• 검색 결과가 없습니다.

• Two- Beam Interference

N/A
N/A
Protected

Academic year: 2022

Share "• Two- Beam Interference"

Copied!
33
0
0

로드 중.... (전체 텍스트 보기)

전체 글

(1)

Chapter 11. Optical Interferometry Chapter 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

(2)

The Michelson Interferometer

The Michelson Interferometer

(3)

The Michelson Interferometer The Michelson Interferometer

Hecht, Optics, Chapter 9.

(4)

The Michelson Interferometer The Michelson Interferometer

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

Assuming that the beamsplitter is 50%

reflecting, 50% transmitting, the interference pattern is

(5)

The Michelson Interferometer The Michelson Interferometer

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,

(6)

The Michelson Interferometer The Michelson Interferometer

Hecht, Optics, Chapter 9.

m = mmax at the center, since θ = 0

(7)

The Michelson Interferometer The Michelson Interferometer

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

(8)

Variations of the Michelson Interferometer Variations of the Michelson Interferometer

Twyman-Green Interferometer

(9)

Twyman-Green Interferometer Twyman-Green Interferometer

Guenther, Modern Optics

(10)

Mach-Zehnder Interferometer

Mach-Zehnder Interferometer

(11)

Laser

CCD mirror

PZT mirror Spatial filtering

& collimation

Beam splitter

2f 2f

Imaging lens

monitor

Test sample

Mach-Zehnder Interferometer

Mach-Zehnder Interferometer

(12)

렌즈 표면의 변화(동영상)

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

(13)

Stokes Relations Stokes Relations

Ei is the amplitude of the incident light.

The amplitudes of the reflected and transmitted beams are given by

From the principle of reversibility

Stokes relations

r = eiπ r’

(14)

Multiple-Beam Interference in a Parallel Plate

Multiple-Beam Interference in a Parallel Plate

(15)

Find the superposition of the reflected beams from the top of the plate.

The phase difference between neighboring beams is

Given that the incident wave is

Δ

δ = k Δ = 2 n

f

t cos θ

t

Multiple-Beam Interference in a Parallel Plate

Multiple-Beam Interference in a Parallel Plate

(16)

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

Define

Therefore

Multiple-Beam Interference in a Parallel Plate

Multiple-Beam Interference in a Parallel Plate

(17)

The Stokes relations can now be used to simplify the expression

Multiple-Beam Interference in a Parallel Plate

Multiple-Beam Interference in a Parallel Plate

(18)

The reflection irradiance is given by

Multiple-Beam Interference in a Parallel Plate

Multiple-Beam Interference in a Parallel Plate

(19)

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

The transmitted irradiance is given by

(20)

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

Minima in transmitted irradiance and maxima in reflected irradiance occur when Minima in reflected irradiance and maxima in transmitted irradiance occur when

(21)

The Fabry-Perot Interferometer:

High-Resolution Air-Spaced 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.

(22)

The Fabry-Perot Interferometer The 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

(23)

The Fabry-Perot Interferometer:

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

High-Resolution Air-Spaced

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

varied.

(24)

The Fabry-Perot Solid Etalon The Fabry-Perot Solid Etalon

For analysis of laser spectra, we typically use

solid 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.

(25)

The Fabry-Perot Interferometer:

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

Fringe profiles – The Airy function

The 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

(26)

The Fabry-Perot Interferometer:

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

Fringe profiles – The Airy function

The coefficient of finesse characterizes the 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

(27)

The F-P Interferometer: Resolving Power The F-P Interferometer: Resolving Power

The 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

(28)

Resolving Power Resolving 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,

(29)

Resolving Power Resolving 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 possible

and the coefficient of finesse should be as large as possible

(30)

Free Spectral Range Free Spectral Range

Consider two wavelengths λ1 and λ2 such that the mth order of λ2 coincides with the (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

(31)

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 4

2 2

2 / 1

F F

π π

δ

π = =

Figure of merit for F-P interferometer

2 π F

( ) ( ) Δ Δ λ λ

minfsr

(32)

The F-P Interferometer: Spectral Resolution

The F-P Interferometer: Spectral Resolution

(33)

T

diode

, λ

diode

37.84 °C

1535.737 nm

37.94 °C

1535.747 nm

38.05 °C

1535.757 nm

38.73 °C

1535.821 nm

Etalon FSR is 10 GHz, scan shown corresponds to 10.67 GHz in idler

frequency.

Etalon fringes display excellent contrast.

Solid Etalon Used to Monitor Laser λ Scanning

Solid Etalon Used to Monitor Laser λ Scanning

참조

관련 문서

Modern Physics for Scientists and Engineers International Edition,

12 that절 내의 주어를 문장의 주어로 쓰고 that절 내의 동사를 to부정사로 써 서 수동태 문장을 만들 수 있다... 13 반복되는 the reason을 관계대명사 which로 바꿔주고

We need to move away from the discussions over BM patent from the perspective of intellectual property to discussing how competition policy should be

웹 표준을 지원하는 플랫폼에서 큰 수정없이 실행 가능함 패키징을 통해 다양한 기기를 위한 앱을 작성할 수 있음 네이티브 앱과

_____ culture appears to be attractive (도시의) to the

The key issue is whether HTS can be defined as the 6th generation of violent extremism. That is, whether it will first safely settle as a locally embedded group

The “Asset Allocation” portfolio assumes the following weights: 25% in the S&P 500, 10% in the Russell 2000, 15% in the MSCI EAFE, 5% in the MSCI EME, 25% in the

1 John Owen, Justification by Faith Alone, in The Works of John Owen, ed. John Bolt, trans. Scott Clark, "Do This and Live: Christ's Active Obedience as the