chapter 11. optical interferometrychapter 11. optical …optics.hanyang.ac.kr/~shsong/11-optical...

Chapter 11. Optical InterferometryChapter 11. Optical Interferometry

Last Lecture• Two- Beam Interference• Young’s Double Slit Experiment• Virtual Sources• Newton’s Rings

This Lecture• Michelson Interferometer• Variations of the Michelson Interferometer• Multiple- beam interference• Fabri- Perot interferometer

The Michelson InterferometerThe Michelson Interferometer


Hecht, Optics, Chapter 9.


Consider the virtual images Q’1 and Q’2 of thepoint Q in the source plane. The optical pathdifference for the two virtual image points is

Assuming that the beamsplitter is 50% reflecting, 50% transmitting, the interference pattern is


For the bright fringes

For the dark fringes

If δr = π as is usually the case because the beam 2 from M2 undergoes an external reflection at the beamsplitter, then Δr = λ/2 and

Bright fringe :

Dark fringe :

Separation of the fringes is sensitive to the optical path difference d.Near the center of the pattern (cosθ ~ 1), as d varies,


Hecht, Optics, Chapter 9.

m = mmax at the center, since θ = 0


Assume that the spacing d is such that a dark fringe is formed at the center

For the neighboring fringes the order m is lower

Define another integer p to invert the fringe ordering

m = mmax : p = 0

m = mmax – 1 : p = 1

Variations of the Michelson InterferometerVariations of the Michelson Interferometer

Twyman-Green Interferometer

Twyman-Green InterferometerTwyman-Green Interferometer

Guenther, Modern Optics

Mach-Zehnder InterferometerMach-Zehnder Interferometer

Laser

CCD

mirror

PZT mirror

Spatial filtering& collimation

Beam splitter

2f 2f

Imaging lens

monitor

Testsample

Mach-Zehnder InterferometerMach-Zehnder Interferometer

렌즈 표면의 변화(동영상)

Ac 0V0V -> 40V 40V -> 0V

Stokes RelationsStokes RelationsEi is the amplitude of the incident light.The amplitudes of the reflected and transmitted beams are given by

From the principle of reversibility

Stokes relationsr = eiπ r’

Multiple-Beam Interference in a Parallel PlateMultiple-Beam Interference in a Parallel Plate

Find the superposition of the reflectedbeams from the top of the plate.The phase difference between neighboring beams is

Given that the incident wave is

Δ= kδ tf tn θcos2=Δ


The reflected amplitude resulting from the superposition of the reflected beams from the top of the plate is given by

Define

Therefore


The Stokes relations can now be used to simplify the expression


The reflection irradiance is given by



The transmitted irradiance is given by


Minima in transmitted irradiance and maxima in reflected irradiance occur when

Minima in reflected irradiance and maxima in transmitted irradiance occur when

The Fabry-Perot Interferometer:High-Resolution Air-Spaced


Inner surfaces polished to flatness of λ/50 or better, coated with silver or aluminum films with thickness of about 50 nm. The metal films are partially transmitting. The outer surfaces of the plates are wedged to eliminate spurious fringe patterns.

The Fabry-Perot InterferometerThe Fabry-Perot Interferometer

The transmitted irradiance is given by

Maxima in transmitted irradiance occur when

For the air space nf = 1, and the condition for maximum transmission is



The fringe pattern will shift as the wavelength of the light is scanned or as the thickness of the air gap is varied.

The Fabry-Perot Solid EtalonThe Fabry-Perot Solid EtalonFor analysis of laser spectra, we typically usesolid etalons. The solid etalon is a piece of glass or fused silica. The two faces are flat and parallel to each other to λ/10 or better. Each face has a multi-layer dielectric coating that is highly reflective at a given wavelength.

The Fabry-Perot Interferometer:Fringe profiles – The Airy functionThe Fabry-Perot Interferometer:

Fringe profiles – The Airy functionThe transmitted irradiance for Fabry-Perot interferometer or etalon is given by

Use the trigonometric identity,

We obtain the transmittance T, the Airy function,

: coefficient of finesse

The Fabry-Perot Interferometer:Fringe profiles – The Airy functionThe Fabry-Perot Interferometer:

Fringe profiles – The Airy functionThe coefficient of finesse characterizesthe resolution of the Fabry-Perot device

The fringe contrast is given by

As F increases (due to increasing r)the fringe contrast increases, the transmittance minimum goes closer to 0, And the fringe thickness decreases.

r = 0.2

r = 0.5

r = 0.9

The F-P Interferometer: Resolving PowerThe F-P Interferometer: Resolving PowerThe resolving power of the Fabry-Perot device is directly related to the full-width-at-half-maximum (FWHM)

The minimum resolvable phase difference between lines with different wavelengths is

δc

Δδ = 2δc

Resolving PowerResolving Power

The phase difference for particular angle θt for two different wavelengths is given by

For small wavelength intervals, λ1 − λ2 = Δλ

Since we are at a fringe maximum,

Resolving PowerResolving Power

The resolving power is defined as

The fringe number m is given by

To maximize the resolving power we need to look near the center of the pattern, cosθt ~ 1, the plate spacing t should be as large as possibleand the coefficient of finesse should be as large as possible

Free Spectral RangeFree Spectral Range

Consider two wavelengths λ1 and λ2 such that the mth order of λ2 coincides withthe (m+1) order of λ1.

The free spectral range is when

Near the center of the fringe pattern,

In terms of frequency ν ,

To avoid overlapping fringes, it is necessary that

The Fabry-Perot Interferometer:The Finesse

The Fabry-Perot Interferometer:The Finesse

The resolving power is proportional to the order m,while the free spectral range is inversely proportional to the order m. The finesse is the ratio of the fringe separation to the fringe FWHM

2/422

2/1

FF

ππδ

π==≡

Figure of merit for F-P interferometer

2Fπ( )

( )minλλ

Δ

Δ fsr

The F-P Interferometer: Spectral ResolutionThe F-P Interferometer: Spectral Resolution

Tdiode, λdiode

37.84 °C1535.737 nm

37.94 °C1535.747 nm

38.05 °C1535.757 nm

38.73 °C1535.821 nm

Etalon FSRis 10 GHz, scan showncorrespondsto 10.67 GHzin idler frequency.

Etalon fringes display excellent contrast.

Solid Etalon Used to Monitor Laser λ ScanningSolid Etalon Used to Monitor Laser λ Scanning

chapter 11. optical interferometrychapter 11. optical …optics.hanyang.ac.kr/~shsong/11-optical...

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