foueier optics课件cx coherence of optical wav.ppt

1、Chapter X Coherence of optical waves,Picked from Joseph W. Goodman, Statistical optics, Wiley Classics Library Edition Published 2000,Reasons for studying coherence,Statistical properties of light play an important role Complete statistical model is too complex Second-order averages-coherence functi

2、ons are commonly satisfactory enough,Energy spectral density,If u(t) is a Fourier transformable function, then Energy spectral density of u(t,Power spectral density,If u(t) is not Fourier transformable but the truncated function does have a transform Power spectral density of u(t,Spectral Density of

3、 a Random Process,Let random process U(t) consists of samples u(t), together with a measure of their probabilities Energy and power spectral densities,Time autocorrelation function,Given a single known time function u(t), the time autocorrelation function of u(t) is defined by,Statistical autocorrel

4、ation function,Given a random process U(t), its statistical autocorrelation function is defined by,Autocorrelation function power spectral density,For a process that is at least wide-sense stationary, the autocorrelation function and power spectral density form a Fourier transform pair,The importanc

5、e of autocorrelation functions,It offers an experimental means for ultimately determining the power spectral density of the signal It provides an analytic means for calculating the power spectral density of a random process model described only in statistical terms,Cross-correlation,Time-average cro

6、ss-correlation function Cross-correlation function of two random processes U(t) and V(t,Cross-spectral density functions,Cross-correlation function cross- spectral density,For jointly wide-sense stationary random processes U(t) and V(t,Monochromatic complex signals,Real-valued monochromatic wave The

7、 complex representation of this signal Phasor amplitude,Difference between spectra of real and complex forms,F u(r)(t)= F u(t),From u(r)(t) to u(t), we double the strength of the positive frequency component and entirely remove the negative frequency component,Nonmonochromatic complex signals,Real-v

8、alued nonmonochromatic signal u(r)(t) with Fourier transform Following exactly the same procedure used in the monochromatic we can represent u(r)(t) by complex u(t,u(t) is called the analytic signal representation of u(r)(t,Fourier integral representation of u(t,u(r)(t) u(t,Hilbert transform,Cauchy

9、principal value,The integral transformation is known as the Hilbert transform of u(r)(t,Important properties of the analytic signal,Construction of analytic signal,Hilbert transforming filter,Signal with narrowband spectrum,Real-valued function u(r)(t) Nonmonochromatic but narrow band,Narrowbandslow

10、ly varying signal,Such a signal may be written in terms of a slowly varying envelope A(t) and a slowly varying phase (t) Corresponding analytic signal,Complex Envelopes or Time-Varying Phasors,Complex envelope is defined by this time-varying phasor amplitude For any signal (wideband or narrowband),

11、we can write the analytic signal representation in the form,Two types of coherence,Temporal coherence Ability of a light beam to interfere with a delayed (but not spatially shifted) version of itself Amplitude splitting Spatial coherence Ability of a light beam to interfere with a spatially shifted

12、(but not delayed) version of itself Wavefront splitting Temporal+spatial coherence mutual coherence function,Temporal coherence,Analytic signal u(P,t) has a complex envelope A(P,t) with a finite bandwidth Finite time duration A(P,t) remains relatively constant during the interval A(P,t) and A(P,t+)

13、are highly correlated, or coherent Coherence time,Michelson Interferometer,Pattern of interference,Phenomena of interference,At zero pathlength a large central peak in the interferogram The mirror is displaced from the zero-delay position drop in the fringe depth The relative delay grows large enoug

14、h the addition of elementary fringes is nearly totally destructive,Mathematical Description of the Experiment,Intensity incident on the detector can be written as where K1 and K2 are real numbers determined by the losses in the two paths and u(t) is the analytic signal representation of the light em

15、itted by the source,Expansion of this expression,Autocorrelationself coherence,Optical intensity Autocorrelation function is known as the self coherence function of the optical disturbance,Intensity on detector,In this abbreviated notation we write the detected intensity as,Complex degree of coheren

16、ce,Normalized version of the self coherence function The detector intensity is given by,Cosine form of ID,Complex degree of coherence can be written in the following general form Assuming K1= K2 = K, we can express the interferogram as,The formulation is typical of the structure of the interferogram

17、,In the vicinity of zero relative pathlength difference, h0 The interferogram consists of a fully modulated cosine,Change of interferogram with h,When pathlength difference h is increased Amplitude modulation Fringes may suffer a phase modulation Visibility of a sinusoidal fringe,Relation of visibil

18、ity and complex degree of coherence,Look back on temporal coherence,2hvisibilityrelative coherence of the two beams Visibility0pathlength difference exceeds the coherence length or relative time delay exceeds the coherence time The concept of temporal coherence has to do with the ability of two relatively delayed light beams to form fringes,Autocorrelation function Power spectral density,Let a complex-valued random process U(t) have autocorrelation function U(t) We call the Fourier transform of U(t) power spectral dens