

PP 1 

PP 2



The distance h 1 

from the primary principal plane (P) to 

the on-axis entrance point of the lens (V), and the 

distance h 2 

from the secondary principal plane (P’) to the 

on-axis exit point (V’), are: 

Sign convention! 

(d l 

is the thickness of 

lens, f is focal length.)



Each paraxial focal point of a thick lens is located at a 

distance f from the nearer of the two principal planes. 

The lensmaker’s equation for a thick lens immersed in air (n 0 

= 1) is



The paraxial object-image relation and Newton’s equation 

still hold if the object and image distances s , s’ are 

measured from the principal planes 

A rule-of-thumb for ordinary glass lenses in air is that the 

separation P-P’ roughly equals one-third the lens thickness 

V-V’.

Thick Lenses

Thick Lenses

Thick lenses, lens system and cardinal points in paraxial optics

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?