1 Introduction

Using fast frequency-chirped laser light provides a direct means to excite particles from one state to another. As a laser sweeps through an atomic resonance, it can drive all of the atom’s population into the excited state via rapid adiabatic passage [1]. Because the sweep typically includes a broad range of frequencies, this laser pulse can be used to adiabatically transfer a broad range of particles into the excited state, for example, all atoms in a room-temperature gas. There have been a number of experiments using nanosecond frequency-chirped laser light in atomic physics, including: coherently controlling ultracold collisions [2, 3], coherently controlling photoassociation on the nanosecond timescale [4–6], controlling atomic motion with counter-propagating beams [7–9], and decelerating molecules using the AC stark effect [10–12]. Other practical applications of rapidly modulated laser light in atomic physics include the manipulation of hyperfine levels with a single pulse [13, 14], control of a single atom with shaped rising pulses [15–18], and induced coherent transient effects [19].

There are numerous techniques for generating controlled frequency-chirped laser light. In a given application, the timescale is an important determination of which technique must be used (see [20] for a summary of various techniques). Fiber-coupled lithium-niobate waveguide devices, such as phase and intensity modulators, allow experimenters to modulate the laser light at speeds up to 20 GHz with relatively low voltages. There have been a number of exciting experiments at longer wavelengths (e.g., 1550 and 1064 nm) that have used these types of laser systems to generate frequency-modulated light [21–28]. At shorter wavelengths (e.g., 780 nm), the usable laser power through the waveguide is limited due to the possibility of photorefractive damage. Recent work has shown that it is possible to create intense arbitrary waveforms on sub-nanosecond time scales at 780 nm by amplifying the output of a system with intensity and phase modulators [20]. The amplification was done by a double-pass tapered amplifier system.

Alternatively, we have developed a laser system that generates a rapid frequency chirp (\(\sim\)1 GHz in a few nanoseconds) that operates at a large controllable detuning [29]. Light from an external-cavity diode laser is passed through a phase modulator. While the light is in the phase modulator, the light is modulated at \(\sim\)7 GHz. This large frequency is frequency modulated with a radio frequency signal (e.g., 100 MHz, modulation frequency). This creates frequency-modulated sidebands. The 7 GHz frequency is used to generate the large controllable detuning, and in our system, it can be tuned from 3 to 7.5 GHz. The modulation frequency is used to generate the frequency chirp. In our experiment, this frequency is limited to about 200 MHz which is due to bandwidth limitations in a number of devices. In the final step, the light is passed into an injection-locked laser, where it is injection-locked to a single frequency-modulated sideband.

This laser system can be used to adiabatically control a transition in situations, where it is difficult to directly lock a laser to that transition. We plan to use this system as an inexpensive means to generate laser light on the D1 \(F=2 \longrightarrow F'=2\) line of \(^{87}\)Rb for studying ultracold collisions with frequency-chirped light [29, 30]. In another possible application of the system, the technique may provide a means to study the velocity dependence of the adiabatic rapid passage (ARP) force as discussed in [9] in a quick yet robust manner. To study the ARP force, two counter-propagating frequency-chirped lasers are needed. The relative detuning between these lasers needs to be quickly adjusted while remaining phase locked [9]. This may be achieved by duplicating the technique discussed in this paper while locking to a single seed laser.

While we initially discussed the operation of our laser system in [29], here, we will discuss how we have extended the system. We show, in real time, how a 1.2 GHz in 5 ns frequency chirp detuned far from a reference point can be realized. In addition, we will explore how we can tweak the system to generate novel pulsed, shaped, frequency-chirped laser light. Finally, we will explore what happens when we use this laser system in a simple atomic physics experiment.

2 Fast frequency light at large detuning

The details of the operation of our laser system are discussed in [29]; however, we will summarize them here. Laser light from an external-cavity diode laser (ECDL) is passed through a lithium-niobate phase modulator (EOSpace PM-0K5-10-PFA-PFA-780, see Fig. 1). As the light passes through, it is phase modulated with a \(\sim\)7 GHz signal generated by our RF system. The modulation creates frequency sidebands separated by \(\sim\)7 GHz. The light out of the phase modulator is passed into a free-running Fabry–Perot diode laser (Injection Locked Diode Laser—ILDL) through an optical isolator. The temperature and current of the ILDL are tuned to lock to a specific sideband while simultaneously filtering out all the other sidebands; resulting in locking to a single sideband. The fast GHz signal is frequency modulated with a range up to 1 GHz. The frequency modulation changes the frequency of the sideband and directly leads to the chirp.

Fig. 1
figure 1

Schematic of the laser system. The top schematic shows our optical system. Light from an ECDL is passed into an optical fiber. The light is modulated by a lithium-niobate phase modulator (EOM). For various diagnostics, we have also placed a lithium-niobate intensity modulator (INS) before the phase modulator (see Sect. 3.1); in our typical configuration, the intensity modulator is not in the optical setup. The light out of the fiber is used to injection lock the ILDL after passing through an optical isolator. We use three tools to analyze our modulated beam: (1) an optical heterodyne with a reference, whose frequency is set \(\sim\) 6.8 GHz away from the primary ECDL; (2) a Fabry–Perot cavity (1.5 GHz FSR); and (3) to examine the time dependence of the injection lock, we send light off a diffraction grating, through a long optical path, into a multimode fiber, and finally onto a photodiode. The majority of the light from the ILDL passes through a Rb cell. Fluorescence is collected via an avalanche photodiode (Thorlabs APD120A). The bottom schematic shows our RF system. An arbitrary waveform generator is used to drive the voltage input of a VCO set to generate a nearly 3.5 GHz signal. That signal passes through an RF switch (RFS), which is toggled by a signal from the arbitrary waveform generator. The resultant signal is frequency doubled and amplified

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The RF system is used to generate the \(\sim\)7 GHz modulated signal. First, we generate a frequency modulated \(\sim\) 3.5 GHz signal with a voltage-controlled oscillator (Mini-Circuits ZX95-3360-S+). The RF signal passes through an RF switch (Mini-Circuits ZFSW-2-46) is doubled (Mini-Circuits ZX90-2-36-S+), and amplified (Mini-Circuits ZJL-7G+, Mini-Circuits ZX60-v82-S+, and Mini-Circuits ZX60-v62+). We estimate that we generate a \(\sim\)7 GHz sine wave with an intensity of \(\sim\)22 dBm. We can coarsely tune the amplitude of the system by controlling the drive voltage for the amplifiers, which allows us to tune the amount of power in a given sideband. Next, we modulate the frequency 7 GHz signal by \(\sim\)1 GHz, by introducing modulation via the \(V_\mathrm{{tune}}\) input of the VCO. We can modulate the VCO input frequency at speeds of up to \(\sim\)200 MHz. The modulation is generated by a two-channel arbitrary waveform generator (Rigol DG4102). The other channel of the arbitrary waveform generator is used to drive the RF switch.

Typically, the seed beam has 0.3–0.5 mW of laser power after exiting the phase modulator. The amplitude of the 7 GHz signal is chosen, such that it minimizes the 0th order mode; this is a modulation index of \(\beta \sim 2.4\). At this modulation index, we detect less than 2% of the laser light in the 0th order mode. After the laser exits the fiber, it passes through a prism pair (Thorlabs PS875-B) to roughly mode match with the ILDL. Almost all of the 0th order mode is filtered in the injection locking process, as discussed in [29]. This is supported by measurements we took of an unmodulated seed beam. We measured the locking bandwidth of the laser to be between 1 and 3 GHz, depending on parameters.

The frequency-chirped laser light is analyzed in a few different ways. To explore the frequency dependence of our laser, we make use of a Fabry–Perot cavity and an optical heterodyne with a reference laser ECDL that is frequency-referenced to an atomic transition 6.8 GHz away from the master laser (note, because of the large range in the frequency chirp and relative stability of our external-cavity diode lasers, in many applications, it is not even necessary to lock the external-cavity lasers). The system is tuned, so that the selected injection locked sideband has its frequency overlapped with the reference laser. We can measure the lower frequency heterodyne with an optical beat frequency of <2 GHz. We use a fast photodiode (Thorlabs DET025AFC/M) to collect light from the heterodyne system and an oscilloscope (Tektronix MSO70404C) to collect and process the data.

We show a typical optical heterodyne for a fast frequency chirp in Fig. 2. These data directly confirm in real time the Fabry–Perot data that were taken in [29]. We generated the frequency chirp by driving the \(V_\mathrm{{tune}}\) input of the VCO with an 80 MHz sine wave and selecting only a portion of the wave. The heterodyne shows that the lock is holding throughout the chirp. We determine the instantaneous frequency of the chirped laser light using two different techniques. For a rough determination of the frequency, we map out the mid-point crossings in the heterodyne signal and then determine the frequency difference between adjacent midpoints (\(\Delta t\)). The frequency is then \(1 / 2\Delta t\). For a more exact determination of the frequency, we map the phase of the oscillation. To do this, we took the arccosine of the ratio of the heterodyne signal and its amplitude (maximum height of the oscillation). We had to account for the fact that the arccosine function only provided us with a phase from 0 to \(\pi\). Next, we fit the phase to a high-order polynomial. Then, we took the derivative of the phase to find the frequency. The shape of the frequency nearly matches the expected sinusoid; however, it is skewed and is not a direct match.

Fig. 2
figure 2

Analysis of a rapid frequency chirp detuned from the seed laser. The top figure shows the optical heterodyne with a reference laser that has a frequency set about 6 GHz away from the master laser and an injection locked laser which is locked to the \(+\)1 sideband of the modulated master laser. The modulation frequency is itself modulated with an 80 MHz sine wave, which gives rise to a frequency modulation. The bottom plot shows the frequency of the light as a function of time. We determine this using two different techniques, as discussed in the text. The solid line is the frequency as determined by mapping the phase. The points show the frequency generated by finding the times that the top graph passes through the midpoints of the oscillation. This is an example of a 1.2 GHz chirp in 5 ns

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Initially (as discussed in [29]), we locked the reference laser exactly at 6.8 GHz away from the seed laser. However, the measured optical beat was complicated by its frequency passing through zero frequency. If the frequency passes through zero frequency during the chirp, the heterodyne signal becomes difficult to analyze, mainly due to the sign change in the difference frequency. To obtain a cleaner optical beat, we tuned the reference laser, so that it was \(\sim\)1 GHz away from the center of the frequency chirp. The net result was an optical beat pattern with frequencies ranging between 0.5 and 2.0 GHz. In much of the data taken in this paper, the reference laser was set about 6 GHz away from the master laser.

3 Extending the capabilities of our frequency-chirped laser light

We have been able to develop two interesting capabilities for our frequency-chirped laser system. We can create a defacto laser pulse by turning on/off the RF signal that generates the sideband. We can also use the system to do some simple shaping of the frequency chirp.

3.1 Pulsing the laser light

We have added a switch in the RF system to be able to pulse the signal on/off quickly. When the modulation turns off, this causes the sideband in the master laser to turn off. As the ILDL is injection locked to the sideband, when the sideband turns off the injection-lock also turns off. The ILDL’s intensity remains on the entire time, even though it is toggling between a locked and unlocked state. Although the laser is still on, for many applications in atomic physics, this unlocked far off-resonant light (typically \(\sim\)1 nm) should have negligible spontaneous scattering [31].

We characterized the pulsing of our laser in two different ways. First, we explored the frequency in real time by examining the heterodyne between the pulsed ILDL and the reference laser. Typical data is shown in Fig. 3. Because of the optical beat signal, it was difficult to extract the pulse shape using this technique. It was further complicated by noise in the heterodyne system, which may have been an indication that the laser was in between being fully locked and partially locked.

To explore the system in a different way, we passed laser light from the ILDL off a diffraction grating. Then, we passed the first-order light off the grating into a multi-mode fiber with a photodiode placed at the other end. The beam path from the grating to the fiber was approximately 1 m. When the laser was locked, the light passes through the fiber. The extracted pulse shapes collected using both techniques were consistent (Within 3%).

Fig. 3
figure 3

Analysis of a rapid pulsed frequency chirp. The top figure shows the optical heterodyne with a reference laser that is \(\sim\)6 GHz away from the ECDL and the injection-locked laser, whose frequency is pulsed on by driving the RF switch in our electronics setup. It should be noted that the intensity of the ILDL is constant through the entire process; only the frequency of ILDL changes. The bottom figure shows the frequency of the pulse as a function of time when the optical beat is detectable

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Independently, we have also incorporated an electro-optical intensity modulator (EOSPACE AZ-0K5-10-PFA-780-UL) before the phase modulator. We can directly switch the master light using the intensity modulator independently of the RF switch, similar to what has been done in [9, 20, 32]. There are benefits of using the intensity modulator over the RF switch. The primary reason for our group is speed. Another benefit with using the intensity modulator is the ability to control the shape of the pulse, which can be used in concert with a tapered amplifier as discussed in Ref. [20]. In this work, we use injection locking as a means to amplify the light out of the modulator and as such there is no benefit for having the ability to pulse shape the beam, because the injection-locked laser can only be in two states locked and unlocked.

There are also drawbacks of using the intensity modulator. The intensity modulator also generates a frequency chirp; however, it can be compensated for as discussed in [32]. Another drawback is that the extra modulator severely attenuates the intensity of the ECDL light. We measured <100 \(\upmu\)W out of the optical system with both the phase and intensity modulator in it. It should be noted that the entrance power is limited to 5 mW at 780 nm to prevent photorefractive damage in the modulators. We were able to injection lock to the first sideband with the reduced parameter space; however, it is unlikely that we would be able to lock to higher order peaks with this configuration. Another relevant consideration is the cost comparison between the intensity modulator and the RF switch. While both the RF switch and the intensity modulator are fairly inexpensive, the cost ratio between them is \(<1/100\).

In essence, both techniques do the same operation. When the intensity modulator is used, the laser is completely turned off. When the RF switch with phase modulator is used, only the sideband is turned off. All the seed lights are detuned entirely out of the ILDL\('\)s locking range. We used the intensity modulator as a means to test the RF-switch/phase modulator setup. We conducted the same set of experiments with the intensity modulator as when we modulated the RF switch directly. As expected, the behavior was similar between the different cases, which is further verification that RF switch mechanism works as discussed.

3.2 Shaping the frequency

Pulse shaping the frequency of lasers on the nanosecond time scale has applications in atomic physics [5, 18], telecommunications, optical signal processing, and microwave photonics [33, 34]. The state-of-the-art laser systems have been made using complex single-sideband generators that are multiplexed to yield high-bandwidth arbitrary waveforms [35]; these laser systems require sophisticated electronics.

Recent experiments involving coherently controlling photoassociation have used shaped frequency-chirped laser light to increase molecule production rates [5]. The arbitrarily-shaped, frequency-chirped laser pulses were generated using the technique, as discussed in [36]. The same group has recently developed a more direct technique for generating shaped, frequency-chirped laser light [20]. One of the major drawbacks in both these techniques is that the frequency dependence of the light is generated by controlling phase modulation. As the frequency is the derivative of the phase, small changes in the phase modulation can cause significant changes to the frequency.

In our apparatus, we are able to modulate the frequency of the light directly. To do this, we program an arbitrary waveform generator to drive the \(V_\mathrm{{tune}}\) input of our VCO in our RF system. We have been able to demonstrate a variety of arbitrary shapes using this method. Figure 4 shows how we can generate the ramp plus arctan shape; a shape similar to this one was recently found to lead to optimal control of ultracold photoassociation of atoms into molecules [5]. To generate this shape, we use three line segments which are arranged, such that the shape of the segments roughly match the expected shape. We manipulated the location of the points (voltage and time) that connect the lines to achieve the desired shape. Figure 4 shows how the function fits to the ramp plus arctan shape. For comparison, we also fit the data to a sine wave to show how that mathematical function does not fit the data. We have begun exploring the possibility of using an optimization algorithm to match the input signal (i.e., output of the arbitrary waveform generator) to the actual shape of the frequency chirp; this idea has already been implemented in a similar system [20].

Fig. 4
figure 4

Demonstration of arbitrary-shaped, frequency-chirped laser light. The top figure shows the voltage signal that is put into the \(V_\mathrm{{tune}}\) input of the VCO. The signal was designed to go from \(-1\) to \(+1\) V; the signal is multiplied by an overall gain by our arbitrary waveform generator to provide the desired chirp range. The middle curve shows the optical heterodyne between a reference laser that is continuously locked to a Rb transition that is \(\sim\)6 GHz away from the ECDL and the injection-locked laser that has its frequency chirped with the desired shape (P.D. Signal refers to the photodiode signal). The bottom curve shows the extracted frequency (Freq.) as a function of time (solid black line). We fit the extracted frequency to two different functions: an arctan curve plus a ramp (dashed red line) and a sine wave (dotted–dashed blue line). The arctan plus ramp function provides a better fit of the data

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We have tried to make a series of more advanced pulse shapes. The overall system is limited to a bandwidth which is \(\sim\)200 MHz. As such, it is difficult to make rapid frequency-chirp shapes with sharp corners that require high harmonics, such as the exponential pulse.

4 Frequency-chirped laser light in a simple atomic physics experiment

As a test of our laser system, we explored using it to excite atoms in a glass cell held at room temperature. Similar work has already been done on much faster timescales and with higher intensities [37–39]. Linearly polarized laser light exiting the ILDL passes through a telescope, where its profile is adjusted, so that the beam is roughly collimated with a width of \(\sim\)1 mm. The beam passes through a room temperature glass Rb cell. The fluorescence generated by the atoms that are illuminated with the laser is spatially filtered and collected by an avalanche photodiode. We recorded the fluorescence generated by the frequency-chirped laser for a series of laser parameters.

Figure 5a shows a typical fluorescence signal. As the light passes through the glass cell, it generates an increasing amount of fluorescence. When the laser shuts off, the remaining excited state atoms decay and emit photons. We fit an exponential curve to the second half of the fluorescence data. The fit yielded a lifetime of 26 ns, which is the expected lifetime for \(^{87}\)Rb [40]. For each decay curve, we determine the amplitude of the exponential curve and plot it as a function of laser power (see Fig. 5b). The chirp rate for the data is 650 MHz in 10 ns, and the FWHM of the pulse is 6.3 ns.

Fig. 5
figure 5

Fluorescence signal from an atomic Rb gas at room temperature generated by excitation with frequency-chirped laser light. The laser was chirped 650 MHz in 10 ns and is roughly centered around the atomic transition (5S\(_{1/2}\) F \(=\) 1 \(\longrightarrow\) 5P\(_{3/2}\,F=\) 2); the FWHM of the pulse is 6.3 ns. a Typical real-time fluorescence signal from the atoms as a function of time. The solid line shows the exponential fit of the data; the fitted lifetime of the decay is 26 ns. For each decay, we record the the amplitude of the exponential curve (peak fluorescence). In part (b), we plot the peak fluorescence as a function of the peak intensity. The fit function shape is generated from the numerical solutions of the Optical Bloch equation. We included two parameters in the fit. There is an amplitude parameter, which is directly proportional to the number of atoms. The other parameter is proportional to the saturation intensity

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In the most ideal case, following the Landau–Zener theory, when the pulse width is significantly smaller than the spontaneous emission rate the excitation rate approaches 100% for large intensity. In this experiment, this would manifest itself in a peak fluorescence signal that would saturate or flatten off as the excitation rate approaches total excitation. This would occur for moderate laser powers. In practice, this is more complicated. There are a number of things that can affect the curve shape, including but not limited to the pulse width is smaller than but still comparable to the spontaneous emission rate, the Doppler shift for a given atom, the fluorescence from atoms in the tails of the Gaussian pulse where the intensity is not maximum, the atom is not a two level atom, and bandwidth considerations of our detector (50 MHz).

Nonetheless, we compared the peak fluorescence data, as shown in Fig. 5b to theory. To do this, we solved the optical Bloch equations for a two-level system excited by frequency-chirped laser light. In the simulation, the atoms are illuminated with a square pulse that has a rise time of 1.4 ns and an FWHM of \(w=6.3\) ns (the pulse is completely turned off after 10 ns). The detuning is set to

$$\begin{aligned} \delta (t) = C_\mathrm{R} t + \delta _0 + \delta _\mathrm{c}, \end{aligned}$$
(1)

where \(C_\mathrm{R}\) is the chirp rate (\(2\pi \times 650\) MHz/10 ns), \(\delta _0\) is \(2\pi \times -325\) MHz, and \(\delta _\mathrm{c}\) is the center detuning, which is an adjustable parameter in the simulation. The calculation provides us with the probability that the atom is in the excited state after the pulse for a given peak Rabi rate (\(\Omega _m\)) and center detuning (\(\delta _\mathrm{c}\)). We assumed that the probability of the atom being in the excited state after the pulse is turned off is proportional to the peak fluorescence. This provided us with a two-parameter function for fluorescence based on the peak Rabi rate and center detuning, \(g = g(\Omega _m,\delta _\mathrm{c})\). In order to account for the atom\('\)s velocity, we did a weighted average with the Maxwell–Boltzmann distribution over the center detuning (i.e., the center detuning is associated with the Doppler shift for a specific atom in the thermal gas), resulting in a function for the fluorescence to be:

$$\begin{aligned} g(\Omega _m) = \int \exp \left( {-\frac{mc^2\delta _\mathrm{c}^2}{2kTf_0^2}}\right) g(\Omega _m,\delta _\mathrm{c}) \mathrm{d}\delta _\mathrm{c} \end{aligned}$$
(2)

where m is the atomic mass and T is the temperature. The net result is a single value function that depends solely on the Rabi rate.

We fit the experimental data show in Fig. 5b with the simulation results. We chose two fit parameters: an amplitude parameter which is the proportionality constant between florescence and excitation probability and a parameter that is related to the saturation intensity. The second parameter is essentially the conversion between the Rabi rate and laser power. Our simple model matches the data well (see Fig. 5b), which is an indication that the frequency-chirped system is working as expected. For our model parameters that are used in the simulation, \(\Omega _m = 30\gamma\) and \(\delta _\mathrm{c} = 0\), the excitation probability is 87%.

5 Conclusions and discussion

Spontaneous emission is a key limiting factor in this process within this parameter space. In the future, we will increase the speed of the laser technique discussed in the paper, use shorter pulses, and higher intensities to drive near 100% of the atoms into the excited state; this will likely be achieved with modest improvements to the laser system as we have shown in our numerical models. When working with ultracold samples, the current laser power is enough for most applications, as one can focus the beam down to the spot size of the ultracold gas. When working with an atomic or molecular beam of a room temperature gas, it is difficult to reach the adiabatic condition for a large sample of atoms, because the atoms fill out a large volume and thus require larger laser beams. In such cases, additional amplification is required. We are currently exploring optimizing the shape of the frequency chirp to improve the excitation probability. This experiment would be similar in principle to [15–18]. In those experiments, the intensity of the beam was modulated. Alternatively, we will explore what happens when the shape of the frequency chirp is optimized.

In conclusion, we have further developed our laser system to generate frequency-chirped laser light at large controllable detuning. We have shown that we can generate short pulses (<3 ns) with rapid chirp rates (>200 MHz / ns). We have also developed the ability to generate arbitrary-shaped frequency chirps. Finally, we have shown that this laser system can be used to do atomic physics experiments that agree with theoretical predictions. There are a number of disadvantages to using our laser system; the primary one is speed. The shortest pulse we have been able to generate is \(\sim\)2 ns. The frequency range of the chirp is limited to about 1 GHz. We are not able to directly shape the intensity of the laser pulse; we are limited to turning the light on and off. However, there are three main advantages of this system over what has been done before (e.g., [20]). (1) The injection locked stage amplifies the laser light out of the modulator and the output power can be made to be relatively high (e.g., >50 mW). In many situations, this power is already high enough. If further amplification is needed, the relatively high power will only require a single pass through a tapered amplifier. In addition, though an injection-lock laser system plus tapered amplifier might be more expensive then a single tapered amplifier, making everything single pass may be technically easier. (2) In our system, the input voltage that controls the modulation is nearly proportional to the modulation frequency, which differs from direct phase modulation where frequency depends on the derivative of the voltage signal. (3) Our frequency-chirp laser system can be done at a large controllable detuning.