Following from their coherent broadband nature, femtosecond pulses allow to control quantum systems in unique ways that cannot be achieved otherwise [1–8]. The corresponding control utilizes the fact that the transition probability to a given state results from the interferences among all the initial-to-final quantum pathways leading to this state that are photo-induced by the broad coherent spectrum of the pulse. By spectrally shaping the femtosecond pulse [9,10], i.e., manipulating the amplitude, phase, and/or polarization of its different frequency components, one can manipulate these interferences and control transition probabilities and state populations.

Over the past decade, among the processes over which such femtosecond coherent control has been demonstrated to be very effective are multiphoton processes in atoms and molecules [8,11–33]. The multiphoton processes, which have been rationally controlled based on the identification of the excitation pathways and their interference mechanism, include non-resonant and resonance-mediated processes of two-photon absorption [8,11–17], three-photon absorption [8,18–21] and Raman transitions [22,23] as well as the process of coherent anti-stokes Raman scattering (CARS) [24–28]. In the context of the present work, an important aspect of the CARS process is the fact that, as opposed to the other processes, it leads to the (directed) emission of stimulated coherent radiation in new frequencies (other than the ones of the exciting femtosecond pulse). Beyond the fundamental scientific interest in quantum control of matter by light, controlling multiphoton processes is of importance for applications of nonlinear spectroscopy and microscopy. Additionally, the control over directed emission of coherent radiation is also of technological significance for standoff detection of materials [34,35].

Here, we study and demonstrate for the first time femtosecond coherent control over the generation of coherent deep-ultraviolet (UV) radiation in an atomic resonance-mediated (2+1) three-photon excitation. The excitation is induced by phase shaped near-infrared (NIR) femtosecond pulse. The overall process can be viewed as an atomic resonance-mediated (2+1) third-harmonic generation process. The UV radiation is emitted due to the build-up of a transient polarization between the excited and ground states of the system. Depending on the excitation pulse spectrum, the coherent UV emission is either predominantly due to a single excited real state that is accessed resonantly or due to a manifold of virtual states. The former leads to a narrowband UV emission, while the latter leads to a broadband UV radiation. The intensity and phase of each emitted UV frequency ωUV results from the interferences between all the three-photon excitation pathways that lead to a total excitation energy of ω UV. Following from the broad coherent spectrum of the excitation pulse, these three-photon pathways include pathways that are on resonance with the intermediate resonance state as well as pathways that are near resonance with it. The resonance-mediated nature of the excitation provides a much higher degree of control over the emitted UV radiation as compared to a completely non-resonant excitation. The present work utilizes previous works on femtosecond coherent control of resonance-mediated (2+1) three-photon absorption [18–20], also extending them from a case of a single real final state to a case of a continuous manifold of multiple (virtual) final states. Resonance-mediated three-photon excitation in atomic vapors has actually been employed very successfully in the past for efficient and tunable generation of short-wavelength radiation, however it has been used only for generating narrowband (nanosecond or picosecond) UV and VUV coherent radiation by narrowband NIR and VIS excitation (for example, see [36–46]); The present broadband nature of the excitation introduces a completely new dimension to the process.

 

Fig. 1. The generation of coherent broadband UV radiation via resonance-mediated (2+1) three-photon excitation in Na. Several sets of three-photon pathways that are on resonance (δ=0) or near resonance (δ≠0) with the intermediate state |r〉≡4s are shown. The inset shows the two NIR excitation pulse spectra with (thick gray line) and without (thin black line) a resonant access to |v _R〉≡7p.

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Consider the atomic resonance-mediated (2+1) UV generation process, induced by a NIR femtosecond pulse, depicted in Fig. 1. It involves an initial ground state |g〉 and an excited state |r〉 that are of one symmetry, and a manifold of excited states |v_j〉 that are of another symmetry. The NIR excitation pulse spectrum is such that all the |g〉-|v_j〉 and |r〉-|v_j〉 couplings are non-resonant, except for the |r〉-|vR〉 coupling that, according to the considered case, is either resonant or non-resonant. Additionally, the excitation spectrum contains half the |r〉-|g〉 transition frequency (ω_r,g/2). Hence, the irradiation with the NIR broadband pulse leads to a resonance-mediated three-photon excitation from |g〉 to a broad continuous range of final energies followed by a stimulated de-excitation back to |g〉 emitting a coherent broadband UV radiation. In the time domain picture, the UV emission results from a time-dependent dipole moment induced by the NIR pulse. This dipole moment is given by:

where a_j(t) is the amplitude of any atomic eigenstate |j〉, µ _{j, j′} and ω_j,j′ are the transition dipole moment and transition frequency between states |j〉 and |j′〉, respectively.

Within the framework of a 3^rd order time-dependent perturbation theory, this temporal dipole moment is given by:

where a ⁽ⁿ⁾j(t) is the n^th-order correction to the amplitude of the state |j〉 given by:

with E(t) being the temporal electric field of the NIR excitation pulse and the zero-order corrections being a ⁽⁰⁾ _j=δ_j,_g, i.e., the system lies initially at the ground state. Eq. (1) can be transformed to the frequency-domain to give the emitted spectral UV field:

where ℘ is the principal value of Cauchy and E(ω) is the (NIR) spectral field of the excitation pulse. It is given by E(ω)≡|E(ω)|exp[iΦ(ω)], with |E(ω)| and Φ(ω) being, respectively, the spectral amplitude and phase at frequency ω. For the (not shaped) transform-limited (TL) pulse, which is the shortest pulse for a given spectrum |E(ω)|, Φ(ω)=0 for any ω. The quantity µ ² _r,g is the |g〉→|r〉 effective non-resonant two-photon excitation coupling, while D ^(UV) _R (ωUV) and D ^(UV) _nonR stand, respectively, for the |r〉→|vR〉→|g〉 coupling via |vR〉 and for the |r〉→|v_j〉→|g〉 coupling via all the other vj states. They are given by

where Γ_vr is the linewidth of |v_R〉. The spectral intensity emitted at a UV frequency ωUV is IUV (ωUV)∝|EUV (ωUV)|². The total UV yield is Y_UV=∫^∞ _-∞ IUV (ωUV)dωUV.

As illustrated in Fig. 1, Eqs. (4)–(8) reflect the fact that the (complex) spectral field EUV (ωUV) at each emitted UV frequency ω_UV results from the interferences between all the three-photon pathways starting from |g〉 and reaching the final excitation energy that corresponds to ωUV. Each such pathway is either on resonance or near resonance with the intermediate state |r〉, having a corresponding detuning δ. It involves a non-resonant absorption of two photons with a two-photon transition frequency ω_r,g-δ and the absorption of a third complementary photon of frequency ωUV-(ω_r,g-δ). The term A ^(2+1)on-res(ωUV) [Eq. (6)] interferes all the on-resonant pathways (δ=0), while the term A ^{(2+1)near-res}(ωUV) [Eq. (7)] interferes all the near-resonant pathways (δ≠0) with a 1/δ amplitude weighting. Hence, in general, each emitted UV frequency, ω_UV, acquires its own amplitude and phase depending on the complex spectral field of the NIR excitation pulse, E(ω). One should note that, due to the resonance-mediated nature of the (2+1) excitation, this applies also for an excitation with a transform-limited NIR pulse due to the interference between the on- and near-resonant pathways.

When the excitation pulse spectrum allows resonant access to the |vR〉 state by three-photon pathways, the emitted UV spectrum contains the corresponding frequency ω_UV=ω_vR,g. Since the coupling component D ^(UV) _R (ωUV) [Eq. (9)] associated with |vR〉 has a narrow response around ωUV=ω_vR,g, with a magnitude that is much larger than the magnitude of the other coupling component D ^(UV) _nonR [Eq. (10)] associated with all the other | v_j〉 states, the emitted UV spectrum consists of a dominant narrowband part centered around ω_vR,g that is superimposed on a broadband part of much smaller magnitude. The corresponding total UV yield is then proportional to the spectral intensity at ω_vR,g, i.e., Y_UV ∝ I_UV (ω_vR,g) ∝ |A ⁽²⁺¹⁾(ω_vR,g)|². This is actually proportional to the population excited to |vR〉 by the femtosecond pulse [18,19]. When there is no resonant access to |vR〉, i.e. A ⁽²⁺¹⁾(ωUV) is zero for ωUV around ω_vR,g, the D ^(UV) _R(ωUV) coupling is effectively of non-resonant nature and becomes effectively independent of the emitted ωUV, similar to D ^(UV) _nonR. The UV emission spectrum is then broadband with no narrowband component.

The above excitation scheme is physically realized here with atomic sodium (Na) (see Fig. 1), having the 3s ground state as |g〉, the 4s state as |r〉, and the manifold of p-states as |v_j〉 with 7p as |vR〉. The transition frequency ω_r,g≡ω_{4s ,3s}=25740 cm^-1 corresponds to two 777-nm photons and the frequency ω_vR,r≡ω _{7p ,4s}=12801 cm^-1 corresponds to a 781.2-nm photon. The sodium interacts with phase-shaped linearly-polarized femtosecond pulses having a NIR spectrum centered around 777 nm with ~5-nm (FWHM) bandwidth (~180-fs TL duration). The pulse energy is 10 µ J and the peak intensity of the TL pulse is about 10 ¹⁰ W/cm² (the focused beam radius is 0.25 mm). Experimentally, atomic sodium vapor of 3.7×10 ¹⁷ cm^-3 density in a heated cell at 900 K with 10-Torr Ar buffer gas is irradiated with such NIR laser pulses, after they undergo shaping in an optical setup incorporating a pixelated liquid-crystal spatial light phase modulator [9,10]. The effective spectral shaping resolution is 2.05 cm^-1 per pixel. Following the interaction with the NIR pulse, the coherent UV radiation emitted in the propagation direction of the NIR beam is measured using a UV-spectrometer coupled to a time-gated camera system. The corresponding overall UV spectral resolution is 35 cm^-1 (0.23 nm), with 5.8-cm^-1 spectral width of each camera pixel.

As a first control work on resonance-mediated (2+1) generation of coherent UV radiation, we have chosen to demonstrate here phase control over the total emitted UV yield Y_UV with shaped pulses having simple spectral phase patterns Φ(ω) of a spectral π-step at variable position ω _step. As previously shown, this group of spectral phases is highly effective in controlling two-photon absorption [11–17] and resonance-mediated (2+1) three-photon absorption [18–21]. The phase control over the UV yield is studied in two cases: when the excitation spectrum allows access to the 7p state (via various three-photon pathways) and when it does not (see above). The latter is implemented by blocking the low-frequency end of the excitation pulse spectrum. The corresponding excitation spectra for both cases are shown in Fig. 1(inset). The corresponding UV spectra for the above two cases, measured with the TL excitation pulse having the appropriate spectrum, are shown in Figs. 2(a2) and 2(b2). As seen and explained above, an access to the 7p state leads to a UV spectrum that is dominated by a strong narrowband component located around ω_UV=ω_7p,3s=38541 cm^-1 (259.46 nm) with a measured width equal to the UV spectral resolution [Fig. 2(a2)], and when the access to the 7p state is blocked the UV spectrum is purely broadband [Fig. 2(b2)]. Here, it is measured to be of 130±6 cm^-1 bandwidth (FWHM) around 38679±3 cm^-1 (258.54 nm).

The experimental (circles) and theoretical (lines) results for controlling the total UV yield Y_UV with shaped femtosecond pulses having π-step spectral phase pattern for the cases of the 7p state being accessible or inaccessible are presented in Figs. 2(a1) and 2(b1), respectively. Each trace is normalized by Y_UV induced by the corresponding TL pulse. The theoretical results are calculated numerically using Eqs.(4)–(10), using a grid with a bin size equal to the experimental shaping resolution. As can be seen, there is an excellent agreement between the experimental and theoretical results, confirming our theoretical description and understanding.

 

Fig. 2. Experimental (circles) and theoretical (solid lines) results for the total UV yield generated by shaped NIR pulses with spectral π phase step, when the excitation pulse spectrum (a1) allows and (b1) blocks the resonant access to 7p (see Fig. 1). The UV yield is shown as a function of the position of the NIR π phase step. The traces are normalized by the yield generated by the corresponding transform-limited (TL) pulse. The UV spectra generated by the NIR TL pulses in the two cases are shown in panels (a2) and (b2).

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Consider first the case when 7p is resonantly accessed. As explained above, the total yield Y_UV in this case is proportional to the population excited to the 7p state P_7p. Indeed, the TL-normalized trace shown in Fig. 2(a1) for Y_UV reproduces the one measured for P_7p with similar NIR excitation [18,19]. The total UV yield is experimentally controlled from 3% to about 200% of the yield induced by the TL pulse. The strong enhancement occurs when ω_step=ω _7p,4s=12801 cm^-1. As previously identified for the resonance-mediated three-photon absorption [18,19], it originates from a change in the nature of the interferences between the positively-detuned (δ>0) and negatively-detuned (δ<0) near-resonant 3s-7p three-photon pathways. With the TL pulse they are destructive, while with a π-step at ω _7p,4s they are constructive. The physical reason for this proportionality between Y_UV and P _7p is the coherent superposition of the 3s and 7p states that is created by the excitation and survives also after the pulse is over, leading to a long-lived time-dependent dipole moment. This dipole moment induces the UV emission at frequency ω_7p,_3s. The decay of this 3s-7p UV emission is, on one hand, due to the inhomogeneous Doppler broadening (free induction decay [47]) and, on the other hand, due to the dephasing of the 3s-7p coherent superposition following collisions with the Ar buffer gas. Based on previous collisional dephasing measurements conducted for various excited states of Na other than the 7p state [48–50], we conclude that under our experimental conditions the decay of the UV radiation is dominated by the inhomogeneous Doppler broadening, with an estimated decay time of tens of picoseconds. Hence, in this case, the overall result is an ultrashort UV pulse of small integrated energy, followed by a very long quasi-monochromatic radiation at ω_7p,3s with large integrated energy.

In the other case when the 7p is inaccessible [Fig. 2(b1)] the total UV yield Y_UV is not dominated by any single frequency. Similar to the above enhancement with ω _step=ω _7p,4s when the 7p is accessed, a π-step at a position ω_step enhances here the UV amplitude at frequencies around ωUV=ω_step +ω _4s,_3s, while it affects the amplitudes of other frequencies in a complicated way (leading to attenuation or enhancement) according to the above theoretical description. Thus, no enhancement of the total UV yield Y_UV beyond the TL level is observed in Fig. 2(b1) for any ω_step. The corresponding control is from about 10% to 100% of the UV yield induced by the TL pulse. Under our experimental conditions, the integrated energy of the broadband UV emission is about 1% of the NIR pulse energy for a TL excitation.

In summary, we have studied and demonstrated for the first time basic phase control over the resonance-mediated (2+1) generation of coherent UV radiation by shaped NIR femtosecond pulses. Depending on the NIR excitation spectrum, the coherent UV emission is either predominantly due to a single excited real state that is accessed resonantly or due to a manifold of virtual states. The former leads to a narrowband UV emission, while the latter leads to a broadband UV radiation. Controlling the directed emission of short-wavelength coherent radiation is important for nonlinear spectroscopy and microscopy as well as for standoff detection of materials. The presented scheme can generally be extended to the vacuum-ultraviolet (VUV) spectral range by changing the excitation pulse spectrum to the visible (VIS) range and choosing an atomic system whose state energies fit the resonance-mediated (2+1) VIS excitation. Then, further development of the scheme can be a basis for the production of shaped femtosecond pulses in the VUV range where no current pulse shaping technique [9,10] is applicable. This will greatly extend the variety of molecules to be coherently controlled as most of the molecular electronic transitions are in the UV and VUV spectral range.

This research was supported by The Israel Science Foundation (Grant No. 127/02), by The James Franck Program in Laser Matter Interaction and by The Technion’s President Fund.