Evaluation of Performance of the Toronto Ultra-Cold Atoms Laboratory s Current Axial Imaging System

Page 1 5/7/2007 Evaluation of Performance of the Toronto Ultra-Cold Atoms Laboratory s Current Axial Imaging System Vincent Kan May 7, 2007 University of Toronto Department of Physics Supervisor: Prof. Joseph H. Thywissen

Page 2 5/7/2007 0. Abstract: The current axial imaging system for the Toronto Ultracold Atoms laboratory was studied using the ZEMAX ray-tracing software in order to assess its performance on-axis, off-axis, and tolerance in the placement of the focal plane. Its performance was compared with identical setups with different lenses from the current AC254-075-B. It was found that the GPX25-80 had the best overall performance in terms of on-axis and off-axis performance. 1. Introduction: 1.1 Theoretical Introduction: 1.1.1 Lens Terminology Figure 1: Diagram of lens terminology 1 The operation of a lens depends on the interaction of light rays with its surfaces, as well as a set of imaginary points and surfaces defined by the lens s shape. Optical axis For a system without decentered optical components, the optical axis is the line along which a beam from the center of the object field propagates through the system. A point lying on the optical axis is said to be an on-axis object. All other points on the object to be imaged are called off-axis objects. Principal points/plane The front principal plane is located where the rays coming from the front of the lens would meet the continuing rays exiting the back of the lens if refraction for each ray occurred at a single point. The back principal plane is defined similarily for rays coming from the back of the lens. In reality, due to spherical aberrations, this plane is really an aspheric surface, however close to the optical axis, it approaches a plane. The front principal point is the point on the principal plane through which the ray passing through the optical axis (the paraxial ray) crosses. The location of the principal point/plane is important for determining the location of the focal length using geometric ray tracing. Focal Length The focal point of the lens is where rays coming from infinity converge to a point behind the lens. For an ideal lens with no aberrations, the plane at which rays from the object form the sharpest image is called the image or focal plane, and it intersects the focal point at the optical axis. In the definition used by the ray tracing program and analysis, the image plane is defined as the plane perpendicular to the optical axis located at the on-axis focal point. In the present study, only lenses that form real images are considered.

Page 3 5/7/2007 The effective focal length (EFL) is measured from the back principal point to the focal point. In practice, it is easier to measure the back focal length, which is the distance from the back surface of the lens to the focal point. Object plane Figure 2: Illustration of formation of real image at the focal plane from a 2-D object located on the object plane 2 The object plane is the plane at which the 2-D object to be imaged is located. Rays depart this plane and are focused by the imaging system at the focal plane. Lens aperture In some scenarios, part of the lens is obscured in order to limit the distribution of rays entering the lens. This may be done to eliminate rays skirting the edge of the lens, which are more prone to off-axis aberrations. The radius of the aperture of the lens is defined as the radius of lens from the optical axis to the where the marginal ray passes through the lens. F-number and numerical aperture The f-number is defined as the ratio between the focal length and the radius of the aperture of the lens. It is written in the form of f/#, where # is the ratio of the aperture size to the focal length. For example, if the aperture is 11.7 mm in radius and the EFL is 69.9 mm, the f-number would be f/5.97. Figure 3: Diagram of calculation of numerical aperture. Φ is the diameter of the aperture. A similar definition is the numerical aperture, which is the sine of the half-angle made by the marginal 1 rays leaving the lens. For small numerical apertures, NA ~. 2 f # 1.1.2 Aberrations Aberrations are factors that decrease the performance of an optical system due to moving the point to be imaged off-axis. One of the main criteria for measuring the performance of a lens is its off-axis performance, specifically, how quickly the performance degrades as points in the object field are moved away from the optical axis. The main types of aberration considered in this study are spherical aberration and astigmatism.

Page 4 5/7/2007 Spherical aberrations Figure 4: Perfect lens without spherical aberration 3 The spherical curvature of the front and back surfaces of lenses introduce aberrations that prevent off-axis rays from focusing at the focal point. This arises from Snell s law: n = 1 sin θ 1 n2 sin θ 2 Where n 1 and n 2 are the refractive indices of the two media that bound the surface, and θ 1 and θ 2 the entering and exiting angle of the ray with respect to the normal perpendicular to the surface at the point where the ray enters. For small angles, Snell s law is simply n 1θ 1 = n2θ 2. This allows small approaching angles (i.e., on-axis and near-on-axis points in the object field) to focus at the focal point. The approximation above is due to the Taylor expansion of sine about θ =0. 3 5 7 θ θ θ sin θ = θ + +.... 3! 5! 7! For off-axis rays, the third-order term in the series expansion causes a shift in the focal length. Figure 5: Lens with spherical aberration 4 This may cause off-axis rays to focus closer than the focal length (positive spherical aberration) or farther (negative). Astigmatism Objects in focus have rays emanating from the same point on the object plane focusing at the same point in the focal plane (cf. Figure 2).

Page 5 5/7/2007 Figure 6: Lens with astigmatism, with separate tangential and sagittal focal lines 5 With off-axis points, again due to the 3 rd order aberration term in the sine expansion, rays traveling in the plane that includes the optical axis (the tangential plane) focus at a different location than rays in the perpendicular (or sagittal) plane. The effect of this is that a point in the object plane becomes a defocused spot, elongated in the direction (sagittal or tangential) that is farthest from the focus. Correction of aberrations There are several ways to correct 3 rd order aberrations. Aspheric lenses can be used, which are shaped so that the lens surface more accurately reflects the sine term in Snell s law to at least the 3 rd order expansion. However, this involves an expensive manufacturing process and was not considered in the current study. A second lens may be added that corrects aberrations. Figure 7: Contrast in spherical aberration with a singlet and doublet lens (Melles-Griot s 01 LPX 023 and 01 LAO 014 respectively) 6 A biconvex lens of low index of refraction is attached to a meniscus glass of higher index of refraction, in what is called a doublet lens. The aberrations caused by the refraction at each len interface cancel out, thus reducing the overall level of aberration present if only two plano-convex lenses were used. The doublet lenses used also canceled out chromatic aberration, that is, the change in focal length with different wavelengths of light, hence the name achromat for the type of lens used. A third way of correcting aberration is by using a lens whose index of refraction changes within the glass.

Page 6 5/7/2007 Figure 8: Effect of a gradient indexed lens. The gradient shading in the Gradium lens reflects a decrease in index of refraction along the optical axis 7 Within these lenses, the index of refraction varies according to depth along the optical axis, so that offaxis rays are not refracted to the same extent as on-axis rays, correcting positive spherical aberration. The lenses studied in the current analysis were Gradium lenses manufactured by LightPath Corporation. 1.1.3 Diffraction and its effects on performance Even without the presence of aberrations, a lens s performance is fundamentally limited by effects of diffraction. Diffraction results from the diverging of waves passing through an aperture Figure 9: Rays diffracting from a narrow aperture 8 When rays exit from an aperture, each point on the aperture can be treated as a point source of circular waves. These waves interfere with each other creating a diffraction pattern. An object point appears on the image plane as an Airy disc, the solution of the differential equation for Fraunhofer (far-field) diffraction. Figure 10: Airy disc fringe pattern The intensity of Fraunhofer diffraction is characterized by a 1 st order Bessel function of the form: π d 2λ J1( sin θ ) 2 ( ) 0 ( λ, as described by the Airy disc as illustrated above. Here λ is the I θ = I ) π d sin θ wavelength of light and d the diameter of the circular aperture of the lens.

Page 7 5/7/2007 This equation has its first zero of intensity at an angular separation of sin θ = 1. 22. Since d 1 sin θ, the diameter of the spot is approximately d = 2.44λ f /#. 2 f # When two Airy disc images of two points in the object plane are brought close enough so that the peak of one corresponds to the first zero of the other, then the two points are said to satisfy the Rayleigh criterion: λ Figure 11: Illustration of Rayleigh criterion 9 When this happens in a real optical system, the resolution is considered to be diffraction-limited as opposed to limited by aberrations. 1.1.4 Resolution The principal criterion used for evaluating the various imaging systems was resolution as quantified by the modulation transfer function (MTF). MTF is a measure of how well contrast is preserved between the object and the image. Figure 12: Measuring MTF with an optical system Figure 13: Above: Input signal and its MTF at various spatial frequencies. Below: Output signal and its MTF at the same spatial frequencies. Note how the MTF decreases from 1.0 for the low spatial frequency pattern to 0.1 for the highest spatial frequency pattern. In the above diagram a square wave signal is input

Page 8 5/7/2007 instead of a sinusoidal one but the principle is similar since the square wave is a sum of Fourier sinusoidal components, thus through Fourier analysis the sinusoidal MTF from the square wave response can be recovered and vice-versa 10 To make an MTF measurement, a sinusoidal pattern of light and dark areas is used as the object. MTF is more precisely defined as the ratio of the difference between the high and low light intensities and the sum of the high and low intensities, of the outgoing signal: MTF( visibility) = I I max max + I I min min This requires that the input maximum and minimum intensities are normalized to 1 and 0 respectively. The MTF is plotted as a function of spatial frequency the number of fringes of light and dark (or cycles) per millimeter. At higher spatial frequencies the intensity of light input into the imaging system varies significantly within a small distance and in general it is therefore harder to distinguish contrast. Imaging systems that have high resolving power are able to maintain a high MTF even at high spatial frequencies, these correspond to higher resolutions. The measure for MTF-based resolution used in this evaluation is the spatial frequency at which MTF falls to 50%. Diffraction-limited MTF If the optical system is diffraction-limited, then the MTF follows the curve: 2 f 2 11 MTF( x) = [arccos( x) x 1 x ], where x =, with f the spatial frequency and π f ic = 1 λ f /#. For the ideal system without aberrations, the MTF as a function of spatial frequency would follow this curve. The reciprocal of spatial frequency, or f -1 gives a measure of resolution in millimeters. Comparison with MTF=50% spatial frequency 1 f ic Diffraction-limited MTF as a function of spatial frequency 0.8 MTF 0.6 0.4 Spatial frequency at MTF = 0.5 0.2 Spatial frequency at Rayleigh criterion 0 0 0.2 0.4 0.6 0.8 1 1.2 f/f ic Figure 14: Plot of the diffraction-limited MTF equation as defined in the previous section, as well as the spatial frequencies corresponding to MTF = 0.5 and the Rayleigh criterion.

Page 9 5/7/2007 For diffraction-limited systems (which occur for on-axis objects for many of the candidate lenses 2 2 studied), solving the equation: MTF( x) = [arccos( x) x 1 x ] = 50% gives x = 0.404, or π equivalently, f = 0.404f ic. In comparison, the Rayleigh criteria radius: d = 2.44λ f /# r = 1.22λ f /#, leads to a spatial frequency of f 1 0.89 = = = 0. f ic r λ f /# 82, twice the spatial frequency of that when MTF=50%. Equivalently, the inverse spatial frequency f -1 gives a measure of resolution twice that given by the Rayleigh criterion. 1.2 Technical Introduction 1.2.1 Current Setup of Axial Imaging System Design of current axial system Figure 15: Setup of current axial system The current axial system is composed of two achromatic 75-mm focal length 25.4 mm diameter AC254-075-B lenses from Thorlabs. Between the atoms to be imaged and the first lens, there is an approximately 3 mm-thick wall of Pyrex that separates the vacuum from the ambient air in which the lenses are located. The two achromatic doublet lenses were chosen in order to minimize aberrations over a simple two-surface biconvex lens, and by having two identical lenses, facing the opposite direction, this can further reduce aberrations. The first lens introduces positive spherical aberrations, while the second lens introduces negative spherical aberration that cancels it out. Figure 16: Identical achromats reducing spherical aberration 12 The lenses that will be studied are three other Thorlabs lenses: AC254-100-B, AC508-100-B, and AC254-150-B, and two Gradium lenses: GPX25-80 and GPX40-80. The data for these lenses can be found in the appendix. The results to be presented include only the AC254-100-B, AC254-150-B, and GPX25-80, for the other lenses did not show any improvement above the current AC254-075-B. 1.2.2 Software used The software used was Zemax, a geometric ray-tracing program. Optical systems can be modeled using the software, which traces rays from the object plane given parameters such as index of refraction, surface curvature, apertures, and wavelength. The software also keeps track of the phase of the wavefront as it enters and exits the lenses. Distortions of the wavefront over the focal plane lead to aberrations visible in the image. It has tools for displaying the MTF as a function of the spatial frequency given the

Page 10 5/7/2007 specifications of the lens. Specifications are requested from the lens companies, or downloaded as Zemax files and imported into the program. Previously, a software package called CODE V, published by Optical Research Associates (ORA), was used for analysis. It was replaced by Zemax because Zemax could handle Gradium lenses at the wavelength of light used (775 nm), while the version of CODE V used (7.61), could not. 1.2.3 Performance criteria The following criteria were used to gauge the performance of the various lenses in the imaging system. On-axis performance Figure 17: In this analysis, the resolution is measured for an on-axis object, with all distances held fixed. This analysis consists of measuring the resolution of a point object on the object plane using the inverse spatial frequency at MTF=50%. The on-axis performance characterizes the maximum performance obtainable from the imaging system. On-axis back focal length adjust tolerance Figure 18: Illustration of on-axis back-fl adjust tolerance This analysis consists of measuring the resolution of an on-axis field point when the back focal length is adjusted away from the ideal focal length determined by Zemax. This characterizes how precisely the focal length must be adjusted in order to obtain performance comparable to that determined in the on-axis analysis section. Off-axis performance Figure 19: Illustration of off-axis analysis The analysis of off-axis performance consists of measuring both the sagittal and tangential MTF-based resolutions of point objects located at 0.3 mm intervals from on-axis to 2.1 mm off-axis. The ratio of the sagittal and tangential resolutions gives the astigmatism of the imaging system. The off-axis performance characterizes how compact and close to the optical axis the object must be to obtain resolutions comparable to on-axis performance.

Page 11 5/7/2007 Effect of refocusing for off-axis objects Figure 19: Illustration of off-axis analysis without refocusing This analysis is similar to the off-axis performance analysis, except that the optical system is not refocused for each off-axis point object. Instead, the focal length is kept at the distance for optimal focus for an on-axis point. The effect of refocusing for off-axis objects is important for characterizing the resolution of small atom clouds that drop across the field of view while the camera is still in focus onaxis. Object-to-(First)-Lens Distance Figure 20: Illustration of Object-to-lens distance analysis This analysis consists of measuring the resolution of an on-axis field point when the distance between the object and the first lens is adjusted, all other distances within the setup remaining fixed to the distances determined in the appendix except for the focal length, which is refocused for every change in the objectto-lens distance. This analysis characterizes how precisely the two lenses as a unit have to be located to obtain performance comparable to that determined in the "On-axis performance" section above. It also gives an indication of how deep the depth of field is, that is, how close imaged objects have to be to the object plane to be highly resolved on the focal plane. Interlens Distance Figure 21: Illustration of interlens distance analysis This analysis consists of measuring the resolution of an on-axis field point when the distance between the two lenses is adjusted, all other distances within the setup remaining fixed to the distances determined in the appendix except for the focal length, which is refocused for every change in the interlens distance. This analysis characterizes how precisely the two lenses as a unit have to be placed relative to one another to obtain performance comparable to that determined in the "On-axis performance" section above.

Page 12 5/7/2007 2. Observations 2.1 Setting up Focus Initially, the lens data is entered, so that the following layout is obtained in Zemax: Figure 22: 3-D view of surfaces in current axial system. The leftmost two surfaces represent the section of Pyrex glass exposed to the rays from field object The location of the best focus is calculated with Zemax by first changing the back focal length (surface 6 s thickness) to a marginal ray height solve: Zemax sets the thickness to where the ray furthest from the optical axis meets the optical axis. This gives an estimate of the focal length, then the program is instructed to reset the focal length to minimize the wavefront error of the rays hitting the focal plane. 2.2 On-axis performance The focal length is optimized by Zemax following the procedure in the above section, for an on-axis field point. On-axis MTF-based resolution v. lens used in Imaging Setup 14 12.5 12 Resolution (microns) 10 8 6 4 7.78 6.84 8.36 Diffraction-limited resolution 2 0 Current Axial GPX25-80 AC254-150-B AC254-100-B Lens used in System Figure 23: On-axis Resolution for optical system with each of the four lenses A lower value indicates a smaller spot size and hence larger resolution. It appears that out of the four systems, the GPX25-80 has the best on-axis performance, and the AC254-150-B the worst, in fact worse by almost twice the best system. The best system (GPX25-80) is better than the current axial system by approximately 13%.

Page 13 5/7/2007 2.3 On-axis back focal length adjust tolerance The resolution was compared as the focal length was adjusted from 36 microns forward from the ideal focal length to 36 microns farther from the ideal. On-axis Back Focal Length Adjustment vs. MTF-based Resolution 20 18 Resolution (microns) 16 14 12 10 8 CurrentAxial (microns) GPX25-80 (microns) AC254-100-B (microns) AC254-150-B (microns) 6-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 Focal length adjust (mm) Figure 24: On-axis back focal length adjustment for the four lenses All systems need to be focused to a precision of at least 0.03 mm or else performance suffers greatly. Although the GPX25-80 had the best off-axis performance, it drops off the second-quickest when the focal length is adjusted, after the current axial system. At 0.036 mm, the best system is the one with AC254-100-B. The AC254-150-B is the most tolerant of having an unfocused beam. 2.4 Off-axis performance To measure the off-axis MTF-based resolution, the field point is set to the desired off-axis distance. Then, the rays from the off-axis distance are vignetted, that is, Zemax is told to ignore rays that do not make it to the focal plane due to failing to pass through a lens component. This is generally only a concern for off-axis rays since by default Zemax sends rays from the field point that fill the entire first lens pupil. The lens is then refocused as in the preceding section. This process is repeated for all field points from 0 mm to 2.1 mm going up by 0.3 mm intervals, for all four lenses tested. The data obtained showed a marked contrast between sagittal and tangential resolutions for all lenses, especially at higher off-axis distances. Hence, to obtain a single measure of overall resolution, the geometric mean of these two values at each off-axis distance was taken:

Page 14 5/7/2007 Geometric Mean of Sagittal and Tangential MTF Resolutions vs. Off-axis distance Geometric mean of Resolution (microns) 70 60 50 40 30 20 10 0-0.5 0 0.5 1 1.5 2 2.5 Off-axis dist (mm) CurrentSystem Geom mean GPX25-80 Geom mean AC254-100-B Geom mean AC254-150-B Geom mean Figure 25: Geometric mean of the two directions of MTF resolution for the four lenses Apparently, although the current axial Imaging system performs well in relation to the other lenses onaxis, in terms of off-axis performance, it rapidly degrades in performance compared to the other lenses. By 2.1 mm off-axis, its performance has degraded approximately 8 times. AC254-150-B degrades the least in percentage terms compared to its on-axis performance, but it starts off at a poorer resolution onaxis. GPX25-80 offers the best resolution at all off-axis distances. In terms of astigmatism, the ratio of the sagittal and tangential resolutions was taken. However, since some lenses had better sagittal resolution than tangential resolution, the absolute value of the natural logarithm of the ratios between the two resolutions was taken, in order to allow comparison of ratios without having some values over zero and others under zero. ln (Resolution sag. / Resolution tang.) Off-axis refocused: Absolute value of the Ln of ratio between sagittal and tangential resolution 0.8 0.6 0.4 0.2 0-0.2-0.5 0 0.5 1 1.5 2 2.5 Off-axis dist (mm) Figure 26: Astigmatism for the four lenses CurrentAxial GPX-25-80 AC254-100-B AC254-150-B

Page 15 5/7/2007 Lower numbers refer to lower levels of astigmatism. The data shows that the current axial system has consistently worse astigmatism than the other three systems studied, with the AC254-150-B offering the least astigmatism at off-axis distances. 2.5 Effect of refocusing for non-off-axis objects For this section, each field point was analyzed without readjusting the focal length, leaving it at the focal length ideal for an on-axis point. Geom mean of Sagittal and Tangential MTF resolutions, without refocusing 150 Resolution (microns) 100 50 CurrentAxial (microns) GPX25-80 (microns) AC254-100-B (microns AC254-150-B (microns) 0-0.5 0 0.5 1 1.5 2 2.5 Off-axis dist (mm) Figure 27: Geometric mean of the two directions of resolution without refocusing Comparison with the previous section shows that the effect of refocusing improves the resolution by slightly more than two times for all the systems studied. Without refocusing, the best two lenses, GPX25-80 and AC254-150-B, are comparable in resolution at a distance of 2.1 mm off-axis. Off-axis unrefocused: Absolute value of the Ln of ratio between sagittal and tangential resolution ln (Resolution sag. / Resolution tang.) 1 0.8 0.6 0.4 0.2 0-0.2-0.5 0 0.5 1 1.5 2 2.5 CurrentSystem GPX25-80 AC254-100-B AC254-150-B Off-axis dist (mm) Figure 28: Astigmatism without refocusing for the four lenses

Page 16 5/7/2007 Without refocusing, the astigmatism gets worse for all systems at high off-axis distances. In fact, the advantages in low astigmatism that the other systems had in comparison to the current axial system disappear at approx. 2.1 mm in off-axis distance, where the current axial system s astigmatism plateaus after 1.5 mm off-axis. 2.6 Effect of Varying the Object to First Lens Distance The resolution was compared as the object-to-first-lens distance was adjusted. 20 Resolution vs. Object-to-lens distance displacement 18 Resolution (microns) 16 14 12 10 8 6-10 -5 0 5 10 15 Displacement of Obj-to-lens distance from Optimum (mm) CurrentAxial GPX25-80 AC254-100-B AC254-150-B Figure 29: Graph of resolution vs. displacement of the object-to-lens distance for each lens from the optimum placement (i.e. that giving the highest on-axis resolution) The AC254-075-B lens demonstrates the fastest degradation. After a movement of 5 mm of the obj-tolens distance from the point that gives the best on-axis resolution, the performance worsens by more than a factor of two. In comparison, the AC254-150-B lens degrades by less than 5% after the same distance. The AC254-100-B degrades by approximately 20% when the distance is decreased by approximately 4.5 mm and less than 5% when the distance is increased by approximately 5 mm. The GPX-25-80's performance decreases by around 20% when the distance is decreased by 5 mm and by around 7% when the distance is increased by 5 mm. The GPX25-80 is therefore the lens that provides the best resolution in absolute terms over a large range (10 mm) in obj-to-lens distances, while the AC254-150-B is the lens that offers the least percentage degradation in this range. 2.7 Interlens Distance For this section, the interlens distance, set initially to 2x the back focal length of the lens, was changed to 0, 1x, 4x, 8x, and 16x the back focal length. The focal length of the system was readjusted at each of

Page 17 5/7/2007 these distances and the resolution based on spatial frequency at 50% MTF was calculated. 13 Resolution vs. Interlens distance 12 Resolution (microns) 11 10 9 8 7 6 0.0000 1 2 (default) 4 8 16 Interlens distance (in focal lengths) CurrentAxial GPX25-80 AC254-100-B AC254-150-B Figure 30: Resolution vs. Interlens distance plot. The analysis in all the other sections have the interlens distance set to 2 x the focal length, as shown in the appendix. All systems show only small changes in resolution after the interlens distance was changed. These changes do not exceed 1.2% in the case of the current axial system and 0.2% in the case of all the other lenses even when the interlens distance is increased by eight times (16 x FL). It is of note that the resolution shows a small increase for large interlens distances for the current axial system and GPX25-80 compared to the original on-axis analysis but a small decrease for the AC254-150-B 2.8 Error Analysis The error in the values on the graphs, in terms of calculation errors caused by approximations used by Zemax are approximately 0.1 cyc/mm when measuring the spatial frequency at MTF=50%. Since the f value of resolution is obtained by taking the reciprocal, the error in resolution is R = 2. With f frequencies in the range of 57 to 146 cyc/mm, the error is in the order of 0.03 µm for resolutions of ~ 17.5 µm and 0.005 µm for resolutions of ~ 7 µm. More important errors are the tolerances, more exactly, how precisely the focal length can be adjusted, as in the previous section. If in practice the focal length can only be adjusted to a precision of 0.03 mm, then the error in on-axis resolution is actually approximately 50% for the current axial system and the GPX25-80, and as low as 10% for the AC254-150-B, according to above graph. The tolerance in interlens distance is high for on-axis objects, so any imprecision in calibrating the interlens distance does not introduce significant percentage errors for any of the lenses. A study was not conducted for the off-axis resolution. Therefore the true test in the error of the analysis performed in this study lies in the precision in the laboratory. 3. Conclusions 3.1 On axis and off-axis The GPX25-80 is the best in overall performance for on-axis and on-axis resolution. The AC254-150-B has the least degradation when the beam is unfocused, and for off-axis points offers the least astigmatism, however it does not offer as high resolution on-axis. The AC254-100-B is the best in terms of Thorlabs lenses in on- and off-axis performance. The current axial system lens (AC254-075-B) has the second-best

Page 18 5/7/2007 performance on-axis, however it quickly degrades off-axis. The best lens recommended depends on how well the lens can be refocused for off-axis points. If the precision is greater than 36 µm, then the GPX25-80 should replace the AC254-075-B. Otherwise, the AC254-150-B offers the best off-axis resolution at across a larger range of focal lengths. 3.2 Object to Lens Distance In terms of the object to first lens distance, the GPX25-80 offers the highest resolution throughout the FL ± 5 mm range that was tested. In this range, the resolution varies by 20%. In comparison, the AC254-075- B s resolution worsens by a factor of two at the ends of the range. Since the atom cloud sizes in the laboratory are in the order of hundreds of microns, the degradation due to depth of field is less than 4% over 1 mm for the AC254-075-B used in the current axial system, which was the worst performing lens of the four studied as is evident from Figure 29. Therefore the primary concern in this area should be the precision in locating the two lenses as a fixed unit rather than the depth of field. 3.3 Interlens Distance The interlens distance was found to not play a major role in the resolution of any of the lenses. By changing this distance to 16x the focal length, the current axial system s resolution changed by only 1.2%, while none of the other lenses saw a change in resolution greater than 0.2% in this range. 3.4 Future Work Future work would consist of measurements in the lab to test the findings in this study. Appendix: Lens Data Surface # Comment Curvature Thickness Glass Semi-Diameter 0 0.0000 35.0000 2.1000 Pyrex 1 window 0.0000 3.0000 PYREX 12.7000 2 0.0000 32.7715 12.7000 3 First lens -0.0024 1.6000 SFL6 12.7000 4 0.0237 5.0000 BAFN10 12.7000 5-0.0271 139.6082 12.7000 6 Second lens 0.0271 5.0000 BAFN10 12.7000 7-0.0237 1.6000 SFL6 12.7000 8 0.0024 69.8041 12.7000 9 0.0000 0.0000 2.1775 Figure 31: Lens data from Zemax for the current axial system with AC254-075-B: The semi-diameter refers to the distance from the optical axis to the marginal ray. Surface 5, the interlens distance, is set to be twice the focal length. Surface 2, the Pyrex glass to first lens distance, is set to the distance that allows unitary magnification for on-axis points. The semi-diameter of Surface 9 is automatically set, and corresponds to how far the marginal rays from Surface 0 (the object surface) land away from the optical axis at surface 9 (the focal plane). The following are the data for the other three lenses studied: Surface # Type Curvature Thickness Glass Semi-Diameter 0 STANDARD 0.0000 35.0000 2.1000 1 STANDARD 0.0000 3.0000 PYREX 12.5000 2 STANDARD 0.0000 43.0948 12.5000 3 GRINSUR8 0.0000 4.0000 G32SFP 12.5000 4 STANDARD -0.0165 160.2646 12.5000 5 GRINSUR8 0.0165 4.0000 G32SFN 12.5000 6 STANDARD 0.0000 80.1323 12.5000

Page 19 5/7/2007 7 STANDARD 0.0000 0.0000 2.1142 Figure 32: Lens data from Zemax for the current axial system with GPX25-80 # Comment Curvature Thickness Glass Semi-Diameter 0 0.0000 35.0000 2.1000 1 Pyrex window 0.0000 3.0000 PYREX 12.7000 2 0.0000 60.0158 12.7000 3 First lens 0.0039 1.5000 SFL6 12.7000 4 0.0186 4.0000 LAKN22 12.7000 5-0.0150 194.1106 12.7000 6 Second lens 0.0150 4.0000 LAKN22 12.7000 7-0.0186 1.5000 SFL6 12.7000 8-0.0039 97.0553 12.7000 9 0.0000 0.0000 2.1164 Figure 33: Lens data from Zemax for the current axial system with AC254-100-B # Comment Curvature Thickness Glass Semi-Diameter 0 0.0000 35.0000 2.1000 1 Pyrex window 0.0000 3.0000 PYREX 12.7000 2 0.0000 107.5223 12.7000 3 First lens 0.0008 3.5000 SFL6 12.7000 4 0.0112 4.0000 LAKN22 12.7000 5-0.0120 289.1339 12.7000 6 Second lens 0.0120 4.0000 LAKN22 12.7000 7-0.0112 3.5000 SFL6 12.7000 8-0.0008 144.5669 12.7000 9 0.0000 0.0000 2.1150 Figure 34: Lens data from Zemax for the current axial system with AC254-150-B References

1 From Melles-Griot Catalogue p. 1.29, Melles-Griot Inc 1999. 2 Lens (Optics) Wikipedia, http://en.wikipedia.org/wiki/lens_(optics) 3 From Melles-Griot Catalogue p. 1.12, Melles-Griot Inc. 1999. 4 From Melles-Griot Catalogue p. 1.12, Melles-Griot Inc. 1999. 5 From Melles-Griot Catalogue p. 1.13, Melles-Griot Inc. 1999. 6 From Melles-Griot Catalogue p. 1.18 Melles-Griot Inc. 1999. 7 Gradium Brochure, LightPath Technologies. 8 McHugh, Sean, cambridgeincolour.com 9 Nave, C.R. Hyperphysics: Rayleigh Criterion. http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/raylei.html 10 Image from Atkins, Bob, Modulation Transfer Function - what is it and why does it matter?, http://www.bobatkins.com 11 Melles-Griot Catalogue p. 3.4 Melles-Griot Inc. 1999. 12 From Melles-Griot Catalogue p. 1.19 Melles-Griot Inc. 1999.