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Research within the Optical Frequency Standards Project at NRC is concerned with the accurate measurement of the frequency of electromagnetic radiation in the optical region of the spectrum and with the development of frequency-stable optical sources. Optical frequency standards are important for a number of applications, for example: dimensional metrology, atomic and molecular spectroscopy, and precise time keeping.
Light waves and radio waves are both forms of electromagnetic radiation which differ only in the period of oscillation of the electromagnetic field. In a vacuum, both can be represented by transversely oscillating electric and magnetic fields as shown in the Fig. 1. It is possible to use conventional electronics to count the number of cycles per second or, in other words, to measure the frequency of the electromagnetic wave, for frequencies up to approximately 100 billion cycles per second (100 GHz). However, no electronic device exists which is capable of counting the oscillations of optical radiation where the frequency is in the range of tens to hundreds of trillions of cycles per second (THz). Another, less direct technique is required.
Until a few years ago, the only way of measuring optical frequencies was through a device called a frequency chain. These chains were made up of specialized microwave oscillators and a number of lasers extending from the far infrared to the visible. In its simplest form, at each step in the chain, a nonlinear device was used to produce a harmonic of the frequency of a particular link in the chain (a microwave source or laser) such that that harmonic was approximately equal to the frequency of the next link in the chain (another microwave source or laser). See Fig. 2. The frequency difference was kept small so it could be measured by conventional electronics. Often, a servo system, which generated a control signal proportional to the offset of the frequency difference from some desired value, was used to control the frequency of one of the lasers or oscillators. More links in the chain were added by repeating this process a number of times until the chain extended from a frequency reference in the radio region, such as an atomic clock, to the optical frequency to be measured. Each new measurement required the construction of a new frequency chain and several years of effort. As a result, only
a handful of optical frequencies were ever measured with these devices. At NRC, a conventional frequency chain was used in 1997 and 1998 to measure the frequency of an optical transition in a single trapped and laser cooled strontium-88 ion (single trapped ion standard) at 445 THz (674 nm) to an accuracy of 200 Hz or 5 parts in 1013 (provide a link to "Optical frequency standard based on a single trapped ion"). This chain, which is shown in Fig. 3, included a total of two microwave oscillators and six lasers.
Recently, a much simpler method of measuring optical frequencies has appeared as a result of developments in femtosecond laser technology and nonlinear fibre optics. It is now possible to measure with a single device and with unprecedented accuracy the frequency of almost any stable optical source. This device is known as an optical frequency comb.
A mode-locked laser emits a regular train of short light pulses separated in time by some repetition period Trep. For example, a high-repetition-rate mode-locked Ti:sapphire laser can emit a pulse of light every nanosecond (1 ns=10-9 second), or at a repetition frequency of ƒrep= 1 GHz (ƒrep=1/Trep). See Fig. 4. The electromagnetic field of the carrier light wave which makes up these pulses oscillates at a much higher frequency. For example, the light emitted by a Ti:sapphire laser is centred at a wavelength of about 800 nm and has an oscillation frequency of 375 THz. Therefore, during the period of time from one pulse to the next, the carrier wave oscillates approximately 375,000 times. If one were able to fix the laser's repetition frequency to some value and fix the exact number of cycles of the carrier between pulses, then the frequency of the carrier wave would be known. This is in essence what is done by an optical frequency comb.
Fig. 4 shows the electric field of the carrier for two consecutive pulses. The envelope of the pulse travels inside the laser cavity at the so-called group velocity, while any particular point on the carrier wave travels at the phase velocity. As a result of dispersion within the laser components, the group and phase velocities are not the same and therefore, the carrier waveform under the pulse envelope does not appear the same from one pulse to the next. If you were able to travel
through the laser cavity while sitting on top the pulse envelope, you would see the carrier wave pulling ahead of you so that in one round trip through the laser cavity, the carrier wave would have advanced under the pulse envelope by a certain number of whole cycles plus some amount, x. That is, in the period of time from one pulse to the next, the carrier wave would advance ahead of the pulse envelope by a distance of a *λc + x, where a is a small integer and λc is the wavelength of the carrier wave. In other words, in the period between pulses, the phase of the carrier wave advances under the pulse envelope by an amount equal to a * 2π + Πceo where Πceo is known as the "carrier/envelope offset" phase and is equal to 2π *x /λc. The carrier frequency is therefore given by ƒc = b * ƒrep+ƒo, where b is an integer of the order of several hundred thousand, and ƒo is the "offset" frequency (less than frep) given by
ƒo = Φceo/(2π) x 1/Trep = ƒrepΦceo/(2π)
If the train of pulses from the mode-locked laser is examined with a high-resolution spectrometer, the spectrum would be seen to consist of a comb of optical frequencies with a spacing equal to the pulse repetition frequency, ƒrep. This is due to the fact that the carrier signal, ƒc, is amplitude modulated, which produces frequency sidebands in the spectrum with a spacing equal to the modulation frequency, ƒrep. The number of sidebands generated is dependent on the pulse duration. For pulses having a duration of say, 50 femtoseconds (1 fs = 1 10-15 s), the spectral full width at half maximum (FWHM) would be approximately 25 nm and the resulting comb would contain over 11 thousand sidebands.
These sidebands are centred about the carrier frequency and therefore, the frequency of any one of these comb elements is given by
ƒ = n x ƒrep + ƒo
where n is an integer of the order of several hundred thousand.
It can be seen from Eq. 2 that the frequencies of all the comb elements are known, provided ƒrep and ƒo can be determined. The pulse repetition period (1/ ƒrep) is equal to the time for the pulse to make one round-trip through the laser cavity. Therefore, ƒrep can be controlled by controlling the cavity length – usually by mounting one of the laser cavity mirrors onto a piezo-electric transducer (PZT). The pulse repetition frequency can be detected with a fast photodiode. A servo system, which controls the PZT, is then used to lock ƒrep to a radio frequency (rf) signal generated by a high quality synthesizer.
Until about 2000, it was very difficult, if not impossible, to determine the offset frequency directly from the output of the mode-locked laser. Even a pulse duration as small as 30 fs is at least ten times the period of the carrier wave, making it difficult to detect and control the phase of the carrier to envelope offset. Control of the offset frequency ƒo, through Eq. 2 requires knowledge of some reference frequency, ƒ, to an accuracy at least as good as that needed in any measurements. Normally such references are not available. As a consequence, the comb from the mode-locked laser could be used only to determine frequency differences, not absolute frequencies. Fortunately, a method has appeared in recent years which modifies the comb and allows it to serve as its own reference. This technique relies on the use of "holey" or "microstructured" optical fibre to broaden the comb of frequencies to over an octave.
At low optical powers, the refractive index of a material is independent of the incident intensity. However, as the intensity is increased, the refractive index of many transparent materials increases. The refractive index can be written as
n = no + n2*I
where I is the intensity and no and n2 are the normal and nonlinear refractive indices, respectively. As a result of the tight confinement of the laser beam in the fibre and the low dispersion properties of microstructured fibre, the pulse intensity remains high as the pulse propagates over many centimetres. This leads to strong nonlinear effects in the pulse shape and spectrum. The most important of these is self-phase modulation. This process is illustrated in Fig. 5. The intensity is highest at the peak of the pulse envelope and therefore, from Eq. 3, that part of the pulse experiences the highest refractive index and as a result, propagates slower than the leading and trailing parts of the pulse. This causes the carrier wave to stretch out on the leading part of the pulse (become shifted to the red) and to pile up on the trailing part of the pulse (become shifted to the blue). As shown in Fig. 5, the resulting spectrum is modulated and broadened.
It is possible to determine the offset frequency, ƒo, if the spectrum is broadened to over an octave. See Fig. 6. For illustration purposes, consider a single comb element at the red end of the spectrum at frequency ƒn= n*ƒrep+ƒo. If this comb element is frequency doubled in a nonlinear crystal, a signal at 2ƒn= 2n*ƒrep+2ƒo is produced. When this signal is mixed on a fast photodiode with a component from the blue end of the spectrum at ƒ2n=2n*ƒrep+ƒo, the frequency of the resulting heterodyne (or difference frequency) beat signal is just 2ƒn – ƒ2n = ƒo. Therefore, if the comb can be broadened to at least one octave, it can serve as its own reference and fo can be found.
It would appear that since frequency doubling is normally a very inefficient process, the signal at ƒo should be extremely weak. However, not just one comb element, but rather, all the comb elements within a band determined by the crystal's phase-matching undergo frequency doubling. In addition, sum frequency generation between nearby modes also takes place and contributes to the total rf signal power at ƒo. The offset frequency can therefore be measured and a servo system can be used to lock ƒo to an rf reference signal. The lock is usually accomplished by controlling the power of the laser which pumps the Ti:sapphire mode-locked laser. Changes in the pump power absorbed by the Ti:sapphire crystal lead to small changes in its chromatic dispersion and, hence, changes in the group and phase velocities and in ƒo.
It is normal practice to phase lock the repetition frequency, ƒrep, and the offset frequency, ƒo, to rf signals provided by high-quality synthesizers, which in turn are locked to a signal provided by a frequency/time standard such as a hydrogen maser. If everything is done properly, the frequency of each of the hundreds-of-thousands of comb elements, as given by Eq. 2, has the same long-term stability and accuracy as the hydrogen maser. Each comb element can then be used as a reference frequency in measurements of other optical frequencies. In practice, this is accomplished by overlapping the laser beam from the source to be measured with the appropriate spectral segment of the comb onto a fast photodiode and measuring the frequency of one of the resulting heterodyne beats, fB. The laser frequency is then given by,
ƒlaser = ƒ ± ƒB = n x ƒrep + ƒo ± ƒB
Since normal optical filters are unable to isolate a single comb element from its nearby neighbours, many heterodyne beats are produced. To determine n and the signs, and therefore, the value of ƒlaser, it is necessary to first measure the approximate value of ƒlaser to an accuracy of several hundred megahertz by some other means. This can usually be accomplished with a commercial wavemeter.
A schematic diagram of the optical frequency comb used at NRC is shown in Fig. 7. The Ti:sapphire laser is used to produce a periodic train of 30-50 fs pulses, centred at 800 nm, at a repetition rate of approximately 700 MHz. An average power of 100-200 mW is coupled by a single-element aspheric lens through a 12 cm length of microstructured fibre. This fibre consists of a 1.8-mm-diameter pure silica core surrounded by an array of small-diameter air holes.
In passing through the fibre, the spectrum of each pulse is broadened to over an octave. A sample spectrum of the output from the fibre is shown in Fig. 8. Clearly, the power is not evenly distributed among the comb elements. The overall spectral width, as well as the locations of the peaks and dips, is strongly dependent on the coupling of the laser light into the fibre, as is expected for such a nonlinear process.
A photograph of the comb in operation is shown in Fig. 9. A diffraction grating was used to disperse the visible part of the output from the comb onto a cardboard screen. The Ti:sapphire laser and the microstructured fibre, which are both contained inside an airtight box to isolate them from acoustic noise and variations in air pressure are shown on the left side of the photograph.
The comb has been used successfully to measure the frequencies of optical sources ranging from 550 THz (543 nm - in the green part of the visible spectrum) to 260 THz (1153 nm – in the infrared).