theory of conformal optics
TRANSCRIPT
Theory of conformal optics Study of classical optics via conformal invariants
Amin Vahdat Farimani
2/17/2015
Gmail: [email protected]
A simple mathematical tool is presented for study of classical optics via conformal invariants. In math, Erlanger program conceives geometry as study of invariants properties under a group of transformations. Theory of conformal optics should be considered as an Erlanger program for study of classical optics when conformal mappings are selected as transformations.
Abstract
A simple mathematical tool is presented for study of classical optics via conformal invariants. In math,
Erlanger program conceives geometry as study of invariants properties under a group of
transformations. Theory of conformal optics should be considered as an Erlanger program for study of
classical optics when conformal mappings are selected as transformations.
Keywords
Conformal optics, refraction, reflection, diffraction, interference, cross ratio, Poincare disk model, unit
disk
Introduction
Felix Klein (1849-1925) in his βErlanger programβ presented a new definition for geometry βstudy of
invariants properties under a group of transformationsβ. Transformations are called conformal
mappings when angles remain unchanged under mappings. As Klein definition for geometry, conformal
optics should be defined. It is study of properties of optical phenomena which are invariant under
conformal mappings. In present paper a variety of optical phenomena will be analyzed by use of one
fundamental axiom.
In section-1 fundamental axiom of the conformal optics will be presented.
In section-2, section-3 and section-4 refraction, reflection and interference phenomena will be analyzed
respectively.
Diffraction is subject of section-5. In all sections the fundamental axiom plays central role. In section-5
an auxiliary concept is presented to develop cross ratio. In other words the Poincare disk model for the
hyperbolic geometry is generalized to a variety of conformal non-Euclidean models. These conformal
models are needed for quantitative analyzing of diffraction phenomenon.
The last section-6 is conclusion.
Section-1 (Fundamental axiom)
When an optical phenomenon occurs, there is at least one non-zero quanta property which is invariant
under conformal mappings from phenomenon scene to the unit disk. (See fig.1) In fig.1 point βSβ
denotes to source of light, π1 and π2 to slots and π3 to position of Geiger counter. Also βCβ refers to
boundary of phenomenon scene.
phenomenon unit disk
s
Z1
Z3
Z2
w1
w2w3
mapping
scene
C
fig.1: Mapping from phenomenon scene to the unit disk
Section-2 (Refraction phenomenon)
L1
L2
media 1
media 2boundary
i
r
L ( normal to boundary )
fig.2: Refraction scene
According to the fundamental axiom we are sure that one non-zero property for refraction phenomenon
exists. Let examine cross ratio (L1 , L2; L) as property of refraction. (See appendix-1)
(πΏ1 ,πΏ2; πΏ) = sinβ πΏ1πΏ sinβ πΏ2πΏ = sin(π β π) / sin(π β π) = sin π sin π 1
Since cross ratio is invariant under conformal mapping then invariance of βsin π sin π β is property of
refraction. It means that
sin π sin π = ππππ π‘πππ‘ 2
Section-3 (Reflection phenomenon)
L1 media 1 media 2
boundary
i
L ( normal to boundary )L ( normal to boundary )
L1
r
fig.3: Reflection scene
According to the fundamental axiom we are sure that one non-zero property for reflection phenomenon
exists. Let examine cross ratio (L1 , L2; L) as property of reflection.
(L1 , L2; L) =sinβ L1L sinβ L2L = sin(Ο β i) / sin(Ο β r) = sin i sin r = constant 3
From symmetry we expect that when L1 is parallel to Ξ£, L2 is also parallel to Ξ£. It means that from
(i = Ο 2) we conclude the(r = Ο 2 ). So from 3 we obtain sin i / sin r = 1 or
π = π 4
Equation 4 is property of reflection.
Section-4 (Interference phenomenon)
boundary
s
L1
L
L2
Fig.4: Interference scene (S shows Light source; L1 and L2 show slots; L shows Geiger counter position)
In fig.4 elements of scene are points. So definition (1-A) from appendix-1 is usable.
L1 , L2; L = (πΏ1πΏ)/(πΏ2πΏ) 5
It is clear that any function of cross ratio is invariant under conformal mappings. Let examine logarithm
of cross ratio as property of interference.
ln L1 , L2; L = ln(πΏ1πΏ) β ln(πΏ2πΏ) 6
According to the fundamental axiom, property of phenomenon must be quanta. So logarithm of the
cross ratio must be quanta.
ln L1 , L2; L = k Ξ· , k = 0, 1, 2, 3,β¦ 7
Note that Ξ· is a parameter that depends only on specifications of light. (For example energy of light)
We prefer not define Ξ· as wave-length because of dependency of length to angle in present paper.
Ξ·=βπ πΈ (= π) 8
Where "β" is Plank constant,"πβ is speed of light, βπΈ" is energy of light and "Ξ»" is wave-length of light.
From 6, 7 and 8 we have
ln(πΏ1πΏ) β ln(πΏ2πΏ) = π.βπ πΈ 9
Equation-9 proposes a definition for length between points πΏ1 and πΏ . It should be noted that in theory
of conformal optics scale of length is not arbitrary at all. So, interference phenomenon has a property
that is described in equation-9. It is clear that logarithm scale for length yields wave formula of
interference 11.
Length of(πΏππΏ) = ln(πΏππΏ) 10
From 9 and 10 we obtain
Length of(πΏ1πΏ) β Length of(πΏ2πΏ) = π.βπ πΈ 11
What happens when right side of equation-9 is constant? Since L1 , L2; L is complex, let distinct two
cases.
Case (a): Constant-phase
Dashed lines in fig.5 are circles pass through points L1 , L2 and their phases are constant.
s
L1
L
L2
Fig.5: Constant-phase diagram for L1 , L2; L
s
L1
L
L2
Fig.6: Constant-amplitude diagram for L1 , L2; L
Case (b): Constant-amplitude
Curved lines in fig.6 are circles that donβt intersect each other.
It should be noted that:
- Each dashed line in fig.5 is perpendicular to all curved lines of fig.6.
- Each curved line in fig.6 is perpendicular to all dashed lines of fig.5.
- Dashed lines in fig.5 may be considered as electric field or magnetic field and curved lines in fig.6
as position of Geiger counter. In present paper we do it.
- Dashed lines in fig.5 may be considered as possible paths between points L1 , L2 (for example
fluid flow in fluid mechanics) and curved lines in fig.6 as co-potential points.
Now let L1 move toward L2 in fig.6. We expect a descriptive diagram for diffraction phenomenon.
(fig.7)
s
L
=L1 L2
Fig.7: Descriptive diagram of diffraction phenomenon (for single slot)
For all points on curved lines in fig.7 L1 , L2; L is equal to zero. In other words for K = 0 in equation-9
all curved lines obtained.
Unfortunately quantum property of light is disappeared. So interference phenomenon does not lead us
to a quantitative theory of diffraction. A quantitative theory of diffraction will be presented in the next
section.
Section-5 (Diffraction phenomenon)
In this section a diffraction theory is extracted from the fundamental axiom of conformal optics. At first
we need to study conformal non-Euclidean models in the unit disk. (See appendix-2) According to the
fundamental axiom there is a property that remains unchanged under a conformal mapping from
diffraction scene to the unit disk. (fig.8)
phenomenon unit disk
s
Z1
Z3
Z2
w1
w2w3
mapping
scene
C
Fig.8: Mapping π€(z) from single-slot diffraction scene to the unit disk (z1z2 is single-slot)
In appendix-2 π€(L1), w(L2); w(L3 ) or in abbreviation w1 , w2; w3 is calculated when w3 locates on
the unit circle. (fig.9)
w1 , w2; w3 = w1w3 w2w3 = [πππ πππ’π‘π(w1w3) πππ πππ’π‘π(w2w3)] ππΙ΅ 12
Ο΄ is phase of w1 , w2; w3 (See appendix-2 fig.10)
Since πππ πππ’π‘π w1w3 = 1 then
w1 , w2; w3 = [1 cos(π cosπ)[1 β tanπ tan π cosπ ] 13
From paper point of view each π-line refers to a field-line, electric or magnetic field. Anyway Fraunhofer
approximation for diffraction will be extracted from equation-13 when βπβ is small enough.
If π β 0 then we obtain from equation-13
w1 , w2; w3 = (1 + tan(π cosπ) tanπ)ππΙ΅ 14
Let define π½ = π cosπ
tan(π cosπ) tanπ = π sinπ tan π½
π½ 15
π½ β 0, tanπ½ β sinπ½. Now let calculate logarithm of w1 , w2; w3 in equation-14
πΏπ w1 , w2; w3 = π sinπ sin π½
π½+ πΙ΅ 16
We need real part of equation-16 because Geiger counter measures real parameters. When you move
on a π-line real part of πΏπ w1 , w2; w3 represents electric field.
π πππ[πΈ w1 , w2; w3 ] = πtan π½
π½ sinπ = πΈ = πΈ0 sinπ 17
For a period of oscillation we obtain from equation-17
πΌ = πΈ2 = πΈ02 2 =
1
2(π2π πππ2π½) 18
Equation-18 is the Fraunhofer approximation of diffraction phenomenon. Note that π2 2 should be
measured via Geiger counter as πΌ0.
For large enough of βπβ diffraction model deviates from equation-18. It will be in general form as
follows:
I = [πΏπ 1
cos π cos π [1βtan (π cos π) tan π)] ]2 19
The cross ratio and related diffraction should be defined for four points as follows:
w1 , w2; w3 , w4 =w1w3
w1w4
w2w4
w2w3=
1+tan (π cos π) tan π
1βtan (π cos π) tan πππΙ΅ 20
I = {π πππ πΏπ w1 , w2; w3 , w4 }2 = [πΏπ
1+tan (π cos π) tan π
1βtan (π cos π) tan π]2 21
Now concentrate on equation-16. What is the meaning of imaginary part of logarithm? For many
reasons we have to define phase of πΏπ w1 , w2; w3 as time. (See appendix-2, fig.10)
Ο΄ β π‘ 22
So when βπβ is small enough, πΏπ w1 , w2; w3 should be represented as an interval of space-time in the
Minkowski diagram. In other words πΏπ w1 , w2; w3 is an interval of space-time between events
w1 and w2. From this point of view we look at the unit disk as a space-time diagram. A space-time
diagram in the unit disk will be presented in other paper.
Section-6 (Conclusion)
We analyzed a variety of optical phenomena by use of just one axiom named fundamental axiom. Also a
bridge made between classic optics and conformal mapping via fundamental axiom. In appendix-1 cross
ratio for three points is defined mathematically. Although cross ratio should be developed for more than
three points we didnβt use developed form in present paper to hold simplicity. A theory of diffraction is
derived for single slot and it is shown that Fraunhofer equation of diffraction is obtained when slot is
small enough.
In appendix-2 a novel geometrical method is presented to understanding diffraction phenomenon. In
fact Poincare model in the unit disk is generalized to π βmodels. Although we donβt pay any attention
these models are as beautiful as Poincare model. (For example triangle inequality is valid in π βmodels)
In the end of section-5 we refer to Minkowski diagram. A novel representation of space-time in the unit
disk is predicted by the conformal optics. This space-time diagram must be locally in accordance with
the Minkowskian metric. (In fact it is done by author before and is presented in other paper)
Finally, conformal optics presents a geometrical approach to understanding optical phenomena.
Appendix-1 Cross ratio
In math, cross ratio for three points is defined by two ways as below
1-A
L1 , L2; L = L1L L2L
L1 , L2 and L , are three points in the complex (or projective) plane and donβt lie on a line.
2-A
L1 , L2; L = sinβ L1L sinβ L2L
L1 , L2 and L , are three lines in the complex (or projective) plane and intersect each other in unique
point.
These two different definitions show duality of points and lines in projective plane. The most important
property of cross ratio (or any function of cross ratio) is invariance under conformal mappings.
Cross ratio for four points is also defined by two ways as below:
1-B
w1 , w2; w3 , w4 =w1w3
w1w4
w2w4
w2w3
Where wi a point in the complex plane
2-B
w1 , w2; w3 , w4 =π ππβ w1w3
sinβ w1w4 sinβ w2w4
π ππβ w2w3
Where wi a line in the complex plane
Appendix-2 Conformal non-Euclidean models in the unit disk
w1
w2
w3w6
w5
w4
d o
Fig.9: Conformal models in the unit disk (w5w6 ππ π
2β ππππ and ow2 = ππ )
w1
w2
w3
o
Fig.10: Phase of (w1,w2;w3) is equal to angle shown at vertices w3
In fig.9 two lines π βline and π
2βline are shown. Note that
π
2β ππππ known as Poincare-line. Each
π βline belongs to related π βmodel and each π βmodel represents related π βgeometry. So Poincare
model is a special case of a π βmodel. In other words the hyperbolic geometry is special case of
π βgeometry. Let define π βmodel so that all its lines cut the unit circle at angle π. Without any
restriction suppose that π€(π§) is determined by three points π€1 ,π€2 πππ π€3 such that π€1 is the center
of the unit disk. (See fig.9) Simple calculations yield:
πππ πππ’π‘π w1w3 = πππ πππ’π‘π w1w6 = πππ πππ’π‘π w1w5 = πππ πππ’π‘π w1w4 = 1
πππ πππ’π‘π w2w3 = [1 β tanπ tan π cosπ ] cos(π cosπ)
πππ πππ’π‘π w1w2 = π
πππ πππ’π‘π w2w5 = 1 + π
πππ πππ’π‘π w2w6 = 1 β π
πππ πππ’π‘π w2w4 = [1 + tanπ tan π cosπ ] cos(π cosπ)
πππ πππ’π‘π w3w5 = 2 1 + sin(π³ + π)
πππ πππ’π‘π w3w6 = 2 1 β sin(π³ + π)
πππ πππ’π‘π w4w5 = 2 1 + sin(π³ β π)
πππ πππ’π‘π w4w6 = 2 1 β sin(π³ β π)
sinπ³ = π cosπ
ππ = 1 (2 cosπ)
We call βwiwj" cross functions of the unit disk. Cross function for three points or more than three points
can be calculated by these functions. From physical point of view these functions are important. For
example w2w3 leads us to the Fraunhofer approximation of diffraction phenomenon when
πππ πππ’π‘π w1w2 be small enough.