Document info 13.

Principle of SuperpositionThursday, 10/19/2006

Physics 158Peter Beyersdorf

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Class Outline

Linear vs. Non-linear optics

Phasor addition of waves

Interference

Beats

Interference Patterns

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Linear Optics

All of our analysis so far has assumed the material polarizability is a linear function of the applied electric field

Saturation effects with high fields cause this assumption to be wrong. This regime is called non-linear optics

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n =!

(1 + "!)

P

E

P = !0!"1E + "2E

2 + "3E3 + . . .

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Superposition

If we are in the small field limit where linear optics prevails, the field at any given point in space and time is the sum of the field from every individual wave present at that point.

The fact that fields can be added is called the principle of superposition.

The irradiance at a point where multiple waves exist is determined from the square of the sum of the fields.

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L1

L2

Phasor Addition of Waves

Consider the interference of two waves that have traveled different paths in a Mach-Zehnder interferometer that has 50-50 beamsplitters

What is the output irradiance I1 in terms of the input irradiance Io and the path lengths L1 and L2?

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Io

I1

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Phasor Addition of Waves

What is the output irradiance I1 in terms of the input irradiance Io and the path lengths L1 and L2?

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L1

L2Io

I1

E1 =!r1t2e

ikL1 + t1r2eikL2

"E0

r1 = r2 = !1/"

2

t1 = t2 = 1/"

2!"#$

E1 = !12

!eikL1 + eikL2

"E0

E1 = !12eik(L1+L2)/2

!eik(L1!L2)/2 + e!ik(L1!L2)/2

"E0

E1 = !eik(L1+L2)/2 cos!k(L1 ! L2)/2

"E0

I1 = E!1E1 = cos2

!k(L1 ! L2)/2

"I0

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Conservation of Energy

If the output I1 isWhat happens to the input irradiance when I1≠I0?

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L1

L2Io

I1

!"#$

I1 = cos2!k(L1 ! L2)/2

"I0

r1 = !r2 = !1/"

2

t1 = t2 = 1/"

2E2 =

!r122e

ikL1 + t1t2eikL2

"E0

I2 = E!2E2 = sin2

!k(L1 ! L2)/2

"I0

I2

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Graphical Phasor Addition

Consider the transmitted field through a Fabry-Perot cavity using graphical addition of phasors. How does the output intensity depend on the length?

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!L=2L

LE1

E2

E3E4E5

Etot

Each successive round trip of the cavity decreases the amplitude and adds a phase to the light

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Coherence

Up until now we have only dealt with monochromatic waves which have (by definition) a well defined frequency and are coherent. For summing waves of coherent light the electric fields add

Many sources of light are incoherent, that is they may have a particular center frequency, but the phase of the light is varying in an essentially random way over short time scales (the coherence time). If we try to observe interference effects for incoherent light, the phase difference that affects the observed intensity is averaged out during the observation interval. When summing waves of incoherent light the intensity adds

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L1

L2

Incoherent Illumination Example

Consider the interference of two waves that have traveled different paths in a Mach-Zehnder interferometer that has 50-50 beamsplitters

What is the output irradiance I1 and I2 in terms of the input irradiance Io when illuminated with incoherent light?

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Io

I1

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Incoherent Illumination Example

What is the output irradiance I1 in terms of the input irradiance Io and the path lengths L1 and L2?

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L1

L2Io

I1

I1 = E!1E1 = cos2

!k(L1 ! L2)/2

"I0

I2 = E!2E2 = sin2

!k(L1 ! L2)/2

"I0

For coherent illumination

Averaging over all possible phases (φ=k(L1-L2)/2) gives for incoherent illumination

I1,inc =12!

! 2!

0cos2 (")d"I0 = I0/2

I2,inc =12!

! 2!

0sin2 (")d"I0 = I0/2

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Beats

Consider the effect of having two collinear waves of differing frequencies

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1/fbeat

1/favg

E1 = E0ei!k1·!r!2"f1t

E2 = E0ei!k2·!r!2"f2t

E1 + E2 = E0ei(!k1+!k2)·!r/2!2"(f1+f2)/2t

!ei(!k1!!k2)·!r/2!2"(f1!f2)/2t

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Interference Patterns

When multiple waves overlap and have different directions of propagation a spatial interference pattern will result

Consider the Moire patterns from a pair of striped transparencies for example

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Interference Example

Determine the thickness of the spacer in terms of the number of interference fringes seen

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Summary

Electric fields from multiple waves add via the principle of superposition for linear optics

Irradiance from multiple waves do not add - the fields add first and the irradiance is determined from the square of the fields

Various interference effects can be harnessed for precision measurements

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