Innovation and Development of Study Field. nano.tul.cz

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1 Innovation and Development of Study Field Nanomaterials at the Technical University of Liberec nano.tul.cz These materials have been developed within the ESF project: Innovation and development of study field Nanomaterials at the Technical University of Liberec

2 Miroslav Šulc katedra fyziky FP TU Liberec

3 Outline Coherence Statistical properties of sources Interference of coherent light Interference of partial coherent light Michelson Interferometer Mach Zehnder interferometer The Sagnac Interferometer Newton s rings 3

multiple longitudinal modes Temporal Coherence How long do the light

4 Coherence Factors that compromise coherence: 1. thermal fluctuations 2. vibrational fluctuations 3. emission of multiple wavelengths 4. multiple longitudinal modes Temporal Coherence How long do the light waves remain in phase as they travel? Coherence Length = l 2 /ndl 4

5 Coherence Spatial Coherence Over what area does the light remain in phase? 5

Consequence on the electric field Existence of an Heisenberg inequality analogous to (for a monochromatic wave) Consequences There is no null field at all moments

6 Consequence on the electric field Existence of an Heisenberg inequality analogous to (for a monochromatic wave) Consequences There is no null field at all moments (see there is no particle at rest ) The electromagnetic field in vacuum is not identically null The field is null only on average : existence of vacuum fluctuations 6

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11 Statistical properties of sources Spontaneous emission by a single dipole (atom, ion, ) variance and photon number distribution : depend on pumping antibunching Spontaneous emission by an incoherent ensemble of dipoles (Thermal / chaotic light) bunching (Hanbury Brown & Twiss) 11

12 Statistical properties of sources Laser field (stimulated emission inside an optical cavity) Poissonian distribution N photon state 12

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22 The coherence time is the reciprocal of the bandwidth. The coherence time is given by: 1/ c Dv where Dn is the light bandwidth (the width of the spectrum). Sunlight is temporally very incoherent because its bandwidth is very large (the entire visible spectrum). Lasers can have coherence times as long as about a second, which is amazing; that's >10 14 cycles! 22

23 Orthogonal polarizations don t interfere. The most general plane-wave electric field is:, Re exp ( ) 0 E r t E i k r t where the amplitude is both complex and a vector: The irradiance is: E E, E, E 0 0x 0 y 0z c * c * * * I E0 E0 E0 x E0 x E0 y E0 y E0 z E0 z

24 Orthogonal polarizations don t interfere (cont d) Because the irradiance is given by: c * c * * * I E0 E0 E0 xe0 x E0 ye0 y E0 ze0 z 2 2 combining two waves of different polarizations is different from combining waves of the same polarization. Different polarizations (say x and y): Same polarizations (say x and x, so we'll omit the x-subscripts): Therefore: c I E E E E I I 2 * * 0x 0x 0 y 0 y 1 2 c I E E E E E E 2 * * * 1 1 2Re I I c Re E E I * Cross term! 24

25 Mach-Zehnder Interferometer The Mach-Zehnder interferometer is usually operated misaligned and with something of interest in one arm. 25

26 Mach-Zehnder Interferogram Nothing in either path Plasma in one path 26

27 The Sagnac Interferometer The two beams automatically take the same path around the interferometer. The paths can differ, however, if the device is rotating. 27 The Sagnac interferometer senses rotation.

28 Sagnac Interferometer Math Suppose that the beam splitter moves by a distance, d, in the time, T, it takes light to circumnavigate the Sagnac interferometer. As a result, one beam will travel more, and the other less distance. I E exp( ikd) E exp( ikd) out I sin ( kd) If R = the interferometer radius, and W = its angular velocity: 2 2 d RW T RW(2 R / c) 2 W( R ) / c 2 W Area / c I I k c out 2 0 sin (2 W Area / ) Thus, the Sagnac Interferometer's sensitivity to rotation depends on its area. And it need not be round! 28

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30 Newton's Rings Get constructive interference when an integral number of half wavelengths occur between the two surfaces (that is, when an integral number of full wavelengths occur between the path of the transmitted beam and the twice reflected beam). 30 This effect also causes the colors in bubbles and oil films on puddles.

31 Quantum correlations with an OPO Heidmann et al., Phys. Rev. Lett. 59, 2555 (1987) At the output of an OPO, the signal and idler beams have quantum intensity correlations. Result : 30 % noise reduction (now : over 85 %) 31

32 THANK YOU FOR YOUR ATTENTION 32

Constructive vs. destructive interference; Coherent vs. incoherent interference

Constructive vs. destructive interference; Coherent vs. incoherent interference Waves that combine in phase add up to relatively high irradiance. = Constructive interference (coherent) Waves that combine