32. Interference and Coherence

Transcription

1 32. Interference and Coherence Interference Only parallel polarizations interfere Interference of a wave with itself The Michelson Interferometer Fringes in delay Measure of temporal coherence Interference of crossed beams Coherence Temporal coherence Spatial coherence Albert Michelson

2 Orthogonal polarizations don t interfere. The most general plane-wave electric field is: r r r r r E r t E j k r t, Re exp ( ) 0 where the amplitude is both complex and a vector: r E E, E, E 0 0x 0y 0z The irradiance is: don t forget the complex conjugation! c r r c I E E E E E E E E * * * * x x y y z z

3 Orthogonal polarizations don t interfere (cont d) Because the irradiance is given by: c r r c I E E E E E E E E * * * * x x y y z z combining two waves of different polarizations is different from combining waves of the same polarization. Different polarizations (e.g., x and y): c I 2 E E E E I I * * 1x 1x 2y 2y 1 2 Same polarizations (e.g. both have x polarization): c * c I *,, 1, 2, 1, 2, 2 Etotal x Etotal x E x E x E x E x 2 Therefore: I I I 1 2c Re E * 1E2 This cross term can give rise to very dramatic effects. This is what is called a cross term.

4 The irradiance when combining a beam with a delayed replica of itself has fringes. Suppose the two beams are E 0 exp(jt)and E 0 exp[jtt)]. That is, a monochromatic wave and itself delayed by some time t : 0 I 2I 1cos[ t] * 1 Re E1 E2 I2 I I c I 2I cre E exp[ jt] E exp[ j( tt)] * I cre E exp[ jt] 0 0 2I c E cos[ t] How do we vary the delay of a beam? Fringes (as a function of delay) I - t

5 Varying the delay on purpose Simply moving a mirror can vary the delay of a beam by many wavelengths. Mirror Translation stage Input beam E(t) Output beame(t t) Moving a mirror backward by a distance Lyields a delay of: t 2L/c Note the factor of 2. Light must travel the extra distance to the mirror and back! Since light travels 300 µm per ps in air, 300 µm of mirror displacement yields a delay of 2 ps. Delays of less than 0.1 fsec (10-16 sec) can be generated using this technique.

6 We can also vary the delay using a mirror pair or a corner cube. Mirror pairs involve two reflections and displace the return beam in space: But out-of-plane tilt yields a nonparallel return beam. E(t) Mirrors E(t t) Translation stage Input beam Output beam Corner cubesinvolve three reflections and also displace the return beam in space. Even better, they always yield a parallel return beam: [Edmund Scientific]

7 The Michelson Interferometer The Michelson Interferometer splits a beam into two and then recombines them at the same beam splitter. mirror Suppose the input beam is a plane wave. Then the intensity measured at the output is: beamsplitter mirror input beam delay * j t kz kl I I I cre Eexp j( tkzkl) E exp ( ) out Fringes (as a function of delay): I out DL = L 2 L 1 output I I 2I Reexp jkl ( L) since I I I ( c /2) E k L 2I 1cos( D ) 2

8 The Michelson Interferometer - clarification 0 k L I 2I 1cos( D ) out I 4I If the path length difference is zero, this becomes: out 0 Are we getting more photons at the output than we put in? No! input beam In this expression, the symbol I 0 refers to the intensity in either oneof the two beams at this location. mirror beamsplitter mirror output delay But both beams that reach this point have passed through the 50/50 beam splitter twice, thus reducing their intensity (relative to the intensity of the input beam) by a factor of 4. Thus: I0I in 4

9 The Michelson-Morley experiment to measure the aether The most famous failed experiment of all time The Aether Wind Albert Michelson Michelson s laboratory, Case-Western University, 1887 Nobel Prize, 1907 (first American to win one of the science prizes)

10 Interference of crossed beams r k kcosq zˆ ksinq xˆ r k ˆ q kcosq zˆ ksinq x z r k r r kcosq z ksinq x r k r kcosq z ksinq x k r r * I 2I cre E exp[ j( tk r)] E exp[ j( tk r)] Cross term is proportional to: x * q q q q Re E0exp j t kzcos kxsin E0 exp j t kzcos kxsin Re exp 2jkxsinq cos(2kxsin q) r k Fringes (as a function of x position) I out (x) x

11 Big angle: small fringes. Small angle: big fringes. The fringe spacing, : Large angle: 2 /(2ksin q) /(2sin q) position As the angle decreases to zero, the fringes become larger and larger, until finally, at q= 0, the intensity pattern becomes constant. Small angle: position

12 You can't see the spatial fringes unless the beam angle is very small! The fringe spacing is: /(2sin q) = 0.1 mm is about the minimum fringe spacing you can see with the naked eye: q sin q /(2 ) q 0.5 m/200m 1/400 rad0.15 o

13 The Michelson Interferometer (Misaligned) Mirror Beamsplitter Input beam Suppose we misalign the mirrors, so the beams cross at an angle when they recombine at the beam Mirror splitter. And we won't scan the delay, so the lengths are equal. q x z If the input beam is a plane wave, the cross term becomes: * q q q q Re E0exp j t kzcos kxsin E0 exp j t kzcos kxsin Re exp 2jkxsinq cos(2kxsin q) Fringes (in position) I out (x) Crossing beams maps delay onto position If the path length difference changes, the fringes shift. x

14 The Michelson Interferometer: A question input beam Let s go back, for now, to the well-aligned Michelson interferometer. mirror beamsplitter output If we move the moveable mirror further and further back, do we continue to see fringes forever? DL = L 2 L 1 = large If not, then how far can we go before they disappear? I 2I 1cos( kdl) out 0 mirror delay

15 The Temporal Coherence Timeand the Spatial Coherence Length The temporal coherence time is the time over which the beam wavefronts remain equally spaced. Or, equivalently, over which the field remains sinusoidal with a given wavelength: The spatial coherence length is the distance over which the beam wavefronts remain flat: Since there are two transverse dimensions, we could talk about two different coherence lengths. Instead, we define a coherence area.

16 Spatial and Temporal Coherence Beams can be coherent or only partially coherent (or, even incoherent) in both space and in time. Spatialand Temporal Coherence: Temporal Coherence; Spatial Incoherence Spatial Coherence; Temporal Incoherence Spatial and Temporal Incoherence

17 The coherence time of monochromatic light A nearly monochromatic light source has a large coherence time: E-field ampitude time If we know there is a maximum here......and we wait a time equal to an integer number of periods......then we know that we will find another maximum. For a perfect cosine, the integer could be as large as you want (up to infinity), and this would still be true. Thus, an ideal monochromatic light source has an infinite coherence time. In the real world: highly stabilized lasers can have coherence times on the order of a few seconds. That s amazing! More than cycles!

18 The coherence time of polychromatic light A polychromatic light source has a smaller coherence time. Here s an example: an E-field composed of a superposition of several monochromatic waves, each with a slightly different frequency Start at this maximum. Wait N periods, where N = 1, 2, 3,... What is the value of the E-field at each successive value of N? Is it still a local maximum? The coherence time for a given waveform is the average amount of time one has to wait from an arbitrary starting point before coherence is lost.

19 What determines the coherence time? S() sum of 3 sine waves E field sum of 5 sine waves E field time S() sum of 10 sine waves E field time S() time More spectral components = more rapid loss of coherence

20 The coherence time is the reciprocal of the bandwidth. The coherence time is given by: t 1/ Dv c where Dnis the light bandwidth (the width of the spectrum). Sunlight is temporally very incoherent because its bandwidth is very large (the entire visible spectrum) Short optical pulses also have small coherence times, roughly equal to their duration. Coherence length = (c 0 /n)t c

21 Why are we interested in coherence time? Because the notion is relevant to measurements that we often do. Let us suppose that we take a wave and interfere it with a copy of itself: a wave a time-delayed replica of the same wave If the time delay is zero (t= 0): perfect constructive interference at every point. The net irradiance is large. If the time delay is half the period (t= /): nearly perfect destructive interference at every point. The net irradiance is zero. If the time delay is the period (t= 2/): constructive interference at every point. The net irradiance is large.

22 What if the delay time tis large? at this time, constructive interference destructive interference here a wave a time-delayed replica of the same wave time delay t If the time delay is large(t> t c ): There is no correlation between the peaks of the wave and the peaks of the time-delayed version. Interference is sometimes constructive, sometimes destructive. The net irradiance no longer depends on the delay t. Interferogram: the pattern formed by the net irradiance as a function of delay. Why is this interesting? One reason: because interferograms are easy to measure using a Michelson interferometer.