For a two-dimensional deposition, the standing wave is formed by crossing two one-dimensional standing waves at 90 across the substrate. With this higher dimensionality, the nodes of the two-dimensional standing wave become more complex, depending on the polarizations of the constituent standing waves and also in some cases on their relative tem- poral phase. There are still zero points in the light intensity at locations on the surface that are an integral num ber of half wavelengths from each standing wave mirror, but examina- tion of an expression for the net electric field amplitude shows that additional nodal patterns can exist, and these can shift around as the temporal phase difference between the standing waves varies One approach to eliminating unwanted variations in the nodal pattern due to temporal phase variations is to generate the two orthogonal standing waves in an actively-stabilized optical cavity. Another approach, which is useful when sta- bility of the nodal topography is desired but absolute position is unimportant, makes use of three laser beams intersecting at 1200. For the present work we have made use of the fact that one can choose a polarization scheme in which the tem- poral phase does not affect the nodal pattern or position. If Section A-A one chooses orthogonal linearly-polarized standing waves, with one polarized perpendicular to, and the other parallel to the plane of the substrate, the resulting intensity distribution 到八人人八 is given by Section b-B I(x,y)=41o(sin'kx+sin'ky) where Io is the intensity of a single incident wave Distance(nm) k=2 /x is the wave vector of the light. and the x- and y-axes are defined to lie in the plane of the substrate. The formed by laser-focused atomic deposition in a two-dimensional standin ensity given by Eq (1)forms a pattern of nodes and peaks wave. The image covers a 4umx4um region of the sample The features on a A/2XA/2 square lattice with the peaks separated by are on a square lattice with spacing 212.78 nm, which is determined by the While the intensity of Eq (1)does not suffer from tem- imensional standing waves oriented at 45 and 135 to the figure. Also shown are two line scans, labeled A-A and B-B, whose locations are poral phase disturbances, it has potential drawbacks that indicated on the AFM image. The vertical scale for the line scans was could in principle cause problems with a deposition experi- determined by the AFM calibration, and the offset was estimated by requir- ment, though these appear to be less severe in practice. First, ing that the integral under the surface equal the total amount of material It might seem that since the intensity does not have cylindri- deposited, obtained from flux measurements. symmetry around the potential minima, the resulting de- posited spots would not be round. However, the intensity is, laser-atom interaction. Chromium atoms entering the in fact, surprisingly symmetric in the regions near the standing-wave field are in their ground state, but are distrib- minima. This can be seen mathematically by converting x uted among a number of degenerate magnetic sublevels. The and y to the polar coordinates (r, e)and noting that strength of the laser-atom interaction for each magnetic sub- I(r, 0)=4l0k-r-(i. e, the 8-dependence drops out) for level varies for different polarizations of the laser, due to kr< 1. Exactly how much effect the non-symmetric regions differences in the Clebsch-Gordan coefficients for the vari- of the potential(where kr is not much less than 1)have on ous transitions between the different magnetic sublevels. I he deposited pattern relative to the symmetric regions is The result is a potentially wide variation in the force on the difficult to predict without a ray-tracing calculation. Nev- atom as a function of space, and also time if the temporal ertheless, the experimental results indicate that there is little phase is not stabilized Despite these potential problems, it appears that a suffi- Second, it must be noted that while the intensity in this ciently symmetric and stable potential exists for a well- configuration does not depend on the relative temporal defined pattern of features to be created. Figure 2 shows an phase, the polarization of the electromagnetic field is com- atomic force microscope(AFM) image of a 4 umX4 um plicated. In fact, the local polarization varies dramatically as region of the two-dimensional chromium pattern deposited a function of x and y over the scale of a wavelength, and on a silicon substrate at room temperature. For this depo exactly what form this variation takes depends on the relative tion, the laser beams for the two dimensions had 1/e- diam- temporal phase. The local polarization of the field can be eters of 0. 13+0.02 mm and each contained a single-beam very important because it can determine the strength of the traveling-wave power of 12+1 mw.The total deposition Appl. Phys. Lett., VoL. 67, No 10, 4 September 1995 Gupta et al. 1379 Downloaded-v14-may-2008to7222.29.123.220.-redIstributionsubjecttoaip-licenseoncopyright;seehttp:/lapl.aiporglapl/copyrightjspFor a two-dimensional deposition, the standing wave is formed by crossing two one-dimensional standing waves at 90° across the substrate. With this higher dimensionality, the nodes of the two-dimensional standing wave become more complex, depending on the polarizations of the constituent standing waves and also in some cases on their relative tem￾poral phase. There are still zero points in the light intensity at locations on the surface that are an integral number of half￾wavelengths from each standing wave mirror, but examina￾tion of an expression for the net electric field amplitude7 shows that additional nodal patterns can exist, and these can shift around as the temporal phase difference between the standing waves varies. One approach to eliminating unwanted variations in the nodal pattern due to temporal phase variations is to generate the two orthogonal standing waves in an actively-stabilized optical cavity.8 Another approach,9 which is useful when sta￾bility of the nodal topography is desired but absolute position is unimportant, makes use of three laser beams intersecting at 120°. For the present work we have made use of the fact that one can choose a polarization scheme in which the tem￾poral phase does not affect the nodal pattern or position. If one chooses orthogonal linearly-polarized standing waves, with one polarized perpendicular to, and the other parallel to, the plane of the substrate, the resulting intensity distribution is given by I~x,y !54I0~sin2kx1sin2ky ! ~1! where I0 is the intensity of a single incident wave, k52p/l is the wave vector of the light, and the x- and y-axes are defined to lie in the plane of the substrate. The intensity given by Eq. ~1! forms a pattern of nodes and peaks on a l/23l/2 square lattice with the peaks separated by saddle regions along xˆ and yˆ at half the intensity. While the intensity of Eq. ~1! does not suffer from tem￾poral phase disturbances, it has potential drawbacks that could in principle cause problems with a deposition experi￾ment, though these appear to be less severe in practice. First, it might seem that since the intensity does not have cylindri￾cal symmetry around the potential minima, the resulting de￾posited spots would not be round. However, the intensity is, in fact, surprisingly symmetric in the regions near the minima. This can be seen mathematically by converting x and y to the polar coordinates (r,u) and noting that I(r,u)'4I0k2r2 ~i.e., the u-dependence drops out! for kr!1. Exactly how much effect the non-symmetric regions of the potential ~where kr is not much less than 1! have on the deposited pattern relative to the symmetric regions is difficult to predict without a ray-tracing calculation.10 Nev￾ertheless, the experimental results indicate that there is little effect. Second, it must be noted that while the intensity in this configuration does not depend on the relative temporal phase, the polarization of the electromagnetic field is com￾plicated. In fact, the local polarization varies dramatically as a function of x and y over the scale of a wavelength, and exactly what form this variation takes depends on the relative temporal phase. The local polarization of the field can be very important because it can determine the strength of the laser-atom interaction. Chromium atoms entering the standing-wave field are in their ground state, but are distrib￾uted among a number of degenerate magnetic sublevels. The strength of the laser-atom interaction for each magnetic sub￾level varies for different polarizations of the laser, due to differences in the Clebsch-Gordan coefficients for the vari￾ous transitions between the different magnetic sublevels.11 The result is a potentially wide variation in the force on the atom as a function of space, and also time if the temporal phase is not stabilized. Despite these potential problems, it appears that a suffi- ciently symmetric and stable potential exists for a well￾defined pattern of features to be created. Figure 2 shows an atomic force microscope ~AFM! image of a 4 mm34 mm region of the two-dimensional chromium pattern deposited on a silicon substrate at room temperature. For this deposi￾tion, the laser beams for the two dimensions had 1/e2 diam￾eters of 0.1360.02 mm and each contained a single-beam traveling-wave power of 1261 mW.12 The total deposition FIG. 2. Atomic force microscope ~AFM! image of chromium features formed by laser-focused atomic deposition in a two-dimensional standing wave. The image covers a 4mm34mm region of the sample. The features are on a square lattice with spacing 212.78 nm, which is determined by the laser wavelength. The standing wave is formed by superimposing two one￾dimensional standing waves oriented at 45° and 135° to the figure. Also shown are two line scans, labeled A–A8 and B–B8, whose locations are indicated on the AFM image. The vertical scale for the line scans was determined by the AFM calibration, and the offset was estimated by requir￾ing that the integral under the surface equal the total amount of material deposited, obtained from flux measurements. Appl. Phys. Lett., Vol. 67, No. 10, 4 September 1995 Gupta et al. 1379 Downloaded¬14¬May¬2008¬to¬222.29.123.220.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://apl.aip.org/apl/copyright.jsp