User:Micatlan/heterodyne holography

Heterodyne holography refers to hologram acquisition in optical heterodyne detection configuration, for example through the use of electro-optic modulators (Pockels cells) or acousto-optic modulators (Bragg cells), to shift the reference laser beam's frequency by a tunable quantity. It permits frequency-conversion of high frequency optical signals to the sensor's temporal bandwidth.

Principle
In the scalar model of light, a single-frequency light wave can be modeled by a complex number $$E$$, which represents the oscillating electric field, or optical field. The amplitude and phase of the field are the absolute value and angle of $$E$$. In optical heterodyne detection, the signal field is non-linearly mixed with a reference optical wave, which is called a local oscillator (LO), in reference to heterodyne detection. Non-linear mixing is a consequence of a detected output signal proportional to the optical field's irradiance. The object field of amplitude $${\mathcal E}$$ and angular optical frequency $$\omega_\mathrm{L}$$ is $$E = {\mathcal E}{\rm e}^{i \omega_\mathrm{L} t}$$. The optical local oscillator field of amplitude $${\mathcal E}_\mathrm{LO}$$ is frequency-shifted by the angular frequency $$\Delta \omega$$ is $$E_\mathrm{LO} = {\mathcal E}_\mathrm{LO} {\rm e}^{i \omega_\mathrm{L} t + i \Delta \omega_\mathrm{L} t}$$. The object field beating against the local oscillator field takes the form $$E + E_\mathrm{LO}$$. Hence the square-law camera sensor array records an interference pattern of the form $$I = |E + E_\mathrm{LO}|^2 = |{\mathcal E}_\mathrm{LO}|^2 + |{\mathcal E}|^2 + {\mathcal E}{\mathcal E}_\mathrm{LO}^*{\rm e}^{- i \Delta \omega t} + {\mathcal E}^*{\mathcal E}_\mathrm{LO}{\rm e}^{i \Delta \omega t}$$. The terms $$|{\mathcal E}|^2$$ and $${|\mathcal E}_\mathrm{LO}|^2$$, are the self-beating (homodyne) contributions. The heterodyne signal is the third term. The fourth term is the twin-image contribution.

Noise reduction to the shot noise limit of the reference wave
In practice, performing digital holography by combining the off-axis and phase-shifting configurations enables filtering of the laser amplitude noise, found in the self-beating contribution in zero-order image. It is then possible to record and reconstruct holographic images at an extremely low signal level. It was demonstrated experimentally that the sensitivity of the method is limited only by the shot noise, in low-light conditions and high heterodyne gain regime (when the optical power of the optical local oscillator is much higher than the object light power).

Preservation of optical phase
Quantitative motion characterization of the lamellophone of a musical box, behaving as a group of harmonic oscillators, under weak sinusoidal excitation. Images of the vibration amplitude versus excitation frequency show the resonance of the nanometric flexural response of one individual cantilever, at which a phase hop is measured. The object field of amplitude $${\mathcal E}$$ and angular optical frequency $$\omega_\mathrm{L}$$ is $$E = {\mathcal E}{\rm e}^{i \omega_\mathrm{L} t + i \psi}$$. The optical local oscillator field of amplitude $${\mathcal E}_\mathrm{LO}$$ is frequency-shifted by the angular frequency $$\Delta \omega$$ is $$E_\mathrm{LO} = {\mathcal E}_\mathrm{LO} {\rm e}^{i \omega_\mathrm{L} t + i \Delta \omega_\mathrm{L} t}$$. The heterodyne signal $${\mathcal E}{\mathcal E}_\mathrm{LO}^*{\rm e}^{i \psi - i \Delta \omega t}$$.

Holographic laser Doppler imaging
cerebral and retinal blood flow

Holographic laser vibrometry
Time-averaged holography methods are commonly used for non-contact, narrowband (single frequency), measurements of out-of-plane vibrations in homodyne [Powel] and heterodyne [Aleksoff] configurations. For an oscillation of interest at angular frequency $$\omega$$ and period $$2\pi/\omega$$, and an exposure time $$\tau_{\rm E}$$ of the optical sensor, the time-averaging condition means that the exposure is much longer than the oscillation period
 * $$\tau_{\rm E} \gg 2\pi/\omega$$

Holography methods are commonly used for non-contact measurements of out-of-plane vibrations [1–3].

These methods exhibit high reliability either in wideband [4–6] or narrowband [2, 7, 8] single point vibration measurements. Wideband methods allow for transient vibration sensing [9] with a temporal resolution given by the reciprocal of sensor bandwidth, while narrowband schemes permit high frequency resolution and better noise-limited sensitivity with respect to wideband approaches.

Absolute measurement of the optical pathlength modulation depth (and hence the out-of-plane vibration amplitude) can be readily derived from the ratio of the first optical sidebands’ magnitude to the non-shifted optical carrier magnitude for narrowband measurements [1]. These sidebands appear with phase modulation of optical waves, as a result of bouncing onto a surface on sinusoidal motion.