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NONLINEAR LASER SPECTROSCOPY Nonlineartechniquessuchasmultiphoton spectroscopy, CARS, DFWM, SHG, and DFG offeradvantagesin resolution and selectivity over conventional methods
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ecause light is not usually affected by the presence of other light beams, our scientific intuition doesn't include experiences in which the color, intensity, or direction of light changes because of other light beams. These effects, however, are becoming increasingly important. Computers based on photonics use nonlinear interactions whereby light beams can control other light beams. In analytical science, surfaceand interface-selective measurements are made with second harmonic generation (SHG) and difference frequency generation (DFG). Degenerate four-wave mixing (DFWM), multiphoton spectroscopy, and coherent anti-Stokes Raman spectroscopy (CARS) are emerging spectroscopic methods with advantages over conventional methods. Optical parametric oscillators are replacing dye lasers. The principles underlying these unusual processes are intimately linked to our intuitive under-
John C. Wright Mitchell J . LaBuda David E. Thompson Robert Lascola Michael W. Russell University of Wisconsin 600 A
standing of light-matter interactions. In this article, we will explore these links, as well as give examples of representative nonlinear analytical applications. First, we need to remember that light is an oscillating electric field, and all lightmatter interactions occur because the field polarizes the material's electron cloud (13). The material's oscillating polarization, in turn, launches new light waves. The functional dependence between input electric fields and the material's induced polarization has linear terms that give the familiar absorption and refraction effects, as well as higher order terms that give the nonlinear effects discussed here. These higher order effects are the result of distortions in the polarization that arise when the input electricfieldsaccroach the same order of magnitude as the fields binding the electrons and nuclei Such large electric fields are easily obtained by focusingpaisedlaser beams The opening artwork chows a
graphical representation of the nonlinear relationship (4) and Equation 1givesits mathematical representation —> - »
...
P(r,t) = x X
—>
-j
Ea(r,t) +
tL
a(r,t)£'b(r,t) +
X Ea(r,t)Ebfr,t)Ec(r,t) + . . . m
Analytical Chemistry News & Features, October 1, 1996
The susceptibility tensors (%(1) »%) are the proportionality constants between the material's polarization and the electric fieldfromall the incident light sources. The %(2) term describes the surface-selective nonlinear effects, and the X(3> term describes most other nonlinear effects. The difference is the result of the symmetry. For isotropic materials, the polarization reverses if the electric field reverses, so terms such as Y(2) don't contribute. In anisotropic situations such as interfaces the polarization is different after field reversal and the %(2) terms dominate In each case these higher order terms describe the distortions of the potarization and these distortions have frenuencies at all of the Fourier rnmnnnents of the inmit lasers T h e new frpnnprirv
components produce light beams with new colors If a Fourier component involves the frefiiTpnries of n-\ inmit lasers causing an wth ontnut th called »-wave mixing . Most nonlinear processes require the experimenter to match the phases of the input lasers and the nonlinear output signal because optimum output intensity occurs when the light launched in different parts of a material is in phase and adds constructively. Phase matching becomes necessary because the wavelengths of the induced polarization and the electromagnetic wave it launches can be different. The polarization wavelength is controlled by the wavelengths of the incoming beams, whereas the launched wavelength is characteristic of the new frequency Phase matching is achieved either by crossing the incoming laser beams so the projection of the polarization wavelength 0003-2700/96/0368-600A/$12.00/0 © 1996 American Chemical Society
Quantitative treatment of Figure 1 recognizes that the atom's state is always a linear combination of the two states *F = ca*Pa + cb*Pb, where ¥ a and **P bepresent hydrogen's Is and 2p states and their amplitudes, c and cb, are changing in time. The four possible products, ca*ca, cb*cb, ca*cb, and cb*ca, characterize the state at any time and represent the populations and coherences. The products define the density matrix elements p , pbb, p b, and respectively. The coherences p h = c *c and o = c *c describe the intera b
i Dea
b
£a
mediate time-dependent states in which the electron clouds slosh to and fro in chrony with the light field Thev are the polarization that launches the lieht waves that all sn^ctrosmnists n -» rSi
i \ a v **'^
Aba^a-OW-'Tba
33) (4>
These equations show that the coherence is proportional to the ratio of the Rabi frequency, ii ba , and the detuning factor, Aba. The polarization can become quite large at resonance and is limited only by the environmentally induced dephasing rate, Tba, which controls the time a material can interact with the light field. .n real lamples, ,he polarizatton must include a summation over all possi-
Analytical Chemistry News & Features, October 1, 1996 601 A
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Figure 1 . Simulated electron probability distribution of hydrogen. Optically induced transition between the 1s and 2p states of hydrogen when there are no dephasing collisions to interrupt the interaction with the light.
ble transitions with the appropriate Rabi frequencies and detuning factors. Because we are concerned with nearly resonant transitions where the resonant transition dominates, we will continue to use the two-state description. However, even at large detunings, strongly absorbing states will contribute to a coherence, which is the origin of the "nonresonant" signal, We can now understand all nonllnear effects simply by extending the linear processes to cases of multiple light-matter interactions. Figure 2 shows energy-level and Liouville diagrams used to follow the evolution of the p^ coherence. The energylevel diagrams show which transitions are resonant with the different light fields, whereas die liouville diagrams show the field-induced evolution of the populations and coherences in the density matrix (9). It is important to remember that the arrows in these diagrams show only the direction of state evolution and do not indicate absorption or emission events. In the Liouville diagram, each interaction changes one of the two states in the subscript of a density matrix element. The initial populations and final coherences are highlighted by boxes, and the evolution between the two is shown for representative processes refraction for a firstorder (or linear) process; SHG for a second-order process (three-wave mixing); 602 A
fluorescence, Raman, and CARS for thirdorder processes (four-wave mixing); and hyper-Raman for a fifth-order process (siswave mixing). Equations analogous to Equations 2-4 for any process are formed from the product of the Rabi frequency/detuning factor ratios for each interaction step of the pathway from the initial state. For example, the CARS process results in a coherence between states d and a. Thus for the shortest pathway (corresponding to the lowest-order and most probable process), the pd coherence is formed sequentially by three interactionsfromthe p population and each interaction introduces the ratio of the Rabi freauency for the transition to the detunine factor for the coherence For example the first transition changes state a to state c so the Rabi frequency is A and the coherence created involves states c and a so the detuning factor is A . Thus
Pda
A A A U3
D3
^
W
C3
where we assume that higher order pathways involving coherences like pdc, pbc, and pcc are all zero. The pda coherence in Equation 5 is identical to one created by a single-photon resonance between states d and a except that it was created after three interactions, each involving a resonance. The Liouville diagram is a particularly powerful representational method. In addition to promoting a straightforward derivation of the appropriate pn, it shows that there may be several pathways to a given coherence. Thus there is one third-order (three-step) pathway for CARS, but there are six fifth-order pathways and even more for higher orders. At very high laser intensities, the higher order processes become significant, but they are treated in the same manner In addition this formalism shows that "conventional" optical processes such as absorption are very closely related to higher order effects Most analytical samples are mixtures, and die net polarization musttiiereforeinclude contributions from each set of transitions of each component in die mixture. The net polarization is determined by die summation over all possibilities
Analytical Chemistry News & Features, October 1, 1996
P=
2J
WXpjjMij + PJjMji- (6)
n - all components i,j = all transitions
where Nn, pj], and u^ are the number density, the final coherence, and die dipole moment associated with the i-j output transition of sample component n. Physically, the output coherence has an appearance similar totiioseof the charge oscillations shown in the center part of Figure 1, except that the oscillations involve states i and jj Because .Pis also proportional to the susceptibility y, we see that Y is proportional lo P-, and thus die susceptibility of a molecule can be calculated directly by determining the appropriate density matrix elements Note that y does sepend on the experimental conditions because it has factors sensitive to both the freauenciee of die incident electricfieldswhich affect A- and the environment (temperature or physical state) of trip
material which can affect the dephasmg r y. If the material's oscillating polarization is known through eitiier Equations 1 or 6, die oscillating output electric field Eout will launch at the samefrequencyand can be calculated using Maxwell's equations. It is then straightforward to calculate output light intensity, Iout, measured in a spectroscopic experiment using -
»
2
c|.E0Ut| 4ut ~~l— =
(7)
Some of the most important spectroscopic effects are understood only when light is quantified and described by photons (10). The intensity of light becomes proportional to the number of photons in a light beam but, remarkably, it does not go to zero if there is no light. The quantum mechanical uncertainty principle requires residual energy, even in a vacuum. Thus any intensity is actually proportional to (n + 1), where the extra factor rf 1 fepresents the zero-point fluctuations. These fluctuations have important consequences because they can participate in die evolution of a coherence such as those seen in Figure 2. In particular, they make fluorescence and Raman scattering possible. However, there are limits to their participation. The zero-point fields can have any frequency and directtonality supported by
the environment, and they can induce energy to be transferred to the photon fields, but no process is able to extract energy from the vacuum. In addition, processes requiring multiple interactions with vacuum fields are too weak to observe.
domly ordered molecules are added. Any signal generated by one molecule is canceled by another of opposite orientation.
Interfaces lack inversion symmetry, the cancellation is incomplete, and a net output dipole can result. These three-wave
First- and second-order effects %m effects. Absorption and refraction are single-photon processes and are controlled by x(1> in Equation 1. The frequency dependence of %m comes from the coherence described by Equation 2. The real part of Equation 2 describes refraction, and the imaginary part describes absorption. If (a>ba - c% ) » r ba , refraction dominates because the real part trr\nir» t r a n o i f i r v n 4-4-
t.
l
L
t\f
a rAarfir-iilat-
flftmnn , , , .
ZOO\ •
W
U
_
1 4-
Two-photon microscopy
nent to enhance other resonances of this Two-photon spectroscopy has formed the same component (24). Similarly, the basis for scanning microscopy in biological mode is selected by fixing a second resosystems (29). A mode-locked 100-fs laser is nance to a particular vibrational mode focused onto a sample with a microscope (25). The scan of the third resonance objective, and two-photon excited fluoresyields a snectrum with strongly enhanced cence at ~ twice the laser frequency is decontrihutions from the modes that are tected using the same objective and a coupled to the chosen vibrational mode. beamsplitter to send the higher frequency An example of this procedure is shown fluorescence to a monochromator and dein Figure 4. Selective enhancements of the tector. The system has the same submispecific vibronic are achieved by tuning the crometer spatial resolution as traditional second resonance to the vibrational states fluorescence microscopes but has much seen in the fluorescence spectrum. A simibetter depth resolution and freedom from lar approach can be used to narrow inhobleaching The depth resolution is a result mogeneously broaded spectra. One resoof the fast numerical apertures of the micronance fixed within the inhomogeneous disscope objectives which define a narrow tribution provides a selective enhancement depth of focus as well as the nonlinear infrom the laser so that spectral scans contensity dependence of two-ohoton fluorestain narrow features from the selected envitence, which ensures that the majority of ronments (26,27) the fluorescence comes from the laser'o focal volume. Hyper-Raman, Other techniques
weak incoherent Raman signals (22). There are two dominant applications of risve used terms 013. hi^ner order tnHn CARS: in fluorescent systems in which the % for spectroscopic analyses; an examstrong directionality allows discrimination ple is shown in Figure 2e (28). In the from incoherent nondirectional fluores- same way that the Liouville diagram in cence, and in hostile environments suchFigure 2d contained the fluorescence and Raman processes, the six-wave mixing as plasmas, explosions, and shock expendiagram in Figure 2e contains the twoments (23) in which complete coherent 4-
photon excited fluorescence process and hyper-Raman scattering. Both are usually used as incoherent processes, although the coherent processes are clear. The interactions associated with £2ac Cnd Q.ci are caused by lasers at those frequencies. There are many pathways from the initial population (aa) to the final output coherence (bd). Those that go through the pdd population give two-photon excited fluorescence and require interaction with a vacuum photon £2bd. Those that avoid the pdd population give hyper-Raman scattering and also require interaction with a vacuum photon Qbd. The oscillations associated with the pbd coherence create the output light signal that is detected. These six-wave mixing methods provide a rich opportunity to extract symmetry information about a sample because the output intensity depends on the polarization of each light beam relative to the symmetry axes of the molecules. The polarization characteristics of this method are complementary to Raman or IR spectroscopy and can provide detailed symmetry information about a molecule.
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6 0 6 A Analytical Chemistry News & Features, October 1, 1996
The photobleaching that hinders singlephoton excited fluorescence is minimized in the two-photon microscope because the photon energy is mich lower than that required to cause bleaching, and any energy absorptiocau thb two-photon energe is confined to the small focal volume. Although
two-photon excited fluorescence is usually very weak, the very high intensities within the focal region can actually saturate the fluorescence from the excited level. Future developments Most work in nonlinear spectroscopy anticipates commercial technologies. In order for nonlinear spectroscopy to make a meaningful contribution, it must offer measurement capabilities that cannot be achieved with other, simpler methods. Research has concentrated on either developing unique capabilities or applying the unique capabilities to research problems. Some of the most intriguing capabilities are the surface and interface selectivity characteristics of nonlinear mixing, such as three-wave mixing, that are based on the even terms in Equation 1. The Doppler-free and line-narrowing capabilities of four-wave mixing methods offer high-resolution capabilities. Multiple simultaneous resonances offer methods in which one resonance can enhance or suooress other resonances so that relationships between resonances can be established Very high signal levels are possible with picosecond lasers because the damage thresholds for a sample improve as the square root of the pulse width and short pulse widths can allow much higher laser intensities within the sample, resulting in an increase in signal level. The development of mode-locked diode-laser-pumped Nd lasers with OPOs could provide the reliable nonlinear excitation source that will allow the unique capabilities of nonlinear spectroscopy to expand out of the research laboratory We believe it is important to develop the suoDorting technologies that will make nonlinear methods convenient powerful and user friendlv as well as to exploit their unique capabilities across the breadth of the analvfrical sHpnrps
This work wss supported by the Analytical and Surface Science Program of the National Science Foundation under grant CHE-9500292.
(4) Wright, J. C. In Lasers in Chemical Analysis, 1st ed.; Hieftje, G. M.; Travis, J. C; Lytle, F. E., Ed;.; Humana Press: Clifton, NJ, 1981; pp. 77-90. (5) Wright, J. C. In Applications of Lasers to Chemical Problems, 1s, 1d.; Evans, T R., Ed.; John Wiley & Sons: New York, 1982; p. 35. (6) Henderson, G.; Rittenhouse, R C; Wright, J. C; Holmes, J. L.J. Chem. Educ: Soffwarre194,5, 2. (7) Henderson, G.Am.J.Phys. 1980,48, 60.. (8) Wright, J. C; Carlson, R. J.; Hurst, G. B.. Steehler, J. K,; Riebe, M. T; Price, B. B.; Nguyen, D. C; Lee, S. H. In.. Rev. Phys. Chem. .991,10,349. (9) Mukamel, S. In Principles of Nonlinear Optical Spectroscopy, 1st ed.; Oxford University Press: Nww York, 1995. (10) Loudon, R. In The Quantum Theory of Light, 2nd cd.. Clarendon Prrss: Oxford, 1985. (11) Corn, R. M.; Higgins, D. A. Chem. Rev. 1994, 94,107. (12) Higgins, D. A; Corn, R. M.J. Phys. Chem. 1993, 97, 489. (13) Lynch, M. L.; Barner, B. J.; Lantz, J. M; Corn, R. M.J. Chim. Phys. 1991, 88, 1271. (14) Huang, J. Y; Shen, Y. R Phys. RRv. 1194, 49,3973. (15) Brosnan, S. J.; Byer, R L. IEEE J. Quantum Electron. .979, QE-13,415. (16) Baumgartner, R. A; Byer, R L. IEEE J. Quantum Electron. 19797 QE-15,432. (17) Morris, M. D;; Wallan, D. J.; Ritz, G. P.. Haushalter.J. F.Anal. Chem. 1978,50, 1796. (18) Williams, S;; Green, D. S.; Sethuraman, S.; Zare, R. N.J. Am. Chem. Soc. 1992,114, 9122. (19) Wu, Z;; Tong, W. G. Anall.hem. 1993, 65,112. (20) Smith, A. P.; Astill, A. G. Appl. Phys. B 1994,58,459. (21) Sappey, A. D. Appl. Opt. 1994,33, ,346. (22) Tolles, W. M.; Nibler, J. W.; McDonald, J. R; Harvey, A. B. Appl. Spectrosc. 1977, 31, 253. (23) Moore, D. S.; Schmidt, S. C; Shaner, J. W. Phys. Rev. Lett. 1983,50,1819. (24) Steehler,J. K.; Wright, J. C... Chem. Phys. 1985,83, 3200. (25) Carlson, R. J.; Wright, J. C. Anal. Chem. 1991,63,1449. (26) Hurst, G. B.; Wright, J. C... Chem. Phys. 1991,95,1479. (27) Hurst, G. B;; Wright, J. C.J. Chem. Phys. 1992, 97,3940. (28) Ziegler, L. D.J. Raman Spectrosc. 1990, 21,769. (29) Denk, W.; Strickler, J. H.. Webb, W. W. Science 1990,248, ,3.
John Wright and his graduate student coReferences authors are working on developing vibra(1) Bloembergen, N. Science 1982,216, tionally enhanced four-wave mixing spec1057. (2) Levenson, M. D.; Kano, S. S. In Introduc- troscopy for performing selective ehemical tion to Nonlinear Spectroscopy, Academic measurements on complex samples. Address Press: New York, 1988. correspondence tt Wright aathe Depart(3) Shen, Y. R. In The Principles ofNonlinear ment of Chemistry, University of Wisconsin, Optics; Wiley Interscience: New York, 1101 University Ave., Madison, WI53706. 1984.
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