1. Geometrical Optics Geometrical light rays Ray matrices and ray vectors
Matrices for various optical components
The Lens Maker’s Formula
Imaging and the Lens Law
Mapping angle to position
Animations from Sergei Popov

2. Leerdoelen In dit college leer je:
Wat de benaderingen zijn van geometrische optica
Wat de belangrijkste karakteristieken van lenzen zijn
Hoe je de stralengang door optische systemen met o.a. lenzen kunt analyseren
Een matrix-methode voor geometrische optica
Toepassing: microscopie, fotografie en bijbehorende eigenschappen
Hecht: 5.1 t/m 5.4; 5.7; 6.1 en 6.2

3. Three Models of Light Photons Rays Waves
The waves are all centered on the light source.
The rays all move away from the source ->
The wavefronts are perpendicular to the rays.

4. Is geometrical optics the whole story?
No. We neglect the wave nature of light.
Also, our ray pictures seem to imply that, if we could just remove all aberrations, we could focus a beam to a point and obtain infinitely good spatial resolution.
Not true! The smallest possible focal spot is the on the order of the wavelength l. Same for the best spatial resolution of an image. This is due to diffraction, which is not included in geometrical optics. ~0 ≈l

5. Arguments in favor of geometrical optics
Remember week 1 in which we discussed three reasons to believe that the ray picture of light is useful:
The straight propagation of light that we see entering through a window.
The camera obscura in which a small aperture produces an image that is upside down.
The very narrow beams that are produced by a laser.

6. Relation between object, image and focus
Gauss Law f f
Magnification:

7. Lensmaker’s formula

8. Types of lenses Lens nomenclature:
Which type of lens to use (and how to orient it) depends on the aberrations (next lecture) and application.

9. Sign convention for R

10. Tracing a few rays (Fig 5.20 Hecht)
Rays through the center of the lens go straight
Parallel rays from the object pass through Fi
Rays through Fo propagate parallel to the axis behind the lens

11. Image formation by a thin lens
The image is formed at the point where these rays cross
A real image is formed if s0 > f
The magnification of the image is given by Mt
For a single, positive lens, the magnification will be negative

12. Virtual Images
A virtual image occurs when the outgoing rays from a point on the object never actually intersect at a point but can be traced backwards to one.
Negative-f lenses have virtual images, and positive-f lenses do also if the object is less than one focal length away.
Virtual image
Virtual image
Object
Object infinitely far away
f < 0
f > 0
Simply looking at a flat mirror yields a virtual image.

13. Combination of 2 lenses spaced < F1, F2

14. Ray Optics
axis
We'll define light rays as directions in space, corresponding, roughly, to k-vectors of light waves.
We won’t worry about the phase. We also ignore reflections.
Each optical system will have an axis, and all light rays will be assumed to propagate at small angles to it. This is called the Paraxial Approximation: sin q  tan q  q

15. The Ray Vector A light ray can be defined by two coordinates:
xin , qin
xout, qout
A light ray can be defined by two coordinates:
its position, x
its slope, q
optical ray q x
Optical axis
These parameters define a ray vector, which will change with distance and as the ray propagates through the optical elements.

16. Ray Matrices
For many optical components, we can define 2 x 2 ray matrices.
An element’s effect on a ray is found by multiplying this matrix with the ray vector.
Ray matrices
can describe
simple and com-
plex systems.
Optical system ↔ 2 x 2 Ray matrix
These matrices are often (uncreatively) called ABCD Matrices.

17. Physical meaning of two of the matrix elements
𝐴= 𝑥 𝑜𝑢𝑡 𝑥 𝑖𝑛 𝑖𝑓 𝜃 𝑖𝑛 = 0
𝐷= 𝜃 𝑜𝑢𝑡 𝜃 𝑖𝑛 𝑖𝑓 𝑥 𝑖𝑛 = 0
angular magnification
spatial magnification

18. For cascaded elements, we simply multiply ray matrices.
Notice that the order looks opposite to what it should be, but it makes sense when you think about it.

19. Ray matrix for free space or a medium
If xin and qin are the position and slope upon entering, let xout and qout be the position and slope after propagating from z = 0 to z.
Xin, qin
z = 0
xout qout z
Rewriting these expressions in matrix notation:

20. Ray Matrix for an Interface
qin n1
qout n2
xin
xout
At the interface, clearly:
xout = xin.
Now calculate qout.
Snell's Law says: n1 sin(qin) = n2 sin(qout)
which becomes for small angles: n1 qin = n2 qout
 qout = [n1 / n2] qin

21. Ray matrix for a curved interface [radius R ]
At the interface, again:
xout = xin.
To calculate qout, we must calculate q1 and q2.
If qs is the surface slope at the height xin, then
q1 = qin+ qs and q2 = qout+ qs
qin n1
qout n2
xin q1 q2 qs R z
z = 0
qs = xin /R
q1 = qin+ xin / R and q2 = qout+ xin / R
Snell's Law: n1 q1 = n2 q2

22. A thin lens is just two curved interfaces:
We’ll neglect the glass in between (it’s a really thin lens!), and we’ll take n1 = 1.
This can be written:
The Lens-Maker’s Formula
Next: justify that f = focal length
where:

23. Ray matrix for a lens
The quantity f, is the focal length of the lens. It’s the most important parameter of a lens. It can be either positive or negative.
f > 0
f < 0
R1 > 0
R2 < 0
R1 < 0
R2 > 0
If f > 0, the lens deflects rays toward the axis.
If f < 0, the lens deflects rays away from the axis.

24. A lens focuses parallel rays to a point one focal length away.
For all rays xout = 0!
A lens followed by propagation by one focal length:
free space lens
Assume all input rays have qin = 0 f
At the focal plane, all rays converge to the z axis (xout = 0) independent of input position, with angle –xin/f.
[Parallel rays at a different angle focus at a different xout.]
Notice we have now proven the Lensmakers’ Law.

25. Concave Spherical Mirror
A concave mirror
A radio telescope
Like a lens, a curved mirror will focus a beam. Its focal length is R/2,
with R the radius of curvature.

26. Ray Matrix for a Curved Mirror
Consider a mirror with radius of curvature, R, with its optic axis perpendicular to the mirror:
qin
qout
xin = xout q1 qs R z
Like a lens, a curved mirror will focus a beam. Its focal length is R/2.
Note that a flat mirror has R = ∞ and hence is described by an identity ray matrix.

27. Consecutive lenses f1 f2
Suppose we have two lenses right next to each other (with no space in between).
So two consecutive lenses act as a single lens whose focal length is computed by the inverse sum.
As a result, we define a measure of inverse focal length, the diopter diopter = 1 m-1

28. A system images an object when B = 0.
When B = 0, all rays from a point xin arrive at a point xout, independent of their angle:
xout = A xin
When B = 0, A is the magnification.

29. The Lens Law From the object to the image, we have: 1) A distance do
2) A lens with focal length f
3) A distance di
This is the Lens Law,
due to Gauss.

30. Imaging Magnification
If the imaging condition,
is satisfied, then:
So:

31. Micro-scopes M1 M2
Image plane #1
Eye- piece
Objective
Image plane #2
Microscopes basically consist of a set of lenses, and:
1. In a microscope, the object is really close and we wish to magnify it.
2. Lens aberrations should be minimized (next lecture).
A microscope is effectively a telescope in reverse. The goal is to magnify the object, not its angular diameter.
Objective
Eyepiece
The microscope objective yields a magnification of from 5 to 100, corresponding to focal lengths of 40 mm to 2 mm.
The eyepiece yields a magnification of 2 to 10, corresponding to various focal lengths and distances.

32. Microscope Terminology
CHOOSING A COMPOUND MICROSCOPE Compound microscopes differ from simple magnifiers in that there are two separate magnification steps that occur instead of one. The objective lens is nearest the subject under observation and provides a magnified real image. The eyepiece magnifies the real image provided by the objective and yields a virtual image appropriate for the human eye. Remember that the eye has its own lens, which relays virtual images onto the retina.Microscopes have been standardized over time to simplify design and manufacture. Most microscopes employ the Deutsche Industrie Norm, or DIN standard configuration, while the Japanese Standard (JIS) is less commonly used. DIN microscopes begin with an object-to-image distance of 195mm, and then fix the object distance (with respect to the rear shoulder of the objective) at 45mm. The remaining 150mm distance to the eyepiece field lens sets the internal real image position, defined as 10mm from the end of the mechanical tube. Objective lens thread is the same for DIN and JIS, that is " (20.1mm) diameter, 36 TPI, 55° Whitworth.
EYEPIECES Microscope eyepieces generally consist of an eye lens unit and a field lens unit. Field lenses are usually plano-convex in shape, and are intended to help gather more light at the real internal image plane. Eye lens units can range from a simple plano-convex lens to a complex lens system consisting of four or five elements. Eyepiece types vary in how well they perform. Huygenian designs employ two plano-convex lenses positioned with both convex surfaces toward the object, and are good for use with lower power achromatic objectives. Ramsden designs, useful for high power achromatic objectives, use two plano-convex lenses with the convex surfaces facing each other. Kellner eyepieces are often called Wide-Field eyepieces. Kellner eyepieces upgrade to an achromatic doublet as the eye lens; Periplan has a three-element eye lens for exceptional correction. Higher quality eyepieces must be used with semi-plan and plan objectives to maintain the flatness of the field.
ILLUMINATION Illumination is critical for effective microscopy. Most DIN microscopes employ some form of sub-stage illumination in which light enters the objective through the sample from below. Lower power objectives, such as 4X or 10X, can accommodate sub-stage and incident (from above the sample) types of illumination due to comparatively large working distances and lower numerical aperture. Higher power objectives, such as 40X and 100X, can realistically only be used with sub-stage lighting techniques.   
QUALITY CORRECTION Three generally accepted levels of correction/quality for microscope objectives are achromatic, semi-planar, and planar. Achromatic objectives - the most common type - have a flat field in about the center 65% of the image. Planar, or plan, objectives correct best for color and spherical aberrations, and display better than 95% of the field flat and in focus. Semi-planar objectives (sometimes called semi-plan or micro-plan) are intermediate to the other two types with about 80% of the field appearing flat.

33. F-number f f f / # = f / d Confusing!! f d1 d2 f / # = 2 f / # = 1
The F-number, “f / #”, of a lens is the ratio of its focal length and its diameter.
f / # = f / d
Confusing!!
For example, a lens with a 25 mm aperture and a 50 mm focal length has an f-number of 2, which is usually designated as f/2 f f d2 d1 f
f / # = 2
f / # = 1
Small f-number lenses collect more light but are harder to engineer.

34. Size of blur in out-of-focus plane
Depth of Field
Only one plane is imaged (i.e., is in focus) at a time. But we’d like objects near this plane to at least be almost in focus. The range of distances in acceptable focus is called the depth of field.
It depends on how much of the lens is used, that is, the aperture.
Object
Out-of-focus plane
Size of blur in out-of-focus plane
Image f
Focal plane
Aperture
The smaller the aperture, the greater the depth of field.

35. Depth of field example A large depth of field isn’t always desirable.
f/32 (very small aperture; large depth of field)
f/32 means f/D = 32, the focal length of the lens is 32 times larger than the aperture diameter
f/5 (relatively large aperture; small depth of field)
wikipedia.org
A small depth of field is also desirable for portraits.

36. Tot slot Wat hebben we gezien: Geometrische optica
Berekeningen met lenzen
ABCD-matrix voor optische systemen
Matrix-berekeningen in geometrische optica
Inleiding in microscopie