Optical heterodyne detection is a method of extracting information encoded as modulation of the phase, frequency or both of electromagnetic radiation in the wavelength band of visible or infrared light. The light signal is compared with standard or reference light from a "local oscillator" (LO) that would have a fixed offset in frequency and phase from the signal if the latter carried null information. "Heterodyne" signifies more than one frequency, in contrast to the single frequency employed in homodyne detection.
The comparison of the two light signals is typically accomplished by combining them in a photodiode detector, which has a response that is linear in energy, and hence quadratic in amplitude of electromagnetic field. Typically, the two light frequencies are similar enough that their difference or beat frequency observed by the detector is in the radio or microwave band that can be conveniently processed by electronic means.
This technique became widely applicable to topographical and velocity-sensitive imaging with the invention in the 1990s of synthetic array heterodyne detection. The light reflected from a target scene is focused on a relatively inexpensive photodetector consisting of a single large physical pixel, while a different LO frequency is tightly focused on each virtual pixel of this detector (a different LO frequency signal focused on a different part of the detector), resulting in an electrical signal from the detector carrying a mixture of beat frequencies that can be electronically isolated and distributed spatially (as we know which part of the detector gives which beat frequency) to present an image of the scene.
History Optical heterodyne detection began to be studied at least as early as 1962, within two years of the construction of the first laser. However, laser illumination is not the only way to produce spatially coherent light. In 1995, Guerra published results in which he used a "form of optical heterodyning" to detect and image a grating with frequency many times smaller than the illuminating wavelength, and therefore smaller than the resolution, or passband, of the microscope, by beating it against a local oscillator in the form of a similar but transparent grating. A form of super-resolution microscopy, this work continues to spawn a family and generation of microscopes of particular use in the life sciences, known as "structured illumination microscopy", Polaroid Corp. patented Guerra's invention in 1997.U.S. Pat. No. 5,666,197; "Apparatus and methods employing phase control and analysis of evanescent illumination for imaging and metrology of subwavelength lateral surface topography"; John M. Guerra, inventor; Assigned to Polaroid Corp.; Sept. 1997.
Contrast to conventional radio frequency (RF) heterodyne detection It is instructive to contrast the practical aspects of heterodyne detection in optical band to radio frequency (RF) band.
Energy versus electric field detection Unlike RF band detection, optical frequencies oscillate too rapidly to directly measure and process the electric field electronically (e.g., 632 nm in wavelength for a visible HeNe laser that appears red, is 4.75×1014 Hz in frequency). Instead optical photons are (usually) detected by absorbing the photon's energy, thus only revealing the magnitude of an optical signal, not the electric field phase. Hence the primary purpose of heterodyne mixing is to down shift the signal from the optical band to an electronically tractable frequency range.
In RF band detection, typically, the electromagnetic field drives oscillatory motion of electrons in an antenna; the captured EMF is subsequently electronically mixed with a local oscillator (LO) by any convenient non-linear circuit element with a quadratic term (most commonly a rectifier). In optical detection, the desired non-linearity is inherent in the photon absorption process itself. Conventional light detectors—so called "Square-law detectors"—respond to the photon energy to free bound electrons, and since the energy flux scales as the square of the electric field, so does the rate at which electrons are freed. A frequency difference between an input signal and a LO signal to a detector appears in the detector output electrical current, only when both signals illuminate the detector at the same time, causing the square of their combined fields to have a cross term or "difference" frequency modulating the average rate at which free electrons are generated.
Wideband local oscillators for coherent detection Another point of contrast is the expected bandwidth of the input signal and local oscillator signal to the detector. Typically, an RF local oscillator is a pure frequency; pragmatically, "purity" means that a local oscillator's frequency bandwidth is much much less than the difference frequency between the input and LO signals. With optical signals, even with a laser, it is not simple to produce a reference frequency sufficiently pure to have either an instantaneous bandwidth or long term temporal stability that is less than a typical megahertz or kilohertz scale difference frequency. For this reason, the same source is often used to produce the LO and the input signals so that their difference frequency can be kept constant even if the center frequency wanders.
As a result, the mathematics of squaring the sum of two pure tones, normally invoked to explain RF heterodyne detection, is an oversimplified model of optical heterodyne detection. Nevertheless, the intuitive pure-frequency heterodyne concept still holds perfectly for the wideband case provided that the signal and LO are mutually coherent. Crucially, one can obtain narrow-band interference from coherent broadband sources: this is the basis for white light interferometry and optical coherence tomography. Mutual coherence permits the rainbow in Newton's rings, and supernumerary rainbows.
Consequently, optical heterodyne detection is usually performed as interferometry where the LO and (input) signal share a common origin, rather than, as in radio, a transmitter sending to a remote receiver. The remote receiver geometry is uncommon because generating a local oscillator signal that is coherent with a signal of independent origin is technologically difficult at optical frequencies. However, lasers of sufficiently narrow linewidth to allow the signal and LO to originate from different lasers do exist.
Photon counting After optical heterodyne became an established technique, consideration was given to the conceptual basis for operation at such low signal light levels that "only a few, or even fractions of, photons enter the receiver in a characteristic time interval". It was concluded that even when photons of different energies (so different frequencies as energy per photon is where is the Planck constant and is the wave frequency) are absorbed at a countable rate by a detector at different (random) times, the detector can still produce a difference frequency. Hence light seems to have wave-like properties not only as it propagates through space, but also when it interacts with matter. Progress with photon counting was such that by 2008 it was proposed that, even with larger signal strengths available, it could be advantageous to employ local oscillator power low enough to allow detection of the beat signal by photon counting. This was understood to have a main advantage of imaging with available and rapidly developing large-format multi-pixel counting photodetectors.
Photon counting was applied with frequency-modulated continuous wave (FMCW) lasers. Numerical algorithms were developed to optimize the statistical performance of the analysis of the data from photon counting.
Key benefits
Gain in the detection The strength of the difference frequency signal can be larger than the strength of the original signal. The strength of the difference frequency is proportional to the product of the amplitudes of the LO and original signal electric fields. Thus the larger the LO amplitude, the stronger the difference frequency strength. Hence there is gain in the photon conversion process. The Poynting vector of the sum of the LO and signal is proportional to the square of the sum:
A photodetector, like a photodiode, is much slower than optical frequency (4.75×1014 Hz at 632 nm in wavelength as a red color), so the detector integrates the Poynting vector over a time window much longer than an optical period (2.1×10−15 second for 632 nm in wavelength). In this sense, we can only consider the average of the Poynting vector over the integration window. By using the product-to-sum trigonometric identity and assuming that the detector is much faster than the difference frequency, the detector output is proportional to the following. (The sum of integrations of terms with at frequencies over a time window much longer than the periods of the frequencies statistically gives zero or a noise in values.)
where . The first two terms are proportional to the average (DC) energy flux absorbed (or, equivalently, the average current in the case of photon counting). The third term is the difference frequency. In many applications the signal is weaker than the LO, thus it can be seen that gain occurs because the energy flux in the difference frequency is greater than the DC energy flux of the signal .
Doppler effect Suppose that a signal light in form of , where is the wavenumber in vacuum, is toward a moving target mirror which position is . The mirror receives the light at time as
where is the refractive index of the main wave propagation medium, is a generally inhomogeneous refractive index of a transmissive optical element i with its length , and is a phase shift by a reflective optical element j (Mirror reflection phase shift).
The target mirror immediately (ignoring the time dilation in the special relativity) reflects the light, so the reflected light is
where in is dropped for simplicity, and with now that the second summation (with the index j) includes a phase shift from the target mirror reflection.
The angular frequency of the reflected light is given by the time derivative of its phase;
where is the mirror velocity at . This is the Doppler-shifted frequency by the mirror in moving w.r.t the light source.
The reflected light is now toward a detector fixed at the location . The detector receives this light at time where is an additional delay by the frequency-shifted light travelling through each transmissive optical element p till the detector. By assuming constant refractive indices and reflection phase changes over a laser frequency range made by the Doppler shift is assumed.In other words, the target mirror velocity is much smaller than the speed of light c.
Without this assumption, the interpretation of the optical heterodyne signal becomes not straightforward because the optical path length from the target mirror to the detector is also affected by the Doppler-effect., the detected light is
where in which counts for the phase changes by reflective optics in the way from the target mirror to the detector. The angular frequency of the detected signal light is given by the time derivative of its phase;
because ( by ignoring the time dilation).For a constant target mirror velocity , the detected light can be expressed as where in is dropped for simplicity.
For the detector close enough to the mirror such that and , it becomes , explicitly showing the doppler-shifted frequency. Note that , , and are retarded quantities (retarded position, velocity, and time) for the light reaching the detector .
Then, with a LO light (that travels through fixed optical elements) at the detector at the time , , the heterodyne detection signal is proportional to