Optical Frequency - Research Projects

 Maintaining the SI unit of length
 Optical frequency standard based on a single trapped ion
 Femtosecond comb

Maintaining the SI unit of length

For almost fifty years, the optical wavelength of light has been used as the reference in high-accuracy measurements of absolute length. Several kinds of light sources can be used, including electrically-excited gaseous discharge tubes and stabilized lasers. In these measurements, a device called an optical interferometer is used to compare the length of an object to the known wavelength of the light source. From 1960 until 1983, it was internationally agreed that the unit of length, the Systeme International (SI) metre, would be defined as 1,650,763.73 wavelengths (in a vacuum) of the orange-red line of krypton 86. Wavelengths of other spectral sources could be calibrated with respect to this krypton line and used as secondary standards. Technical difficulties involved in the use of optical interferometers, however, limited the uncertainty of any length measurement to approximately 1 part in 10¹⁰. In effect, the accuracy to which the metre could be realized in practice was also limited by the same uncertainty.

By the early 1980's techniques for measuring the frequency of light had surpassed those for measuring optical wavelength in terms of accuracy and so, in 1983, it was agreed to redefine the metre in terms of the unit of time, the SI second. The definition chosen was "The metre is the length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second". At first glance, this definition appears to suggest that a very careful measurement of the propagation of light in a vacuum would be necessary in order to realize a practical metre standard. However, from the simple equation

c =

where c is the speed of light (defined), is the wavelength in vacuum, and is the frequency of light, it can be seen that if the optical frequency of some stable light source is measured with an uncertainty of say, 1 part in 10¹², then the wavelength of that source is also automatically known to the same uncertainty. A practical metre standard can therefore be realized in terms of the wavelength of any stable light source whose frequency has been accurately measured.

The Optical Frequency Standards (OFS) Project maintains the SI metre for Canada through the vacuum wavelength of a helium-neon (HeNe) laser, which is stabilized to a transition in the iodine molecule in the red part of the optical spectrum at 633 nm. In the language of spectroscopy, the relevant spectral line is the a₁₃ or i component of the 11 — 5 R(127) transition of the ¹²⁷I₂ molecule. The frequency of this line has been measured with respect to the SI second in a number of national laboratories, including NRC and has a value of

= 473 612 214 705 kHz
= 632.991 398 22 nm

with an uncertainty of 12 kHz or 2.5x10^-11, limited by the day-to-day and device-to-device reproducibility of these lasers. In practice, the neighbouring d, e, f, and g components are also used. There are three of these iodine-stabilized helium-neon (I₂/HeNe) lasers at NRC as shown in the figure above. Their accuracy has been maintained through periodic intercomparisons of their optical frequencies and through comparisons with similar lasers from other national metrology laboratories.

Optical Frequency Standard based on a Single Trapped Ion

Currently, the System International (SI) second is defined as "the duration of
9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom". Cesium atomic clocks have been in existence since the late 1950's and have been gradually improved to the point where modern cesium-fountain clocks (Fountain Project) realize the definition of the second with a relative uncertainty of about 1x10^-15. These clocks use millions of atoms at any one time and it is expected that their accuracy ultimately may be limited by collisions between the atoms.

For approximately twenty years, it has been possible to hold a single charged atom (ion) in a small electromagnetic trap under a high vacuum. One can then slow down the motion of this ion to the point where it is confined to a region of space smaller than one cubic micron (10^-6 m)³. It was quickly realized that such an ion was almost completely isolated from the surrounding environment and could serve as the ultimate frequency or time standard with a reproducibility and stability several orders of magnitude superior to the best cesium time standards. At NRC, the Optical Frequency Standards (OFS) Project has developed an optical frequency standard, based on a single, trapped and laser cooled strontium-88 ( ⁸⁸Sr) ion.

The ⁸⁸Sr ion was chosen because of the availability of the lasers necessary for laser cooling and probing of the "clock" transition. The ion is levitated inside a high-vacuum chamber by a small Paul trap as shown in the figure on the left. The distance between the two end electrodes is approximately 1 mm. A radio-frequency voltage applied between the ring and endcap electrodes results in an electromagnetic force on the ion which keeps it near the centre of the trap. When it is first captured, the ion moves around inside the electromagnetic trap typically at speeds of about 300 m/s. A blue laser is focused onto the ion and used to slow it down through a process called laser cooling. Here, the laser frequency is tuned to the low frequency (red) side of a strong electronic transition at 422 nm. Whenever the ion moves towards the laser, it sees the laser frequency Doppler shifted to a higher frequency, closer to the linecentre of the 422-nm transition. This process is similar to how the pitch of the whistle changes as a train goes by. The ion can therefore preferentially absorb a photon of light from the laser beam when its motion opposes the direction of the light beam. For each photon absorbed, the ion receives a small kick in the direction opposite to its motion. Since the re-emitted photon goes off in a random direction, the ion's velocity toward the laser is reduced. After absorbing and re-emitting several thousand photons, the ion's speed can be reduced to only a fraction of a metre per second. The electromagnetic trap then confines the ion to a much smaller volume of space than it would without laser cooling.

The figure on the right shows an energy level diagram for the ⁸⁸Sr ion. Three lasers are required for the frequency standard: the 422 nm blue laser, required for laser cooling; an auxiliary infrared laser at 1092 nm, required to repump the ion back to the P_1/2 state when it occasionally decays to the long-lived D_3/2 state; and a red laser at 674 nm, whose optical frequency is locked to the S_1/2 - D_5/2 "clock" or reference transition. This latter transition is an electric quadrupole transition and is so weak that an ion excited to the D_5/2 state takes, on the average, about 0.3 s to decay back to the S_1/2 ground state - a very long time on an atomic scale. This results in an extremely sharp spectral line at 674 nm and a correspondingly narrow tuning range for the 674-nm laser over which it is capable of exciting the ion to the D_5/2 state. In fact, the natural linewidth of the S_1/2 - D_5/2 transition is less than 1 Hz, compared to its centre frequency of 445 THz (445x10¹² Hz)!

In order to use this narrow transition and the 674-nm laser in an optical frequency standard, the 674-nm laser must be extremely stable and locked in frequency to the centre of the S_1/2 - D_5/2 transition. Our laser is based on a red diode laser, similar to those used in bar-code scanners, with its optical frequency carefully controlled so that an integer multiple of its wavelength is exactly equal to twice the length of a stabilized optical cavity. A photograph of this cavity is shown on the left. It is approximately 25 cm in length and made from a special glass that expands or contracts very little with changes in temperature. The cavity is temperature-stabilized and placed in a vacuum chamber, as shown in the photograph below, in order to isolate it from acoustic noise.

The vacuum chamber is mounted on a vibration-isolating table located in a concrete bunker. Part of the light from this laser is shifted in frequency to the centre of the S_1/2 - D_5/2 transition by a device called an acousto-optic modulator (AOM). Our current laser has achieved a drift rate of a fraction of one hertz per second and a linewidth of less than 100 Hz - much wider than the natural linewidth of the S_1/2 - D_5/2 transition. It is expected that this laser will eventually have a linewidth below 10 Hz.

The absorption of single photons from the 674-nm laser beam by the strontium ion at a rate of only a few per second would be undetectable by normal means. Any slight decrease in the beam power would be completely masked by the normal power fluctuations present in the laser. A special technique is therefore required to tell when the laser frequency is tuned onto the narrow S_1/2 - D_5/2 transition. This technique is called the method of quantum jumps.

When the ion is excited by the 422-nm laser, it makes millions of transitions every second back and forth between the S_1/2 and P_1/2 levels, so many in fact that the fluorescence photons at 422 nm from one solitary ion can be detected by a sensitive photomultiplier. However, when the strontium ion absorbs a single photon at 674 nm, it jumps to the long-lived D_5/2 state and no longer interacts with the 422-nm laser beam. The fluorescence at 422 nm suddenly stops. Only after the ion decays back to the S_1/2 state does the florescence at 422 nm reappear. These jumps in the fluorescence signal are known as quantum jumps.

The figure above shows a 15-s plot of the measured fluorescence signal. Several quantum jumps can be seen. In our experiment, a computer is used along with the AOM to scan the optical frequency of the 674-nm laser across the S_1/2 - D_5/2 transition. The computer counts the number of quantum jumps in a certain time interval and uses this information to lock the shifted laser frequency to the centre of the transition.

The frequency of the S_1/2 - D_5/2 transition at 674 nm was measured several years ago at NRC using a complicated device known as an optical frequency chain. A value of 4444 779 044 095 400 Hz with an uncertainty of only 200 Hz was measured. This corresponds to a relative uncertainty of just 5 parts in 10¹³. Although this uncertainty is very small, it is several orders of magnitude larger than any suspected systematic offsets or errors in the S_1/2 - D_5/2 transition frequency due to perturbations of the ion. Errors in the locking of the 674-nm laser to the centre of the transition and errors in the frequency chain limited the accuracy achieved in our measurements.

Recent measurements at NRC and the National Physical Laboratory (NPL) in Britain using a new device called an Optical Frequency Comb (The Optical Frequency Comb) have confirmed earlier frequency chain measurements and reduced the uncertainty to 100Hz.

Applications of the Single-Ion Frequency Standard

Soon after the frequency of the S_1/2 - D_5/2 transition was measured; the single ion standard was used in measurements of other important optical frequency standards. These included a standard near 1500 nm, with applications in the field of fibre optic telecommunications, and an iodine-stabilized helium-neon laser standard at 633 nm, which is used worldwide as a practical realization of the SI metre (Maintaining the SI Metre).

The recent development of optical frequency combs (The Optical Frequency Comb) has made it possible to measure the frequency of almost any stable optical laser source to unprecedented accuracy. An optical frequency comb can be used to compare quickly and accurately the frequency of other optical standards to the single ⁸⁸Sr ion standard or to the Cs atomic clock SI realization of the second. It should also be possible to use the ⁸⁸Sr ion as the source of regularly timed "ticks" in a new kind of atomic clock - the optical clock. The optical frequency comb is capable of counting every cycle of the laser locked to the ⁸⁸Sr S_1/2 - D_5/2 transition, no mean feat since there are 445 trillion of these cycles every second. Such an optical clock, with the ⁸⁸Sr ion as the source of ticks and the optical frequency comb serving as the clockwork, is expected to be far superior to the best cesium atomic clocks in terms of reproducibility and stability.

Optical Frequency Comb

The Measurement of Optical Frequencies

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.

Figure 1: The temporal behavior of an electromagnetic wave.

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

Figure 2: A classical fequency chain for measuring optical frequencies

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.

Optical Frequency Combs

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 T_rep. 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 ƒ_rep= 1 GHz (ƒ_rep=1/T_rep). 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

Firgure 3: The NRC classical chain for measuring the optical
transition at 445 THz in a single trapped stronium ion.

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 *λ_c + x, where a is a small integer and λ_c 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 + &#966ceo where &#966ceo is known as the "carrier/envelope offset" phase and is equal to 2π *x /λ_c. The carrier frequency is therefore given by ƒ_c=b * ƒ_rep+ƒ_o, where b is an integer of the order of several hundred thousand, and ƒ_o is the "offset" frequency (less than frep) given by

Figure 4: The electical field of a mode-locked laser

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, ƒ_rep. This is due to the fact that the carrier signal, ƒ_c, is amplitude modulated, which produces frequency sidebands in the spectrum with a spacing equal to the modulation frequency, ƒ_rep. 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

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 ƒ_rep and ƒ_o can be determined. The pulse repetition period (1/ ƒ_rep) is equal to the time for the pulse to make one round-trip through the laser cavity. Therefore, ƒ_rep 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 ƒ_rep to a radio frequency (rf) signal generated by a high quality synthesizer.

It is 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 ƒ_o, 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 can 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

whereI is the intensity and n_o 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.

Figure 5: Results of a numerical simulation of self-phase modulation

It is possible to determine the offset frequency, f_o, 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 ƒ_n= n*ƒ_rep+ƒ_o. If this comb element is frequency doubled in a nonlinear crystal, a signal at 2ƒ_red= 2n*ƒ_rep+2f_o is produced. When this signal is mixed on a fast photodiode with a component from the blue end of the spectrum at ƒ_2n=2n*ƒ_rep+fo, the frequency of the resulting heterodyne (or difference frequency) beat signal is just2ƒn – ƒ_2n = ƒ_o. Therefore, if the comb can be broadened to at least one octave, it can serve as its own reference and fo can be found.

Figure 6: Self-referencing method of determining the value of the offset frequency, ƒo.

It would appear that since frequency doubling is normally a very inefficient process, the signal at ƒ_o 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 ƒ_o. 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 ƒ_o.

It is normal practice to phase lock the repetition frequency, ƒ_rep, and the offset frequency, ƒ_o, 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, f_B. The laser frequency is then given by,

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.

The NRC Optical Frequency Comb

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 20-30 cm length of microstructured fibre. This fibre consists of a 1.7-mm-diameter pure silica core surrounded by an array of 1.3-mm-diameter air holes. In passing

Figure 7: Schematic diagram of the NRC optical frequency comb. AOM – acousto-optic modulator; IF – interference filter; KTP – frequency doubling crystal.

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. Two photographs of the comb in operation are shown in Fig. 9. The photograph on the top shows 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. The Ti:sapphire laser is located in the lower part of the photograph while the fibre is at the top. The high-power green pump beam at 532 nm is clearly visible due to scattering off of air molecules and dust and is located near the bottom edge of the photograph. The photograph on the bottom shows the

Figure 8: The spectrum of the comb after the microstructured fibre. Fibre length = 25 cm; Total power = 170 mW.

microstructured optical fibre. The pulses from the Ti:sapphire laser are centred at a wavelength of approximately 800 nm and appear red. These pulses enter the fibre from the left and as they propagate down the fibre, additional colours are generated through self-phase modulation and the output from the fibre appears yellow-green.

Figure 9: Photographs of the mode-locked laser and microstructure fibre.

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).