Last chapter

6. Optical instrumentation 6. Optical instrumentation

Last chapter

¾

¾ Third-order (Seidel) Optical Aberrations

™

This chapter

¾

¾ Field stop, Entrance window, Exit window

¾ Depth of field, Depth of focus

¾ prism and dispersion

¾ Camera

¾ Magnifier and eyepiece

¾ Microscope

¾ Telescope

Aperture effects on image

Optical instruments

Stops in Optical Systems Stops in Optical Systems

Larger lens improves the brightness of the image

S S’

Two lenses of the same Two lenses of the same focal length (

focal length (f), but f), but diameter (D) differs diameter (D) differs

More light collected More light collected from S by larger from S by larger lenslens

Bundle of Bundle of rays from S, rays from S, imaged at S imaged at S’’ is larger for is larger for larger lens larger lens Image of S Image of S formed at formed at the same the same place by place by both lenses both lenses

Stops in Optical Systems Stops in Optical Systems

• Brightness of the image is determined primarily by the size of the bundle of rays collected by the

system (from each object point)

• Stops can be used to reduce aberrations

Stops in Optical Systems Stops in Optical Systems

How much of the object we see is determined by:

How much of the object we see is determined by:

Field of View Field of View

QQ

QQ’’

(not seen) (not seen)

Rays from Q do not pass through system Rays from Q do not pass through system

We can only see object points closer to the axis of the system We can only see object points closer to the axis of the system

Thus, the Field of view is limited by the system Thus, the Field of view is limited by the system

Stops Stops

• Brightness

– Aperture stop: The real aperture in an optical system that limits the size of the cone of the rays accepted by the system from an axial object point – Entrance pupil: The image of the aperture stop formed by the optical

elements (if any) that precede it.

– Exit pupil: The image of the aperture stop formed by the optical elements (if any) that follow it.

– The aperture stop also is used to control the depth of field and depth of focus for an optical system, and to reduce the effect of optical aberrations.

• Field of view

– Filed stop: The real aperture that limits the angular field of view formed by an optical system

– Entrance window: The image of the field stop formed by the optical elements (if any) that precede it.

– Exit window: The image of the field stop formed by the optical elements (if any) that follow it.

– The field stops are used to control the field of view (the extent of the object plane that is imaged in the image plane) and to control aberrations.

Aperture Stop Aperture Stop

• A stop is an opening (despite its name) in a series of lenses, mirrors, diaphragms, etc.

• The stop itself is the boundary of the lens or diaphragm

• Aperture stop: that element of the optical system

that limits the cone of light from any particular

object point on the axis of the system

Aperture Stops Aperture Stops

Optics, E. Hecht, p. 149

Aperture Stop (AS) Aperture Stop (AS)

OO

EE

EE

Assume that the Diaphragm is the AS of the system Assume that the Diaphragm is the AS of the system

Diaphragm

Entrance Pupil (E

P) Entrance Pupil (E

P)

The entrance pupil is defined to be the image of the aperture The entrance pupil is defined to be the image of the aperture stop in all the lenses preceding it

stop in all the lenses preceding it (i.e. to the left of AS (i.e. to the left of AS -- if light if light travels left to right)

travels left to right)

OO

LL₁₁ EE

EE

EE’’

EE’’

How big does the How big does the aperture stop look aperture stop look to someone at O to someone at O

EE_nnPP –– defines defines the cone of rays the cone of rays accepted by the accepted by the system

system FF₁₁’’

E’E’E’E’ = E= E_nnPP

Exit Pupil (E _x P) Exit Pupil (E _x P)

The exit pupil is the image of the aperture stop in the lenses The exit pupil is the image of the aperture stop in the lenses coming after it

coming after it (i.e. to the right of the AS)(i.e. to the right of the AS)

OO

LL₁₁ EE

EE

E’’E’’

EE’’’’

FF₂₂’’

EE””EE”” = = EE_xxPP

Location of Aperture Stop (AS) Location of Aperture Stop (AS)

• In a complex system, the AS can be found by considering each element in the system

• The element which gives the entrance pupil

subtending the smallest angle at the object point O is the AS

Example, Telescope Example, Telescope

eyepiece eyepiece Objective = AS =

Objective = AS = EE_nnPP O

Example: Find the AS of the Eyepiece Example: Find the AS of the Eyepiece

EE

EE

9 cm 9 cm

ФФ₁₁ = 1 cm= 1 cm

ФФ₂₂ = 2 cm= 2 cm ff₁₁’’ = 6 cm= 6 cm ff₂₂’’ = 2 cm= 2 cm

ФФ_ASAS= 1 cm= 1 cm

1 cm

1 cm 3 cm3 cm

OO

Diaphragm

Example: Eyepiece Example: Eyepiece

Find aperture stop for s = 9 cm in front of Find aperture stop for s = 9 cm in front of LL₁₁.. To do so, treat each element in turn

To do so, treat each element in turn –– find

find EE_nnPP for each (Lens 1, diaphragm, Lens 2)for each (Lens 1, diaphragm, Lens 2)

(a) E(a) E_nnPP ofof Lens 1Lens 1 –– no elements to the leftno elements to the left tan tan µµ₁₁ = 1/9= 1/9

defines cone of rays accepted defines cone of rays accepted

9 cm9 cm

1 cm1 cm OO µµ₁₁

Example: Eyepiece Example: Eyepiece

cm s

s

2 . 1 '

6 1 '

1 1

1

−

=

= +

Find aperture stop for s = 9 cm in front of

Find aperture stop for s = 9 cm in front of Diaphragm. Diaphragm.

(b) E(b) E_nnPP of Diaphragmof Diaphragm –– lens 1 to the leftlens 1 to the left EE

EE

9 cm9 cm 1 cm1 cm

2 . ' =1

−

= s M s

ФФ_DD’’ = 1.2 cm= 1.2 cm EE’’

EE’’

1.2 cm 1.2 cm

Look at the system Look at the system from behind the slide from behind the slide OO

ff₁₁’’ = 6 cm= 6 cm

Example: Eyepiece Example: Eyepiece

Calculate maximum angle of cone of rays accepted by entrance Calculate maximum angle of cone of rays accepted by entrance pupil of diaphragm

pupil of diaphragm

EE’’

EE’’

OO 0.6 cm0.6 cm

9 + 1.2 cm 9 + 1.2 cm

µµ₂₂

tan tan µµ₂₂ = 0.6/10.2 ≈= 0.6/10.2 ≈ 1/171/17

Example: Eyepiece Example: Eyepiece

(c) E(c) E_nnPP of Lens 2of Lens 2 –– 4 cm to the left of lens 14 cm to the left of lens 1

cm s

s 12 '

6 1 '

1 4

1

−

=

= +

4 3 12 ' = =

−

= s M s

ФФ₂₂’’ = 6 cm= 6 cm Look at the system Look at the system from behind the slide from behind the slide

9 cm

9 cm 4 cm4 cm

OO

ФФ₂₂ = 2 cm= 2 cm

Example: Eyepiece Example: Eyepiece

Calculate maximum angle of cone of rays accepted by entrance Calculate maximum angle of cone of rays accepted by entrance pupil of lens 2.

pupil of lens 2.

OO

3 cm3 cm

9 + 12 cm 9 + 12 cm µµ₃₃

tan tan µµ₃₃ = 3/21 = 1/7= 3/21 = 1/7

Example: Eyepiece Example: Eyepiece

8.1 tan µ₃ = 1/7

Lens 2

3.4 tan µ₂ = 1/17

Diaphragm

6.3 tan µ₁ = 1/9

Lens 1

Acceptance cone angle (degrees)

Entrance pupil Component

Thus

Thus µµ₂₂ is the smallest angle is the smallest angle

The diaphragm is the element that limits the cone of rays from O The diaphragm is the element that limits the cone of rays from O Therefore,

Therefore, Diaphragm = Aperture Stop Diaphragm = Aperture Stop in this eyepiecein this eyepiece

Example: Eyepiece Example: Eyepiece

Entrance pupil is the image of the diaphragm in L Entrance pupil is the image of the diaphragm in L₁₁..

9 cm 9 cm

EE’’

EE’’

1.2 cm 1.2 cm

OO

µµ₂₂ = tan= tan^-1-1 (1/17)(1/17)

ФФ_DD’’ = 1.2 cm= 1.2 cm EE_nnPP

Example: Eyepiece Example: Eyepiece

Exit pupil is the image of the aperture stop (diaphragm) in L Exit pupil is the image of the aperture stop (diaphragm) in L₂₂..

EE

EE

9 cm 9 cm

ff₂₂’’ = 2 cm= 2 cm

ФФ_DD = 1 cm= 1 cm

1 cm

1 cm 3 cm3 cm

OO

cm s

s 6 '

2 1 '

1 3

1

=

= +

3 2 6 ' = − = −

−

= s M s

ФФ₂₂’’ = 2 cm= 2 cm

Example: Eyepiece Example: Eyepiece

EE

EE

ff₂₂’’ = 2 cm= 2 cm

ФФ_DD = 1 cm= 1 cm

3 cm 3 cm

OO

ФФ_EExxPP’’ = 2 cm= 2 cm E’E’

EE’’

6 cm 6 cm

E E

P P

Aperture Stops Aperture Stops

Here is an aperture stop (AS) is formed by an aperture after a single lens.

Optics, E. Hecht, p. 150

Aperture Stops Aperture Stops

Here is an aperture stop (AS) is formed by an aperture before a single lens.

Optics, E. Hecht, p. 150

Aperture Stops Aperture Stops

Here is an aperture stop (AS) in a three-lens system. Ray traces are

shown for the chief ray from an object point at the top of the bulb and for a marginal ray from an axial object point.

Optics, E. Hecht, p. 151

The Chief Ray The Chief Ray

The chief ray is directed from the object point to the center of the

entrance pupil. The chief ray will thus always pass through the center of the AS.

W. Stevenson notes

Chief Ray Chief Ray

• for each bundle of rays, the light ray

which passes through the center of the aperture stop is the chief ray

• after refraction, the chief ray must also pass through the center of the exit and entrance pupils since they are

conjugate to the aperture stop

• E

P and E

P are also conjugate planes

of the complete system

Marginal Ray Marginal Ray

• Those rays (for a given object

point) that pass through the edge of the entrance and exit pupils

(and aperture stop).

Ray tracing with pupils and stops Ray tracing with pupils and stops

PP’’

QQ’’ OO

EE_nnPP

Q’’Q’’

P’’P’’

EE_xxPP PP

QQ ASAS TT

Marginal Rays from T,O Marginal Rays from T,O

•Must proceed towards edges of •Must proceed towards edges of EE_nnPP

•Refracted at L•Refracted at L₁₁to pass through edge of ASto pass through edge of AS

•Refracted at L•Refracted at L₂₂to pass (exit) through Eto pass (exit) through E_xxP.P.

LL₁₁ LL₂₂

Chief Ray from T Chief Ray from T

•Proceed toward centre of •Proceed toward centre of EE_nnPP

•Refracted at L•Refracted at L₁₁to pass though to pass though centre of AS

centre of AS

•Refracted at L•Refracted at L₂₂to pass (exit) to pass (exit) through centre of

through centre of EE_xxPP

TT’’ OO’’

Field Stop Field Stop

That component of the optical system that limits the field of view

θθ AA

dd

θθ = angular field of view= angular field of view A = field of view at distance d

A = field of view at distance d

FS

Field Stop Field Stop

The aperture that controls the field of view by The aperture that controls the field of view by limiting the solid angle formed by

limiting the solid angle formed by chief rays chief rays As seen from the centre of the entrance pupil As seen from the centre of the entrance pupil ( ( E E

P P ), the field stop (or its image) subtends the ), the field stop (or its image) subtends the smallest angle.

smallest angle.

In previous example, the lens L

In previous example, the lens L

is the field stop is the field stop

Entrance Window (E

W) Entrance Window (E

W)

The image of the field stop in all elements

The image of the field stop in all elements precedingpreceding itit

Defines the lateral dimension of the object that will be viewed Defines the lateral dimension of the object that will be viewed

Example: Camera Example: Camera

AS AS FS FS

Where is the Where is the entrance entrance window?

window?

Exit Window (E

W) Exit Window (E

W)

The image of the field stop in all elements

The image of the field stop in all elements following following itit

Defines the lateral dimension of the image that will be viewed Defines the lateral dimension of the image that will be viewed Example: Camera

Example: Camera

AS AS FS FS

Where is the Where is the exit window?

exit window?

Angular field of view in

Angular field of view in object object plane plane angle subtended by entrance window (

angle subtended by entrance window (E E

W) W ) at centre of entrance pupil (

at centre of entrance pupil ( E E

P P ) )

Angular field of view in

Angular field of view in image image plane plane angle subtended by exit window (

angle subtended by exit window ( E E

W W ) at ) at centre of exit pupil (

centre of exit pupil ( E E

P P ) )

Exit window and Angular field of view

That component whose

That component whose entrance window entrance window (E ( E

W) W ) subtends the smallest angle at the centre of the subtends the smallest angle at the centre of the entrance pupil (

entrance pupil (E E

P P ) )

Field Stop

Entrance window and Angular field of view

Field of a positive thin lens Field of a positive thin lens

Eye pupil Eye pupil

AS= AS= E E

P P P P

Q Q P P ’ ’

Q’ Q ’

Entrance pupil Entrance pupil

(small) (small)

Ob jec t fi eld

Ob jec t fi eld Image Image field field F F

Object point must be within cone Object point must be within cone (to left of lens) to be seen

(to left of lens) to be seen

αα = field of view in object space= field of view in object space αα’’ = field of view in image space= field of view in image space

FS= FS= E E

W W

α α α α ’ ’

Stops, pupils and windows in an optical system

Stops, pupils and windows in an optical system

AS AS FS FS E E

P P E E

W W E E

W W

E E

P P

α α ’ ’

α α

The f-number The f-number

The f/# or f-number is the ratio of the lens focal length to the diameter of the aperture stop: f/# = f/D.

Optics, E. Hecht, p. 152

Depth of Field Depth of Field

Consider a fixed image plane. The distance in the object space over which object points are in acceptable focus at the image plane is termed the depth of field.

W. Stevenson notes

Depth of Field Depth of Field

1 1 1 1

1

1 1

1 1

1 1

1

d P d P y P

y r r d r y

y y P

y d

m mr y

= = + ⇒ =

−

′ ′

= ⇒ =

− ′

2 2 2

2

2 2

2 2

2 2

22

d P d y P

y r d r y

y y P

y d

m mr y

= − ⇒ =

+

′ ′

= ⇒ =

+ ′

For a point located to the left of the object plane:

For a point located to the right of the object plane:

W. Stevenson notes

Depth of Field Depth of Field

( ) ( )

1 2

1 2 2 2

. :

1 1

2

,

2

Let the acceptable radius of the blur circle in the image plane be y y y Then

Depth of field d d P y Py mr

mr y mr y mr y

Usually y mr and we find Depth of Field Py

mr

Depth of field is inversely pro

′ = ′ = ′

⎡ ⎤

⎡ ⎤

′ ′

= + = ⎢⎣ − ′ + + ′⎥⎦ = ⎢⎢⎣ − ′ ⎥⎥⎦

′

≅ ′



. portional to the diameter of the entrance pupil

W. Stevenson notes

Depth of Focus Depth of Focus

Consider a fixed object plane. The distance in image space over which object points are in acceptable focus at the image plane is termed the depth of focus.

W. Stevenson notes

Depth of Focus Depth of Focus

. :

2 2

. Let the acceptable radius of the blur circle in the image plane be y Then

y r

d P

Depth of focus d P y

r

Depth of focus is inversely proportional to the diameter of the exit pupil

′

′ ′

′ = ′

′ ′ ′

= =

′

W. Stevenson notes