Aaron Rury Research Prospectus 21.6.2009 Introduction: The Helmhlotz equation, ( 2 +k 2 )u(r)=0 1, serves as the basis for much of optical physics. As a partial differential equation, the Helmholtz equation does not lend itself easily to analytical solutions. One of the ways, however, to produce analytical solutions is to impose a symmetry upon the system. Of the possible symmetries, one will be considered in this project, namely cylindrical symmetry. Symmetry is not the only tool used for producing the solutions to Helmholtz s equation. The paraxial approximation must also be invoked. This approximation pertains to the curvature of the field amplitude in the direction of overall propagation of the wave. In both the rectangular and cylindrical symmetries, we will assume that the overall direction of the wave is the z- direction. Then the field amplitude goes like u(r)~exp(-ikz) where k is the wave vector of the wave. When written explicitly in Cartesian coordinates, the Helmholtz equation becomes, 2 u/ x 2 + 2 u/ y 2 + 2 u/ z 2 -ik u/ z=0 The paraxial approximation states that 2 u/ z 2 << 2 u/ x 2, 2 u/ y 2, k u/ z Therefore, the term 2 u/ z 2 can be dropped from the Helmholtz equation and we are left with a somewhat simpler differential equation, the paraxial wave equation 2. Despite this simplification, there is another approximation to be invoked.
In a laser, there is a gain material that transfers pump energy into the desired laser light. Each component or part of the laser material can be thought of a point source of the light. Such point sources emit spherical EM waves. In the far field or Fresnel approximation, these point sources can be treated as producing Gaussian beams. This is very convenient because Gaussian field amplitudes are exact solutions to the paraxial Helmholtz equation 2. Once the far field approximation has been invoked, one can analytically solve the paraxial wave equation. In cylindrical coordinates, the solutions contain associated Laguerre polynomials as well as an exponential term that includes the variable φ. The field amplitude becomes, u mp (ρ,φ,z)~l pm (ρ 2 /2w 0 )exp(ρ 2 /2q*)exp(imφ) 2 with the necessary normalization terms left out for convenience. Such field amplitudes are known as Laguerre-Gauss (LG) beams. Intensity profiles of some LG beams are shown below. This field amplitude resembles very closely the functional form of the electronic wavefunction of hydrogen atom. The angular dependence of this wavefunction is associated with an orbital angular momentum (OAM) as opposed to the intrinsic angular momentum associated with the electron s spin. One can draw the parallel that the angular dependence of this LG beam is actually an OAM of light, or photons. The expressed purpose of this project is to determine the
Intensity Profiles of u mp (ρ,φ,z) influence, if any, of this form of angular momentum on the coupling of light to materials system in general and large, complex molecular systems in particular. Theoretical Development: OAM has been shown to couple to material systems in a variety of situations. Micro-sized beads of dielectric material have been shown to absorb LG beams and OAM via rotations induced in the bead 3. LG beams and OAM have also been used to make elaborate traps for cold and ultra cold atoms 4. Some spectroscopy has even been accomplished on such atomic systems using OAM. What is missing, however, from this work is a large-scale investigation of the role of photonic OAM plays in molecular interactions with light. There is some promise to the idea the OAM of light can influence molecular dynamics. In a recent theoretical study 5, researchers have shown that new transitions are possible based on particular couplings of nuclear motion to the LG beam. The overall wavefunction of the molecule, H 2 + in this case, can be written as
Ψ> = a> ν> χ> c.m.> where a>, ν>, χ>, c.m.> are the electronic, vibrational, rotational and center of mass wave functions respectively. When the transition matrix elements are calculated with the approximated Hamiltonian for electronic transitions, there are new selections rules for overlap of both the vibrational and rotational states on different electronic manifolds. The interaction with the LG and its OAM with the molecule allows for parity switching of the vibrational states depending on the coupling of the light to different orders of the nuclear normal mode vibrations q, or Μ ν ~ <ν f q n ν i > This transition matrix element says that the normal selection rules for vibrational state overlap during electronic transitions, ν = 0, ±2, ±4,, can be amended to include odd changes in the vibrational quantum number. Similarly, the selection rules for rotational state overlaps during such electronic transition no longer remain the same. Due to these changes in selection rules, it may be possible to create new types of initial conditions for molecules on excited state electronic manifolds leading to new dynamics. The remarkable simplicity of H 2+, however, brings this study under scrutiny. It is hard to say that the approximations made for such a simple system will lead to measureable changes in larger, more complex molecules. One might need to include higher order terms in the multipole expansion to completely couple the OAM of the LG beam to the sample. There may also need considerations taken into the reference frame used to describe the changes induced by the OAM. An atomic study of selection rules shows conservation of momentum via both internal and external motions coupled to the light field 6. An extension of this reasoning may help elucidate the complete effect of OAM in such systems.
Because of the new considerations necessary for such experiments, there does not seem to be a theoretical framework in which to describe how all of this will be present in the spectroscopic signal we measure. The production of such a framework is part of the scope of this project, but will be undertaken by a researcher from SRI International. It will be very important to work in tandem with this theorist in order to best use electromagnetic and quantum mechanical theory to understand what we are seeing in the proposed experiments. Experimental Design: The first goal in this project will be to produce pulsed LG beams from visible laser light. Such pulses have been produced by other labs, but not used in spectroscopic studies. To make LG beams, one must use holography. A hologram is calculated using the desired LG beam to create an interference pattern with a reference wave 7. In this case, a plane wave will serve as the reference wave. When the reference beam is diffracted off of the calculated hologram, the output is the desired LG beam. Their charge number denotes the holograms used for making LG beams. That is, a charge-l hologram produces, for a particular reference beam, a LG beam with m=1. A charge-1 hologram is shown below. Because of the form of the transfer function of this hologram, there is a fork present in the hologram. It is this fork that leads to the production of OAM in the incident beam. These holograms can be made in a variety of ways. One way in particular makes it easier to measure the effect of LG beams on the systems of interest. Because holography is so widespread in research, a large number of devices have been designed to produce holographs. Computer generated holographs are especially powerful tools in research. We plan to use computer-generated holograms from a liquid
Charge-1 Holograph with Distinctive Fork 7 crystal spatial light modulator (SLM) to produce the desired charge-n holograms. The SLM also allows for the production of any charge-n holograms up to a large n given the malleability of the liquid crystals. Since the final goal of this project is ultrafast spectroscopy on molecular systems, one must account for the group velocity dispersion introduced in the LG production process. As a dispersive optic, the hologram will diffract different spectral components of the ultrafast pulse at different angles. This introduces a temporal stretching of the pulse and may bring it out of the ultrafast regime. In order for the temporal profile of the pulse to be kept intact, the spectral components must be recombined properly. To properly recombine the pulse temporally, a 4 focal length (4-f) recombination setup will be used 8. Such a setup takes the dispersed pulse and focuses its spectral components onto the Fourier plane. A mirror will be placed in there and as the pulse passes back through the device, it is recombined using a different part of the hologram. The setup is seen pictorially below with the solid and dashed lines as the input and output beams respectively.
Computer-Generated Hologram (CGH) in 4-f setup 8 After we are able to make the desired ultrafast LG beams, the next goal will be to undertake spectroscopic studies. We will be using ultrafast, or femtosecond (fs), and picosecond (ps) pump-probe spectroscopies. This will allow the comparison of signal we measure with signal from the same systems in the literature for noticeable differences. The two techniques we are proposing to utilize to produce signal are the simple pump-probe spectroscopic geometry and transient-grating pump-probe spectroscopy. Each method will give us information on population dynamics during non-reactive photophysical processes in the pertinent molecular systems 9,10. Despite the similarities, there are distinctions between each technique. The simple pump-probe spectroscopic geometry is shown below. A pump pulse, in this case a LG beam, produces an excited state in the sample. This excited state is probed at some later time by another pulse that can be either a LG or normal Hermite-Gauss beam, depending on the demands of the experiment.
Simple Pump-Probe Spectroscopy As the excited state population moves along a pathway, the probe can see how much has reverted back to the initial ground state of the molecule and how much has undergone a nonreactive photophysical change, namely photo-isomerization 11,12,13. The signal produced in such an experiment is a convolution of the third order polarization induced by the pump, P (3) (t) 10, and a local oscillator field in the form of the probe itself. This technique gives us information on the absorptive nature of the polarization induced by the pump. The transient grating pump-probe spectroscopic geometry also produces population dynamics of pertinent molecular manifolds, but does so in a different way. Co- or counterpropagating pump beams, again LG beams, interfere in the sample itself to produce a hologram of excited state sample that the probe is then scattered off from at different time delays. Such geometry is pictured below. The signal is measured in the so-called background free manner, meaning that a local oscillator is not necessary to amplify the signal, as is the case in the simple pump-probe geometry 9,10. This background free measurement also means that phase information in the induced polarization is lost. The signal measured in such a set-up is the modulus squared of P (3) (t).
Transient-Grating Pump-Probe Spectroscopy To measure the signal due to the OAM of the LG beam, phase sensitive detection using a lock-in amplifier is necessary. By modulating the LG pump beams OAM content, we can measure changes due to the OAM induced in the system. The SLM producing the computer generated holograms refreshes at a high enough rate to work with a lock-in amplifier in order to calculate the change in absorbance caused by interactions with the OAM of the LG pump pulses. By using this detection scheme in spectroscopy techniques described, we hope to be able to produce the first evidence of an effect of the orbital angular momentum of photons in large molecular systems. References: 1. Jackson, J.D. Classical Electrodynamics; Wiley and Sons, Inc., Hoboken, NJ: 1999 2. Seigman, A. E. Lasers; University Science Books, Herndon, VA: 1986 3. He, H et al. Phys. Rev. Lett., 75, 826-829 (1995) 4. Tabosa, JWR and Petrov, DV Phys. Rev. Lett., 83, 4967-4970 (1999)
5. Alexandrescu, A et al. Phys. Rev. Lett., 96, 243001 (2006) 6. van Enk, SJ Quantum Opt., 6 (1994) 445-457 7. Heckenberg, N. R. et al. Opt. Quantum Electron. 24 (1992) S951-S962 8. Zeylikovich, I. et al. Opt. Lett. 2007, 32, 2025-2027 9. Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press, New York: 1995 10. Xu, Q and Fleming, G J. Phys. Chem. A, Vol. 105, No. 45, 2001 11. Vogt, G et al J. Chem. Phys., 125, 044513, (2006) 12. Prokhorenko, V. I. et al. Science 313, 1257 (2006) 13. Nuernberger et al. J. Chem. Phys., 125, 044512, (2006)