2.1.2: Ultrashort pulses and longitudinal modes of the cavity
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The most interesting (and most difficult to access) processes take place in this very pico- and femtosecond time range. To get light pulses with the durations in this range, one needs to use modelocked lasers. Here we will briefly introduce the principles of modelocking and discuss its importance for the production of ultrashort light pulses.
We already said that a laser is an amplifier of electromagnetic waves with positive feedback. (The well-known example of this is a microphone-amplifier-loudspeaker system, where bringing the microphone too close to a loudspeaker produces a nerve-wrecking whistle. The oscillation starts, when a bit of noise is picked up by microphone, amplified, emitted by a loudspeaker, picked up by the microphone again, amplified again, and wheeeee!). So, a laser does a similar trick with light. It consists of an active medium, were photons traveling in the gain medium produce their twin brothers via stimulated emission. The light is contained within the laser cavity by (at least) two mirrors, one of which is semitransparent. The leaked out light is the useful laser output, whereas the light that stays in the cavity gives the positive feedback for the amplifier. So, the light travels back and forth in the cavity and is amplified, as shown in fig. 4. Note that although the laser cavity in fig. 4 consists of only two mirrors, more complicated configurations are often employed. However, for the explanation of principles, this simple cavity is sufficient. Such cavity is optically indistinguishable from Fabry-Perot etalon, i.e. it acts as a frequency filter: the only wavelengths that constructively interfere with themselves upon reflection from the mirrors and survive in the cavity are the ones that satisfy the condition:
In other words, for a wave to survive in the cavity, the integer number of wavelengths has to fit in the cavity length. The waves that satisfy this condition are called longitudinal modes of the cavity.
Naturally, the wavelength of light that laser produces also depends on the gain profile of the active medium (i.e. the stimulated emission spectrum). Because the mode frequencies are different, the generation of different modes in the laser cavity starts independently from one another[1], and the mode phases generally are random (fig. 4A). Adding several such fields together and taking the square of the sum, we obtain the intensity of the electric field (light). With random phases, the intensity will be some random periodic function (red line in fig. 4C) with the period that is equal to the double length of the cavity (round trip distance). On the other hand, if we add the modes with identical phases (as in fig. 4B) and take the square of this sum, the resulting intensity will be a train of pulses, where the distances between adjacent pulses – again – are equal to the double cavity length. The physical meaning of such periodicity is clear: this is the distance light needs to cover while travelling around the cavity. After covering this distance, light comes back to the output coupler mirror, through which part of the radiation is emitted to the outer world. Therefore, the distance between the pulses in the train is twice the cavity length, and the repetition frequency of the pulses (called laser repetition rate) can be found by dividing the speed of light by this distance.
The more modes we add, the shorter laser pulses will become, because the extent of space (or time) where all the modes are at their maxima will be shorter. This dependence is visualized in fig. 4D, where the square of the electric field amplitude, resulting from the superposition of 4, 8 and 50 modes with identical phases, is compared. In other words, the more different colours (frequencies) we add, the shorter pulse will become. This can be demonstrated mathematically. For that, let us describe the electric field of light pulse as an oscillating function (imaginary exponent) with a Gaussian envelope, i.e. a product of a Gaussian envelope and an imaginary exponent:
Such pulse shape, graphically depicted in fig. 5, is widely used: even if the pulses of many ultrafast lasers are not exactly described by the eq. , the approximation is quite good for many purposes. Parameter w_{0 }is called the central (carrier) frequency of the pulse, and t is the pulse duration. Instead of t, the temporal length of the pulse at half the maximum amplitude, full width at half maximum, FWHM_{t,} is often used. Equating the Gaussian envelope in eq. to 0.5, we can easily find the relationship between t and FWHM_{t}:
Fig. 2) Electric field (blue line) and amplitude envelope (red line) of a Gaussian pulse. Pulse duration (full width at half-maximum, FWHM, of the amplitude is indicated by the arrow. |
To find the spectrum of the pulse described by eq. , which we can measure by, for example spreading the light of the pulse on a CCD sensor of a camera by a diffraction grating, we take a Fourier transform:
This Fourier integral is easily calculated: you just have to supplement the exponent to a full square, i.e. convert it into an integral of the type , the value of which is known from mathematical physics and is equal to . After collecting all the prefactors, we obtain the spectrum of the pulse described by eq. .
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Here we have denoted . As we can see, the spectrum of a Gaussian pulse is also a Gaussian function, and its spectral width is inversely proportional to the temporal duration, i.e.
The constant on the right-hand side of eq. , which is equal to one, is called time-bandwidth product. Its minimum value is equal to one, in which case we have a “nice” and symmetric Gaussian electromagnetic pulse, which is known as transform-limited pulse because its bandwidth is dictated by Fourier transform. If the frequency components are slightly “out of step” in time, this number will be greater than unity. Eq. also shows that the shorter the light pulse, the broader its frequency spectrum (the more different frequencies are present); the same is illustrated by fig. 4D. Equations - are written down for the electric field strength E, expressed in terms of angular frequency w. This form is more convenient mathematically, however, in the lab we deal with the field intensity, which is the square of the electric field, not the field itself. The frequencies are also usually conventional, rather than angular. When these real-life parameters are plugged into the equations, and the square of the electric field is taken, additional prefactors emerge and for a Gaussian pulse, the product between temporal and spectral widths becomes as follows:
In this product, the value of frequency bandwidth is expressed in Hz; it is measured as full-width at half maximum of intensity; similarly, the temporal duration is full-width at half intensity maximum expressed in seconds.
[1] The mode with every frequency „sees“ only those atoms of the active medium, the stimulated em ission frequencies of which match the frequency of the mode.