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Biophotonics

Geometric Optics

NPTEL Biophotonics 1

Geometric Optics

• Geometric optics treats light as a ray (consisting of corpuscles) which obeys certain laws at interfaces between two different materials.

• The following lecture introduces the geometric optics analysis of lightanalysis of light

• Keywords: Geometric optics, ray theory, basic optical components


Overview

• In this lecture you will learn,

• A historical perspective of optics

• Reflection and refraction

• Total internal reflection

• Planar and spherical optical elements

• Matrix method to analyze optical systems• Matrix method to analyze optical systems

• Some lens systems


Early Days of Optics

• Mirrors dating back to 2000 BC– Pyramid of Sesostris Egypt

– Polished metal

– Metal on glass, alloys

• Burning glass (lens for focusing)– Mentioned in Aristophanes 424 BC

• Refraction studies• Refraction studies– Plato’s Republic (380 BC)

– Ptolemy (100 AD)

• Straight line propagation of light, which is empirically observed, would imply “rays” of light (particle model) travelling obeying some laws. Hero of Alexandria (~ 40 AD) postulated shortest distance path for light rays

4NPTEL Biophotonics

• Roger Bacon (1215 – 1294) suggested using lenses for correcting eyesight– Concepts of refraction appreciated

– Focusing (or bending of light) in curved surfaces

• Tinkering with these lenses results in – Lippershey invents refracting telescope (1608)

– Janssen invents compound microscope


– Janssen invents compound microscope

– Kepler discovers small angle law of refraction (1611)

• Refracted angle is proportional to incident angle

– Snell discovers law of refraction (1621)

– Descartes puts it in terms of sine function (1637)

– Fermat proposed law of least time (shortest optical path) (1637)

• In essence a restatement of Hero’s postulate but with the concept of refractive index

– Refractive index of a material identified as its capacity to bend lightNPTEL Biophotonics 5


• Application of law of refraction leads to the design (and improvement) of several optical components like lenses, microscopes, telescopes etc.

• In addition calculus developed around 1700’s to enable studies of curvilinear surfaces like spherical lenses, studies of curvilinear surfaces like spherical lenses, parabolic mirrors etc


Reflection and Shortest Optical Path

• Law of reflection– Any point P has a conjugate point P’

– Q is the point of observation

– We can view the light ray as emerging from the conjugate point P’

– Shortest path is P’Q

– From the geometry < PON = < NOQ

P

P’

Q

O

N

– From the geometry < PON = < NOQ

– i.e., incident and reflected angles are same

• Reflection will be symmetric with respect to normal

• Note that we derive this result using the shortest path hypothesis


Refraction• Empirical observation that light

bends when moving between different media, e.g. glass

• This means that speed of light in the two media have to be different (why?)different (why?)

• Law of refraction

– Minimize

– Subject to constraint


2

22

1

11 secsec

v

d

v

d θθ +

.2

22

1

11 const

v

vd

v

vd

y

x

y

x =

+

Refraction

• Minimizing the expression under the constraint given in the previous slide provides the famous law of refraction. It is left as an exercise for the more mathematically inclined readers to show this.

Medium 2, v2Q

O2θ d1


Medium 1, v1

P

O

N1θ d2

2

2

1

1 sinsin

vv

θθ =

Refractive Index and Speed of Light• From the observation of light refracting through a glass

slab, one can conclude that light speed in glass must be lower than the light speed in air to explain the shortest path hypothesis in the context of the observed light path

• We define cvnvn == 2211n

cv =

– We will see later that this c is a universal speed of light

• Using snell’s law and the definition above we can show that rays bend towards normal when going from a rarermedium to a denser medium and away from the normal when going from denser to rarer. Denser and rarer refer to refractive index being higher and lower respectively


Total Internal Reflection

• Rays bend towards normal when going from a rarer medium to a denser medium and away from the normal when going from denser to rarer

Medium 2, n2

Medium 1, n1

Q

O

θ

2θ

d

d1

• This implies that when light travels from denser to a rarer medium it will get reflected beyond a critical angle


Medium 1, n1

P N1θ d2

n1 > n2

= −

1

21sinn

ncθ

Total Internal Reflection

• This phenomenon is called total internal reflection

• TIR results in natural phenomena such as mirage. TIR is also the basic concept behind optical

Medium 2, n2

Medium 1, n1

Q

O

θ

2θ

d

d1

basic concept behind optical fibers and is also exploited in certain imaging and molecular sensing techniques to be discussed later


Medium 1, n1

P N1θ d2

n1 > n2

= −

1

21sinn

ncθ

Recap

• Law of reflection and refraction follow from the postulate that light rays travel in the path that minimizes time of flight

• Alternately, minimize the ‘Optical path length’ where OPL = ∫ nds=

OPL = ref. index multipled by geometric path length


∫path

nds

Recap

• Refractive index is related to speed of light in the medium as

• All of geometric optics can be analyzed by applying the laws of reflection and refraction at the boundaries

n

cv =

laws of reflection and refraction at the boundaries (interfaces) of the objects which may be comprised of various geometrical shapes such as a parabolic mirror or a spherical lens


Non-Planar Reflectors

2axy =x

y

af 4/1=

• Parabolic mirrors: Using the law of reflection at various points where the incident light ray strikes the parabolic surface, one can show that all rays will go through the focus shown in the diagram. So the parabolic surface acts as a perfect focusing mirror


Spherical Mirrors

• However, spherical mirrors are easier to manufacture. In the paraxial regime a spherical mirror can be assumed to have a focus of f = R/2. Paraxial approximation

C FR/2

siParaxial approximation considers only rays that are very close to the normal, i.e. small angles where sinθ = θ


so

si

Mirror Equation

• Using the geometrical construction shown in the diagram, tt can be shown that

Rss io

211 =+C F

R/2

si

• This is the mirror equation (Gauss)


so

si

Finite Objects: Magnification

C F

si

• It can be shown that image magnification is given by,

• Ray Diagrams

osR

RM

2+−=


so

• Virtual image from convex mirrors

• Use ray diagrams

• Sign convention:

• Convex: f is negative

• Concave: f is positive

Matrix Method for Geometric Optics Analysis

• Optical systems with several components are easily analyzed by a matrix method. Here, a ray is characterized by the position and the direction as shown in the diagram.

• Transmission through any component is • Transmission through any component is described by a matrix multiplication as follows


y

θ

=

1

1

2221

1211

2

2

θθy

MM

MMy

Matrix Method for Geometric Optics Analysis

• Transmission through a number of components, e.g a lens with focal length f1, free space passage through distance d1, reflection at a mirror followed by free space passage through distance d2 and finally transmission through a lens with focal length f2 can be analyzed by simply multiplying the appropriate matrices in the right order. For the train of optical elements given below, the order. For the train of optical elements given below, the equivalent matrix will be

Mnet = Mn*Mn-1*....*M2*M1


1 2 n-1 n

Some Matrix Representations

• Consider translation in free space through distance d,

• Then y2 = y1+ d*θ1 and θ2 = θ1

• Therefore, the matrix representing translation will be

=

10

1 dM

• Similarly the matrix representing plane reflection will be


−=

10

01M

Spherical Refraction

• The matrix representing spherical refraction as shown in diagram can be found using simple geometry to be,

C FR/2

n1

n1

n2

−= 121

01

n

n

Rn

nnM


so

22 nRn

Spherical Lenses

• Using the previous result for a single spherical interface we can analyze the transmission through a spherical lens as shown in the diagram

t

R1 R2

• One has to multiply the matrix for the spherical interface with radius R2, followed by translation through thickness t, followed by spherical interface with radius R1


R1 R2

Spherical Lens: Focal length

• Assuming refractive index to be same on both sides of the lens, one can show that the equivalent matrix for the lens is where,

• This result is for t = 0 or thin lens

−= 1

101

fM ( )

−−=

12

111

1

RRn

f

• This result is for t = 0 or thin lens


Focusing by Thin Lens

• A consequence of our definition of f in the previous slide is that f > 0 for bi-convex or plano-convex lenses and f < 0 for bi-concave or plano-concave lenses.

• Using the matrix derived in the previous slide, it is straightforward to show that a lens will focus all parallel rays to a single point at a distance f from the center of rays to a single point at a distance f from the center of the lens. Therefore, f in the previous slide is the focal length of the lens.


Diffraction Limit

• According to geometric optics all parallel rays are focused on to a single point. But later on we will see that the wave nature of light implies that this is not possible. There is a limit of how much one can focus light. This limit is called the diffraction limit.


Lens Equation

• Applying the transformation matrix

for a lens to the set of rays from P we get,

y’ = (1 – v/f)y + {u(1 – v/f) + v}θ and

−= 1

101

fM

y’ = (1 – v/f)y + {u(1 – v/f) + v}θ and θ’ = -y/f + (1 – u/f)θ


P P’

u v

Lens Equation

• If point P is to be imaged on to point P’ all rays passing through the lens must pass through P’ irrespective of θ. This means y’ must be independent of q. By rearranging the coefficient of q in the equation for y’, we get the famous lens equation that links the object and image distance with the focal length of the lens

1/f = 1/u + 1/v1/f = 1/u + 1/v


P P’

u v

Lens Magnification

Also, we see that, image magnification = M = y’/y = -v/u. By appropriate choice of f and u, one can create single lens object magnifiers which are the basic stepping stones to optical microscopy.


P P’

u v

Lens Systems• Using the transformation matrix for lens derived in this

lecture, one can analyze lens systems consisting of multiple lenses (convex or concave) with different focal distances.

• Modern microscope lenses consists of several lenses arranged to compensate for image aberrations such as arranged to compensate for image aberrations such as spherical aberration where rays striking at different distances from the lens axis (called the optical axis) focus at different points (due to the error in approximating a spherical surface with a parabolic surface); or chromatic aberration where light with different wavelengths (color) gets focused at slightly different focal points.


Summary

• As we saw in this lecture, geometric optics is the simple application of laws of reflection and refraction and the behavior of an optical component is a function of its geometry (e.g. focusing of rays by a parabolic surface.


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