The unique structure of a protein is encoded in its characteristic sequence of amino acids. Protein folding is the process by which this linear sequence of amino acids collapses into a unique three dimensional structure. The mechanics of this process is one of the most challenging problems in Bimolecular Science, and still remains a mystery for the most part. In order to better understand the transition between the native and denatured states of proteins in a time resolved fashion, several new optical techniques have been developed. Time scales for such conformation changes to take place may range from milliseconds to nanoseconds, which makes this a challenging undertaking. However, optical techniques show much promise for these sorts of noninvasive time resolved measurements.

One newly developed technique involves laser induced conformation changes. Optical techniques have great potential because of the many different ways that laser pulses can be employed to trigger protein folding. Photo-induced electron transfer techniques have been used in pump probe configuration to measure conformation induced absorption changes in proteins Tobjorn Pascher et al (1). Laser methods such as these allow measurement of time resolved absorption spectra at the nanosecond time scale which have been previously unattainable.

Other methods for probing the time resolved protein folding problem include phosphorescence spectroscopy (5), and circular dichroism measurements (6). However, nonlinear optics techniques such as four-wave mixing (FWM) show promise in providing precise time resolved absorption measurements as well as previously unmeasured index changes due to conformation changes. Four-wave mixing is a technique whereby a volume grating (a thick diffraction grating) is written in some nonlinear optical material, such as a protein solution, by two intense laser beams called pump beams. By passing a third beam of smaller intensity through the volume grating at a precise angle in order to satisfy the phase matching condition, it may be diffracted, producing a fourth beam. Measurement of this diffracted beam yields very precise information about index and absorption changes induced in the nonlinear optical medium. If the pump beams can optically trigger protein folding, thereby inducing a volume grating due to the denaturing of proteins, this technique may be successfully used to measure absorption and index changes in proteins on the order of ~ 10^-4 in a time resolved manner.

In this paper we will discuss some preliminary measurements using four-wave mixing techniques to measure index and absorption changes of an aqueous coffee solution. We have preformed our degenerate four-wave mixing experiments using an Nd:YAG pulsed laser at 532 nm. The ultimate goal of these studies will be to replace the aqueous nonlinear reference material with a protein solution on which a similar measurement will be preformed.

2. Sample preparation

The coffee solution with which four-wave mixing was successfully observed was a mixture of 3 cups of water and 2 table-spoons of Foldgers instant coffee. The sample was prepared by bringing the water to a boil, and mixing the instant coffee with the water. After the coffee is thoroughly dissolved, the coffee solution was passed through two 0.7 µm qualitative filters. This reduces the scatter produced by large particles. The solution was then transferred to a 5 mm glass optical cell. The concentration of the sample is more precisely identified by its absorption coefficient of ~0.56 cm^-1 at 532 nm after preparation.

3. Experimental arrangement

All experiments were performed with a Q-switched Nd:YAG laser at a 20 Hz repetition rate. Harmonically generated 532-nm laser light was used for both pumps and probe beams in these four wave mixing experiments. All intensity and signal measurements were taken with Thorlabs Det 100 broad area silicon photo detectors having a rise time of 20-ns. All signal and intensity measurements were averaged over 100 repetitions on a TDS 540 Textronics digital oscilloscope. A 5 mm path length optical cell was used to contain the aqueous coffee solution.

The experimental apparatus for degenerate four wave mixing is illustrated in the figure 1. An unconventional degenerate four-wave mixing (DFWM) technique was implemented whereby an image mask composed of three circular apertures of 3 mm diameter were used to separate the single laser beam of ~8 mm diameter into two pump beams and a probe (2). The lower aperture on the first image mask serves as the probe beam, while the upper two apertures serve as the pump beams. Refer to figure 9 for a schematic. The image mask is quite simple for degenerate four wave mixing, because the Bragg condition, 2L sinq _B = ml reduces to q _B = q _c the first order of diffraction, where represents the period (fringe spacing) of the diffraction grating. Here _c represents the angle of intersection between the pump beams. The image is then passed through a 10-cm lens. Due to the transforming properties of the lens, all of the Fourier transformed aperture functions converge to the same location in the focal plane of the lens, giving rise to a volume grating and diffraction signal. This has proven itself to be a very practical technique for attaining ultra-high light intensities in four wave-mixing experiments.

The particular four-wave mixing geometry chosen is a unconventional one (2) which is particularly powerful because it allows the experimenter to achieve four-wave mixing with very high intensities. Intensity ranges between zero to one gigawatt/cm^2 can be obtained with ease. This is done by using a lens to transform a field pattern created with a mask. The laser beam passes through the mask, creating a known intensity distribution in the focal plane of the lens. The particular mask used to create the field pattern consisted of two circular apertures of diameter a, spaced a distance b apart as shown in figure 2.

Recall from Fourier optics that the resulting image produced by some known aperture function is simply given by equation 1 (1).

In the above equation, A(x,y) represents the electric field as a function of positions x and y in the object plane of the lens. Note that x and y represent positions in the object plane, while x_f and y_f represent the positions in the focal plane of the lens, approximately a distance f from the lens. Naturally f represents the focal length of the lens, and l represents the wavelength of light.

Next it is necessary to apply equation 1 to the problem at hand. To a first approximation, one may assume that the apertures are illuminated uniformly with collimated light. Thus the problem reduces to the Fourier transform of two circular aperture functions, as shown in figure 2.

This may alternately be represented as the convolution of two delta functions spaced a distance B with a single aperture of diameter A. This convolution is displayed pictorially by figure 3, and expressed analytically by equation 2.

We are interested in the Fourier transform of the aperture function A(x,y). Exploiting the convolution theorem we have equation 3.

Notice that the transformation of A(x,y) using equation 1 may be written as equation 4.

Equation 4 now represents the electric field in the focal plane of a lens due to the aperture function A(x,y). For the above result it is assumed radiation incident on the mask is normalized to an amplitude of unity. Thus the normalized intensity at the focal plane of the lens is proportional to the modulus square of the result in equation 4. The intensity can be written as:

Plotting the intensity of the electric field for the experimental setup used can be very helpful in understanding the outcome of an experiment. The following graph shown in figure 4 shows the normalized intensity as a function of position on the x-axis in the focal plane of the lens.

Notice the surprising result that can only be seen when one plots the spatial variation of intensity. The grating from which one will see diffraction only consists of three fringes.

The index gratings in the coffee solution is thermal in nature (2). Knowing this it is useful to correlate the intensity pattern shown in figure 4 with an index change in the water. Figure 5 is a graph of the index of refraction of water Vs temperature for 589.32 nm light (7).

Since we are only interested in the index changes that occur at the focus of the lens, we will limit our investigation of index change to the volume of water about the beam waist. Assuming that this volume of water has a diameter of 20 microns (approximately the waist size) and a depth of the confocal parameter of the beam, approximately 2 mm, our volume of interaction is approximately 8E-7 cm². The change in index corresponding to various absorption coefficients can now be calculated in a straightforward manner. Figure 5 is a graph of index change Vs pulse energy for various absorptivities.

Four wave mixing is a nonlinear optical technique whereby two intense mutually coherent laser beams, called pump beams, are interfered inside of some nonlinear optical material. This interference pattern induces a thick optical hologram in the form of a volume grating inside the nonlinear optical material. By passing a third beam, called the probe, through the hologram, it can effectively be read by measuring the diffracted light. The fraction of the incident probe beam that is diffracted tells us how the optical properties of the material have been changed by the pump beams. This ratio of diffracted light I_d/I_p = h _d is referred to as the diffraction efficiency of the volume grating, which can be related to the induced index and absorption changes in the grating.

In this experiment degenerate four wave mixing was performed using the beam geometry depicted in the above figure. By virtue of the fact that the two pump beams and the probe beams are in separate planes, they are easily distinguished at the output. In this experiment the interaction length of the pump beams, and thus the effective length the grating is limited by the the path length of the optical cell. A 5 mm path length optical cell was used throughout the experiments. The fringe spacing of the volume grating produced inside the sample is given by L = l ₀/2sin( q _c/2), where q _c denotes the crossing angle between the pump beams. The fringe spacing is 14 µm for the given arrangement.

The diffraction process can occur in two different regimes based on the thickness of the grating, L.Classification of the volume grating may be made based on the characteristic interaction length, L₀. The characteristic interaction length, L_0, is defined by L₀ = n L ²/ l ₀, where n is the index of refraction of the sample. Raman-Nath diffraction (the case of the thin grating) is defined by an interaction length which is much smaller than the characteristic length, L ´ L₀. Raman-Nath diffraction results in orders of diffraction described by the grating equation, sin( q _m) = m l ₀/n , where the amplitude of the m-th order of diffraction is given by the m-th order Bessel function. In the Bragg regime of diffraction of a volume grating (where diffraction grating is thick)most of the incident beam is diffracted into the first order. The Bragg regime is generally defined by the case where L ª L₀. For this particular configuration the ratio L/L₀ ~ 10. This means that our volume grating technically falls in the intermediate case of diffraction which is not as easily classified. However, a detailed study preformed by Klein and Cook (3) demonstrated that in the case where L = L₀ the maximum attainable diffraction in the first order is 90%, showing that the higher orders have little contribution. For all practical purposes, this configuration may be treated as Bragg diffraction with little deviation from the exact diffraction relations.

The phase matching direction for the first order of diffraction can be identified by the grating vector of the volume grating. The grating vector is defined as k_g = 2 p /L , where k_g = k₁ -k₂. The phase matching condition of the diffraction signal is expressed as k₄ = k_g + k_3. Substituting, we have that the phase matching direction is k₄ = k₃ + k₁ - k₂. However, notice that only the grating vector formed by the two pump beams k₁ and k₂ were considered. Another grating vector may be formed by k₂ and k₃ which will diffract light in the phase matching direction. As a result it is very important that the ratio between the pump intensities and the probe beam be very small, or that the polarization of the probe be orthogonal to that of the pump beams when using this geometry. The latter solution was implemented during this experiment to entirely extinguish the contribution produced by the grating vector k₃ - k₂. Having the probe polarization of the beam orthogonal to the pump beams also allows us to distinguish diffracted probe radiation from scattered pump radiation despite very large scattered light intensities. (Note, the particular half wave plate used to rotate the plane of polarization of the probe introduced an attenuation factor of 0.2.)

Figure 4 shows the k-vector diagram for the probe, pump beams, and the diffracted signal. This geometry was achieved with the use of an image mask having three circular apertures, each of diameter 3-mm. By changing the distance between the centers of the holes, and the focal length of the lens, one can change the crossing angle between the pump beams quite precisely. For this experiment the center of each aperture was placed 3.81 mm apart forming a right triangle. A 10-cm lens was used to focus the three beams inside the sample, having an angle of intersection of 38 mrad. The waist of the focus as measured in air was found to be approximately 25 µm.

The output diffracted signal was measured in the phase matching direction with a Si photo-diode positioned in the phase matching direction. This geometry of DFWM is designed such that the diffracted signal is spatially separated at the output. The following graph is a plot of signal in the phase matching direction, and diffraction efficiency Vs intensity.

Notice that the diffraction efficiency follows a quadratic curve fit, as expected. This is indicative of index and or absorption changes which are proportional to the pump intensity, since diffraction efficiency can be expressed as:

The only definitive conclusions that can be made without another complimentary experiment is that the index and or absorption changes induced in the coffee solution are proportional to intensity.

Through this study on a reference material we have established an experimental setup which is consistent and easy to align for the observation of degenerate four-wave mixing. Large diffraction efficiencies of ~10% have been achieved, which can yield extremely precise measurement of optical changes induced in a nonlinear optical material. We plan to perform a similar study using a protein solution as our nonlinear optical material of study. By using a non- degenerate CW four-wave mixing configuration, one in which the probe beam is of a different wavelength, and continuous in beam output, we will be capable of measuring the conformation induced absorption changes of proteins. A single intense 10-ns pulse from the pump beams will excite the proteins, causing a conformation induced grating, allowing us to see minute changes in index and absorption with the CW probe beam as the protein denatures. Before this study can be continued, an appropriate protein must be identified and manufactured.

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