Complex numbers/phasors and amplitudes

There was a question last time: since the Re part of a complex field is the physical one, why do we need the full phasor length as the physical field magnitude?

The physical field is the Re part of E any time. Re[ ( )]phys complexE t E t

But the phasor as usually drawn represents t=0. So Re[phasor] is the physical field only at t = 0.

The maximum physical field strength (amplitude) won’t generally be seen t= 0, but instead whenever the phasor rotates across the real axis, given by the phase angle). So the phasor magnitude is the physical field amplitude.

Absorption filters (colored glass)

Transmission vs :

short-pass filter

Multilayer filters and Multilayer mirrors: much sharper and specialized features possible.

Notch filter

3-notch filterlong-pass filter

-Parallel components of E, B are equal at interfaces. -Have two directions in each material except the last.

cos cos j j j j j vac j jk l n k l Phase shifts to the next boundary:

E’s are defined where their arrows start (except Eo). Match || parts at right side of j, left side of j+1 layers. P polarization:

1 1 1

1 1 1

cos cos cos cosj j

j j

i ij j j jj j

i ij jj jj j

e e E E

n nE En e n e

11

1

( ) (0)j jj j j

j j

E EC C

E E

111

1

( ) (0)j jj j j

j j

E EC C

E E

1 1 1 31 1 1 2 2 2 3(0) (0) ( ) (0) ( ) (0)

0o

o

o

E EC C C C C C

E

H

1 31 2 3(0) (0)

0o

o

o

E EC M M C

E

Two-layer example:

11 2 3

1(0) (0)

0 0o

t tC M M C A

r

Multilayer method summary

( ) 1

coscos sin

(0) ( )

sin coscos

jj j

jp polj j j j

jj j

j

in

M C Cn

i

( ) 1

coscos sin

(0) ( )

sin coscos

jj j

jp polj j j j

jj j

j

in

M C Cn

i

cos cos j j j j j vac j jk l n k l

P. If we choose a layer of effective thickness “ l /2”, the matrix will ______:a) have zeros for its diagonal elementsb) have zeros for its off-diagonal elements

(Note: in multilayer technology, layers described as “ l /2” or “ l /4”, etc, l is that in the material, not vacuum. The effective thickness is .)cos j jl

Physics 123 ideas If we choose the thickness to be “ l /4”, the phases of these reflecting rays (due to thickness alone) are ________. a) All in phaseb) Alternating phase

If we choose the thickness to be “ l /4”, and we want a high R, we better choose the n’s to be ___________. Look at the phase of the first three reflections.

a) (1.4, 1.5, 1.8)b) (1.5, 1.4, 1.8)

Physics 123 ideas

Reflectance and coatingsAir-Glass (multilayer theory works for zero layers, too!):Normal incidence

2

G A

G A

n nR

n n

Normal incidence R≈4% across visible for nG=1.5.

Anti-reflectance coatingsSingle l/4 layer, Normal incidence

Air-L-Glass (H):

Normal incidence R≈1% across visible for nG=1.5.

22

2

a g L

a g L

n n nR

n n n

Anti-reflectance coatingsTwo l/4 layers, normal incidence

22 22 1

2 22 1

a g

a g

n n n nR

n n n n

Air-L-H-Glass :

More freedom to chose n’s, so usually better at some design , but

but not so good at ’s away from the design .

L H

Single l/4 Anti-Reflectance Coating: AL Glass(H)light still bounces among lenses in camera, spreading glare

Triple l/4 coating : ALHLGlass(H)

High reflectance multilayersAir-(HL)N-Glass(H) l/4 layers

Better: Air-(HL)N-H-Glass

Computer design can optimize for any application, with different d’s, n’s and ordering

One complication n: depends on .

What is different below for s polarization?a) We divide by n’s instead of multiplyb) The cos’s go to sinesc) The cos’s and n’s switch rowsd) The cos’s move to the n’s row e) There are no cos’s

1 1 1

1 1 1

cos cos cos cosj j

j j

i ij j j jj j

i ij jj jj j

e e E E

n nE En e n e

P polarization: