<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/138b19e2-101c-4e4f-a218-2a25f2d63cd4/Interference.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/138b19e2-101c-4e4f-a218-2a25f2d63cd4/Interference.png" width="40px" /> Interference: Combination of a finite number of waves (Young’s slits)
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<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/cfe5b7ad-b05c-4590-af5e-88bfd25fa065/Diffraction.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/cfe5b7ad-b05c-4590-af5e-88bfd25fa065/Diffraction.png" width="40px" /> Diffraction: combination of an infinite number of wave (wide aperture)
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🗒️ Note: these 2 concepts are related for both:
If we add 2 linearly polarized waves with subscript 1 and 2 we get the following $E$-field
$$ \vec E=\vec E_1 \exp ( i(\vec k_1 \cdot \vec r - \omega_1 t+\epsilon_1))+\vec E_2 \exp ( i(\vec k_2 \cdot \vec r - \omega_2 t+\epsilon_2)) $$
We can write the intensity
$$ \vec E^* \cdot \vec E=|\vec E_1|+|\vec E_2 |+2\vec E_1 \cdot \vec E_2 \cos [(\vec k_1 -\vec k_2)\cdot \vec r-(\omega _1-\omega_2)t+(\epsilon_1-\epsilon_2)] $$
The $\cos$ term is called the interference term, for it to exist
- $\lang \vec E_1 \cdot \vec E_2 \rang \ne 0$, ie interference ways must not be orthogonal
- $\omega_1\simeq \omega_2$, this is because we care about averages and $\cos(\omega_1-\omega_2)t$ quickly tends to 0 if $\omega_1\not \simeq w_2$. We need $\Delta \omega < 1/T$ where $T$ is the time over which we observe
- $\epsilon_1-\epsilon_2$ is constant, again for the average
💎 Conclusion: if these conditions are fulfilled (coherence), then we add waves $E^*E$ otherwise we add intensities
💼 Case: if we assume the frequencies are the same we can do 2 simplifications
Write all of the subscripts in the exponentials as a simple phase:
$$ \begin{aligned} E^*E &=(A_1e^{-i\epsilon_1}+A_2 e^{-i\epsilon_2})(A_1e^{i\epsilon_1}+A_2 e^{i\epsilon_2}) \\&=|A_1|^2+|A_2|^2+2A_1A_2\cos(\epsilon_1-\epsilon_2)
\end{aligned} $$
Regard waves as phasors (amplitude complex plane):
🗒️Note: this can be shown using geometry
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/cef85ac3-2aca-4fb9-aae3-670b4c7dff63/Temporal_coherence.gif" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/cef85ac3-2aca-4fb9-aae3-670b4c7dff63/Temporal_coherence.gif" width="40px" /> Temporal coherence: the measure of the average correlation of the phase of the light wave along the propagation direction.
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<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/1957bd57-cfdd-4eb5-a467-45a5923c630c/Temporal_coherence_time.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/1957bd57-cfdd-4eb5-a467-45a5923c630c/Temporal_coherence_time.png" width="40px" /> Temporal coherence time: $t_c=1/\Delta \upsilon$. If $t>t_c$ interference will not be observed between a wave and its delayed version. The corresponding coherence length is $ct_c$
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2 waves with slightly different frequencies get
$$ t_c\sim 1/\Delta \upsilon $$
where $\Delta \upsilon$ is the difference in frequency
🗒️ Note: can be written as the length $ct_c$
🗒️ Note: the fixed points are due to the slider changing by 0.05 increments, they are a consequence of the graph not of real life
🚷 Why is coherence unlikely
- Doppler broadening: atoms in motion emit light waves at different $\upsilon$ resulting in a spread of frequency (typically forms a Gaussian profile)
- Collisions broadening: atoms colliding ($10$ $\rm ns$) disrupts the continuous wave they emit (typically forms a Lorentzian profile)
- Natural (Heisenberg) broadening: due to uncertainty principle, transitions of electrons lasting $\Delta t$ result in frequency broadening $\Delta\upsilon \sim 1/\Delta t$
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/95e4bf9f-e4e7-452f-80b7-385f48dd3b56/Spatial_coherence.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/95e4bf9f-e4e7-452f-80b7-385f48dd3b56/Spatial_coherence.png" width="40px" /> Spatial coherence: measure of the phase difference, $\phi$, between each point of a wavefront (small $\Delta \phi$ = large spatial coherence)
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