Coherence. This is based on. Chapter 10. Statistical Optics, Fundamentals of Photonics Bahaa E. A. Saleh, Malvin Carl Teich. and

Transcription

1 Coherence This is based on Chapter 10. Statistical Optics, Fundamentals of Photonics Bahaa E. A. Saleh, Malvin Carl Teich and Lecture Note on Laser Technology and Optics Prof. Matti Kaivola Optics and Molecular Materials, Helsinki University of Technology and Coherence and Fringe Localization T. D. Milster and N. A. Beaudry Optical Sciences Center, University of Arizona

2 Coherence Coherence is a measure of the correlation between the phases measured at different (temporal and spatial) points on a wave Coherence theory is a study of the correlation properties of random light which is also known as the statistical optics.

3 Coherence theory Coherence and Fringe Localization, T. D. Milster and N. A. Beaudry,

4 Coherence as Statistics

5 Statistical Properties of Random Light Second order average of a function

6 Mutual coherence function Mutual coherence function

7 Degree of Coherence

8 Coherence and Visibility Young s double pinhole interferometer (YDPI)

9 Temporal Coherence The temporal autocorrelation means the time average at the same position r = r 1 2 Note, we use both notations of G( τ ) = Γ ( τ) and g( τ) = γ( τ) See the next example

10 Degree of Temporal Coherence

11 Coherence Time and Length Coherence time Note, it is different from ½ width, 1/e width, Coherence length Coherence length l c = cτ c Later, we will similarly define the spectral width

12 Temporal Coherence τ = τ = 1/ Δν c ave l = c = c Δ = Δ c 2 τ c / ν λ / λ Actually, what is the definition of Δv, why is satisfied the relation?

13 Coherence Time and Spectral Width Fundamentals of Photonics, Bahaa E. A. Saleh, Malvin Carl Teich where,

14 Coherence Time and Spectral Width FWHM (full width at half maximum)

15 Example : A wave comprising a random sequence of finite wave train. Note, it is not a monochromatic wave A truncated monochromatic wave Is not monochromatic, any more. Δt Δv = 1

16 Example A wave comprising a random sequence of wavepackets decaying exp.

17 Quasi-monochromatic waves Now, start to investigate concretely a relation between the visibility and the coherence from a 2-wavelength light source up to a polychromatic light source.

18 A point source with two wavelengths Young s double pinhole interferometer (YDPI) d << z o OPD = vδt

19 A point source with with two two wavelengths T = 2π/ω m = nλ eq /c, where = = m λeq λa λb c/ n ω

20 A point source with with two two wavelengths Now, under the constraint that d << z o

21 λ eq λ C.L. = 2 λ c λeq = 2 2 Δλ 2 Δν Δ λ = λ λ and Δ ν = ν ν a b a b Coherence length

22 A Polychromatic light source 3 λ s V decreased 5 λ s

23 A Polychromatic light source

24 A Polychromatic light source Fringe visibility

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26 Δy z d o = c Δv dyo Δν Δν sinc = sinc Δl = 0 zo c c c Δl= Δν Δy z d o = c Δv Coherence length Coherence time

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28 Δv m Δv 2Δl =

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30 Remind! Spatial Coherence

31 Spatial Coherence: point source

32 Spatial Coherence: two point source l c ~ λ/θ

33 Basic Spatial Coherence d

34 Basic Spatial Coherence

35 Basic Spatial Coherence Basic properties of spatial coherence

36 Basic Spatial Coherence Now, extend to a continuous source distribution (an ensemble of incoherent point sources) yd OPD = y θ s s s A zs

37 Basic Spatial Coherence Fringe visibility

38 Basic Spatial Coherence Fringe visibility van Cittert-Zernike Theorem : Degree of Spatial Coherence is Fourier Transform (or, Fraunhofer Diffraction Pattern) of the source irradiance distribution Appendix I

39 Example: Spatial coherence length from a circular source

40 L c ~ λ/θ

41 OPD = yd s Since OPDs and z θ = A s = A s S f ys s f θ = d z s θ s θ s s OPD s

42

43 Concept of Coherent area

44 l λ = 500 nm A = 1 mm d = 3 mm V = 0

45 Terminology Used in Coherence Theory Remind! Define the normalized mutual coherence function, or complex degree of coherence

46 Terminology Used in Coherence Theory Normalized mutual coherence function Complex degree of coherence

47 Control of Coherence Making Light Coherent Making Light Incoherent Spatial Filter for Spatial Coherence Wavelenth Filter for Temporal Coherence Ground Glass to Destroy Spatial Coherence Move it to Destroy Temporal Coherence

48 Appendix I van Cittert-Zernike Theorem for Spatial Coherence Chuck DiMarzio Northeastern University

49 Summary of van Cittert-Zernike Theorem for Spatial Coherence

50 Van Cittert-Zernike Theorem: 1 Chuck DiMarzio, Northeastern University

51 Van Cittert-Zernike Theorem: 2 Chuck DiMarzio, Northeastern University

52 Van Cittert-Zernike Theorem: 3 Chuck DiMarzio, Northeastern University

53 Van Cittert-Zernike Theorem: 4 Chuck DiMarzio, Northeastern University

54 Van Cittert-Zernike Theorem: 5 Source Irradiance Far-Field Correlation Function Chuck DiMarzio, Northeastern University