Group leaders: Dennis Mills, APS, Argonne National Lab., Argonne, IL 60439

Seb Doniach, Stanford Univ., Stanford, CA 94309

Science involving non-linear optics and quantum optics with visible radiation has grown tremendously with the development of visible lasers. These fields are quite diverse, covering fundamental questions like the validity of the quantum theory, applied physics such as spectroscopy, and communication and computing technology. Typical nonlinear optical effects in the visible spectrum include: mixing of frequencies (with the important degenerate cases of harmonic generation and 4-wave mixing, parametric amplification, and down conversion), multiphoton absorption, non-classical states of light (squeezed states), and stimulated Brillouin scattering.

In contrast, the study of the non-linear interactions of x-rays with matter is a virtually unexplored area of physics. This is not surprising considering x-ray lasers have not been available in the past and, generally speaking, nonlinear optical effects with x-rays are considerably weaker than in the visible regime. Although several tour de force experiments have been performed in the past demonstrating that non-linear effects can be observed at x-ray wavelengths, they have provided only a glimpse of the possibilities that might exist with the advent of an x-ray laser such as the LCLS. The enormous peak field strengths generated by the LCLS may result in nonlinear effects becoming significant or even dominating the interaction with the sample, and hence complicating the analysis of experimental data. Therefore, it is essential to study, in detail, the interaction of the high peak-power, x-ray pulse with matter so that correct interpretation of experimental results can be made with confidence.

In the table below is a partial listing of possible non-linear processes that might be observed with x-rays.

Perhaps the simplest nonlinear interaction is the simultaneous ionization of several electrons by the same number of photons in a single atom or small molecule. The probability for this can become large considering typical linear photoionization cross sections are about 1Mb (10¹⁸ cm²) and photon densities of10¹⁸ cm² can be achieved in a focus of about 1 mm diameter. The strongest effects are to be expected for the longest wavelengths.

Parametric down conversion of x-rays, i.e. the conversion of a incident photon of energy E_pump to two photons of energy E_signal and E_idler, (E_pump = E_signal + E_idler) is probably the most studied of the above-mentioned processes, having been observed using a laboratory source (Eisenberger and McCall, 1971) after theoretical predictions by Freund and Levine (F+L,1969) and with synchrotron radiation sources (Yoda et al., 1998) and (Adams et al., 1998). However the exceedingly low count rates of these experiments (several counts per minute with an incident flux of 10¹¹ photons/sec) have limited systematic studies of this process and have not allowed careful comparison of the theoretical predictions to the experimental results. Parametric down conversion can be thought of as the nonlinear mixing of a photon of frequency w_pump with that of a vacuum fluctuation w_idler producing a beat frequency w_signal. Therefore it depends only linearly on the incident x-ray intensity, although it is a nonlinear optical process. The observed event rate is proportional to the incident flux, however the increased count rates that would be available at a 4^th generation source would permit a much finer study of the details of this process.

Potential applications of the pairs of correlated photons obtained from parametric conversion are to be found in tests of quantum theory and in materials research.

Experiments that probe the violation of Bell's inequality (a formal representation of the famous Einstein-Podolsky-Rosen paradox concerning the interpretation of the measurement process in quantum physics) are being done with pairs of correlated photons in the visible regime. All of these experiments depend crucially on the quantum efficiency of the detectors since a lack of detection probability means that the pure quantum states in question are mixed with external degrees of freedom. A specific advantage of x-rays over visible photons lies in the possibility of using detectors with almost 100% quantum efficiency whereas in the visible range, the maximum efficiency reported (Kwiat, 93) is just above the minimum requirement for Bell's inequality of 71%.

In materials research, one possible field of application is the use of one photon of each down-converted pair for normalization purposes while allowing the other one go to the sample. Apart from absorption losses in the instrumentation, this allows exact knowledge of the number of photons incident on the sample, instead of a Poisson-statistical number. This may lead to an enormous reduction of the radiation dose on the sample required for a given precision of the measurement. This technique may be particularly applicable to materials that are sensitive to radiation.

Another possible application is in-line interferometry. In this type of experiment, both converted photons, differing in energy and/or polarization state, take the same path through a sample. The difference in dynamical phase incurred by the two photons is then specifically sensitive to the difference in polarization or energy while all other effects cancel out. As in similar visible photon experiments (Ou and Mandel,1988), the phase difference will show up in a coincidence rate contrast.

A second standard technique in the visible regime, non-linear mixing, has recently been shown to be feasible at x-ray wavelengths by Namikawa and colleagues (Namikawa,1994), where they have demonstrated the mixing of optical laser light with x-rays. By generating new waves of different frequencies (such as second harmonics or in four-wave mixing processes), the new waves have a well-defined phase with respect to the original wave(s) and make possible phase-sensitive non-linear measurements. For instance, using heterodyne detection techniques in multiple-wave mixing experiments, it is possible to obtain both the real and imaginary parts of the susceptibility c, whereas homodyne techniques only give information about the magnitude c².

A possible application of the mixing of x-rays with visible laser light is the precise shift of Mossbauer lines from a nuclear resonant monochromator (i.e. an ⁵⁷Fe crystal) by mixing with the beat frequency from two visible light lasers. The advantage over current frequency shifters using nuclear forward scattering on spinning disks (Gerdau, 1996) is a larger energy tuning range and faster tuning rates.

Little work had been done in the area of calculating non-linear susceptibilities at x-ray wavelengths. However it is very encouraging to report that calculation of c⁽ⁿ⁾ in this wavelength range should be a straight-forward extension of the methods that have been developed for optical methods (Mukamel, 1999). These techniques can be applied to resonant and non-resonant situations using sum rules to simplify the calculations. Non-linear x-ray mixing experiments would provide experimental verifications of these calculations.

The LCLS will provide a time structure much more favorable for the observation of mixing effects because its repetition rate and pulse lengths would be well-matched to a visible laser pulse length and repetition rate (the latter is often dictated by thermal effects in the converting crystal).

The application of non-linear x-ray optical effects as a tool for other x-ray techniques might open new possibilities. One such technique might be the x-ray analog of the 2-photon fluorescence confocal microscope. At visible wavelengths this technique utilizes a non-linear, two-photon absorption process to excite the atom of interest. Improved depth resolution occurs because the two photon fluorescence occurs only at the focus of the laser, where the cross-section for the two photon process is highest, and not along the entire path length of the light. Similar experiments might be performed in the x-ray regime with scanning x-ray micro-probe beams. (The depth resolution would be dependent on the details of the focusing optics and will be limited by the convergence angle.) This approach might also lead to reduced background at energies near that of the fluorescence line, hence improving signal-to-noise and reducing the minimum detection level for trace analysis.

Phase conjugation is used in the optical regime to provide an `automatic' correction for distortions by conventional optical elements, for optical data processing, such as in associative memories or image processing, and for spectroscopy (Pepper, 1986). One optical technique which might be used for phase conjugation is four-wave mixing. Nearly degenerate four-wave mixing is used for pulse shaping and pulse shape restoration (Pepper, 1986) in the visible region. In the x-ray regime, applications in spectroscopy (Hendricks and Nienhuis, 1989) and as new optical elements would perhaps be possible. The efficiencies of these effects have yet to be worked out.

Eisenberger, P. and McCall, S.L. (1971) Phys. Rev. Lett. 26, 684.
Adams, B., Fernandez, P., Lee, W.-K., Materlik, G., Mills, D. M., Novikov, D. V., and Schulte-Schrepping, H., (1998) to be published.
Freund, I, and Levine, B.F.(1969) Phys. Rev. Lett. 23, 854.
Pepper, D. M. (1986) "Spektrum der Wissenschaft".
Yoda, Y., Suzuki, T., Zhang, X.W., Hirano, K. , and Kikuta, S. (1998) J. Synchrotron Rad. 5, 980 (1998).
Kwiat, P.G., Steinberg, A.M., Chiao, R. Y., Eberhardt, P.H., and Petroff, M. D. (1993) Phys. Rev. A 48, R867.
Namikawa,K., Uematsu, H. K., Ohi, M., Whang, X. W., Ando, M., amd Itoh, S. (1994) Workshop on Scientific Applications of Coherent X-rays, SLAC Report 437, p133.
Mukamel, S. (1999) private communication.
Hendriks, B.H.W., and Nienhuis, G. (1989) Phys. Rev. A 40,1892.
Ou, Z. Y., Mandel, L. (1988) Phys. Rev. Lett. 61, 54.
Gerdau, E. (1996) in proceedings of a meeting on the XFEL at DESY.

WORKING GROUP REPORTS

Subpicosecond Time-resolved X-ray Measurements

Photon Correlation Spectroscopy and Holography

Non-Linear X-Ray Optical Processes

High Field Physics and Non-linear Quantum Electrodynamics with the LCLS

Content Owner: J. Arthur
Page Master: L. Dunn
Last Update: 16 Nov 1999