Improved ray tracing air mass numbers model

Transcription

1 Improved ray tracing air mass numbers model Sergey N. Kivalov P.O. Box 0355, Brooklyn, New York 11218, USA Received 12 March 2007; revised 5 July 2007; accepted 31 July 2007; posted 31 July 2007 (Doc. ID 80886); published 1 October 2007 An improved ray tracing air mass model to calculate the air mass numbers for the entire zenith angle range is developed. The improved model uses the approach when the trajectory element of light in the atmosphere is approximated by an arc of a circle. This way the angles at the beginning and at the end of the trajectory element can be counted simultaneously. This approach gives the second-order approximation for the real light trajectory with more accurate results than the results of the approaches of Link and Neuzil (Tables of Light Trajectories in the Terrestrial Atmosphere, Hermann, 1969) and the Kasten and Young models [Appl. Opt. 28, 4735 (1989)]. The developed model allows us to avoid the calculation problems of the Link and Neuzil and Kasten models when the zenith angle is close to or equal to 90. As a result, we deliver the new air mass number table for the entire zenith angle range and provide the comparison of the developed model results with the results of the Link and Neuzil and the Kasten models Optical Society of America OCIS codes: , , Introduction In the 18th and 19th centuries, many distinguished scientists devoted their works to both atmospheric refraction and light extinction phenomena. While Cassini s refraction model [1] for the homogeneous atmosphere used the single refraction concept, Oriani [2] attempted to resolve the refraction integral for the real atmosphere, approximating it by the finite series of odd powers of tan z up to the fifth power. This series approximation approach had been developed by Lambert. Oriani s two-termapproximation based on Oriani s Theorem works fine for the zenith angles (z) till 80. However, at degrees higher than 80, the higher powers of tan z have to be considered, and the approximation series does not converge at all. Biot [3] provided the physical background to Oriani s Theorem and showed the high influence of the lower atmospheric layers on the refraction. Forbes [4] noticed that Bouguer [5] was the person to whom we owe the first careful consideration of the varying intensities of absorbed and reflected light and a special application of it to...thetransparency /07/ $15.00/ Optical Society of America of atmosphere. Bouguer showed that light extinction can be represented by a logarithmic function of thickness of a uniformly dim medium. In 1805, in his Celestial Mechanics, Laplace [4,6 8] developed the atmospheric model, which contains a set of concentric strata. He showed that air masses are directly connected with the refraction of light in the atmosphere and that both of these phenomena depend on the atmospheric density profile. Laplace s ray tracing model counts how the curvature of atmosphere affects atmospheric refraction and the path of rays in the atmosphere. His model uses the atmospheric refractive index (n) to change the angle (i) between the tangent to the path of light and the radius of the concentric stratum r h r 0 h. The angle (z) is the zenith angle in the observer s point. The formula for the angle (i) is sin i r n rn sin i r 0n 0 sin z. (1) r h n Laplace introduced the relationships among the air mass element dm, the trajectory element ds, the height element dh, the atmospheric density, and the angle (i) with the following formulas: 10 October 2007 Vol. 46, No. 29 APPLIED OPTICS 7091

2 ds sec idh, dm ds sec idh. (2) Using Laplace s model and the results of observations, Forbes [4] showed how to derive the formula, which connects the air masses M a with the refraction (R) in arcseconds and the zenith angle ( z ): R M a. (3) sin z In the beginning of the 20th century, Bemporad [9,10] improved Lambert s and Oriani s approach, extending the series for the powers of sec z and tan z to approximate the light extinction integral for the angles higher than 80. He calculated the first accurate extinction tables, and he also regarded Laplace s and Bouguer s works as the origin of this research. In 1969, using the same light extinction integral, Link and Neuzil [11] derived the air mass formula and calculated the air mass and refraction tables for the different atmospheric conditions based on the U.S. Standard Atmosphere, 1966 model [12]. They used the methods of numerical calculations and changed variables in the integral to avoid the calculation problems near the horizon. In their ray tracing model, which is similar to Laplace s model, the atmosphere is presented as a set of concentric spherical layers. Each spherical layer has the constant relative density ( ) and the constant refractive index n 1. Figure 1 presents the scheme of Link s model. Starting from formula (2), Link and Neuzil used the following formula for the absolute air mass: h M abs h 0 h h 0 h sec idh 1 r 0 r h 2 n 0 n h h 0 2 h 1 sin 2 i dh sin 2 z 1 2 dh. (4) Starting from the same concept, Kasten [13] derived the improved relative air mass formula, and using the Air Research and Development Command (ARDC) Model Atmosphere, 1959, he calculated his air mass table from it. Instead of the zenith angle (z), this formula uses the solar altitude angle ( ) above the horizon: M a 1 0 H 0 dh r 0 2 r 0 h cos 1 2. To derive this formula, Kasten used the concept of the homogeneous atmosphere for the calculation of (5) Fig. 1. Scheme of Link and Neuzil s ray tracing model, where r 0 is the radius of the Earth; h is the height of the atmospheric layer above the ground; dh, n, and are height, refraction index, and density of the layer, respectively; i and i are angles of light direction at the beginning and at the end of the trajectory element inside the layer; z is the zenith angle; and d is the angular element for the geocentric angle. the scale height (H) and the Lorenz Lorentz equation for the representation of the atmospheric refractive index as a function of air density: n 2 1 n const. (6) Formula (5) does not have an analytical representation, so based on the International Organization for Standardization (ISO) Standard Atmosphere, 1972 model [14], Kasten and Young [15] improved the approximation formula for sea level: 1 M a sin a b c. (7) Following Link and Neuzil, let us introduce R, the integral refraction angle. This is the angle between the tangent to the trajectory in the observer s point and the tangent to the trajectory in the point located at the height h. It is clear from constructions for the geocentric angle that z i R. (8) Theorem 1: In Laplace Link Kasten s model, there are the following relationships among the height element (dh), the angular element (d ), the refraction element (dr), and the angle (i): 7092 APPLIED OPTICS Vol. 46, No October 2007

3 d 1 r h tan idh, (9) d di dr, (10) dr 1 tan idn. (11) n As Young [8] showed in his work, the minus sign in Eq. (11) is important because the integral refraction (R) is a positive and increasing function of h, but the refractive index n(h) is a decreasing function, which leads to the negative derivative: dn dh 0. Both the relative air mass numbers M a and the refraction are widely used in astronomy and atmospheric optics to estimate the extinction and scattering of radiation, which goes through the atmosphere to an observer located at some height above the Earth s surface [11,16 19]. The relative air mass numbers can also be used for energy measurements to estimate the amount of the direct component of solar radiation on the Earth s surface normal to the solar beam, which depends on the atmospheric transparency [20]. In this paper, the ray tracing is done using h, height above the ground level, as an integration variable. For the calculation of the angle i, the standard expression (1) is used. In each finite height interval, Laplace s formula (2) is replaced by the modified formula for the secondorder approximation of the trajectory element in expression (17) that is taken as an arc of a circle with an appropriately chosen curvature. This replacement is described in Section 2. In Section 3, the relative positioning of the real light trajectory and different approximations is described. Based on the curvature analysis, it is shown that the error of the improved model based on the expression (17) is smaller than the error of the standard model based on the formula (2) for the same height interval. Section 4 discusses similarities between different atmospheric models used in the previous papers. It also addresses the differences in the approximations of the U.S. Standard Atmosphere, Section 5 presents a comparison of the improved model results with the standard model results for the different atmospheric model approximations. 2. Improved Ray Tracing Air Mass Model Both Link and Neuzil, and Kasten noticed a problem with their integral calculations when h U 0 and the solar altitude angle 0 (the zenith angle z 2). It happens because their models explicitly use sec i from Laplace s formulas (2). The function sec i actually represents the ratio of the hypotenuse to the adjacent side in the rectangular triangle. So, formula (2) gives the linear approximations of the real light trajectory in the relationships among the dm, ds, and dh. Fig. 2. Scheme of the improved ray tracing model, where r 0 is the radius of the Earth; h is the height of the atmospheric layer above the ground; h is the finite height of the layer; i and i are angles of light direction at the beginning and at the end of the finite trajectory element ( s) inside the layer, respectively; L is the chord corresponding to s; z is the zenith angle; is the finite angular element for the geocentric angle ; and is the central angle of the arc of the circle that approximates s. The improved model s concept is based on the assumption that the trajectory element of light in the atmosphere can be better approximated by an arc of a circle or by its chord (L). This approach gives us the second-order approximation of the real light trajectory, and this way we can simultaneously count both angles (i) and i at the beginning and at the end of the finite trajectory element s. Hence the model counts the curvature of the trajectory. Figure 2 shows the scheme of the improved model. Let us introduce new angles i i i 2 and. By construction, i is the angle between the chord (L) and the radius of the circle r h r h h 2 rotated on the angle 2. The point of the intersection of the radius and the chord can be taken as the center of the chord. Hence, r h r h h 2 and r h r h r h 2. Actually, for the center of the chord, the corresponding radius of the circle and rotation angle will be r h r h h 2 h and 2. It can be shown that asymptotically h h r h 2 4 h and h 8r h, so they become negligibly small with the small h and. That is why the removing of the h and from the model will not affect the model performance when the h and are small. is the central angle of the arc of the circle that approximates s. By construction and formula (8), i R. (12) 10 October 2007 Vol. 46, No. 29 APPLIED OPTICS 7093

4 Because angles R are very small, it can be assumed that the length of s is equivalent to the length of L: L s r circle 2 sin L. (13) 2 So, instead of the estimation of the length of s, the length of L can be estimated. Theorem 2: In the assumptions of the improved model, there are the following relationships among the finite height element ( h), the finite angular element ( ), the radius of the circle r h, and the angle i : tan 2 1 tan i r h r h, (14) h 1 r h tan i h. (15) Expressions (14) and (15) are the improved model s equivalents of expression (9) from the Laplace Link Kasten model theorem. They are presented in the form of the small variations of the height and the angular elements and can be directly used for the numerical calculations of the length of L and consequently for the calculation of the length of s for the entire zenith angle range. The transformation of tan 2 into cos gives the final expression for the length of the chord (L) correspondent to the s: L 2 r 2 h r 2 h 2r h r h cos 4r 2 r 2 h r 2 h cos 2 i h 2 sin 2 i h 2r h r h 4r 2 h cos 2 i h 2 sin 2 i. (16) light trajectory and their errors can be based on the curvature analysis. Really, for the same initial point of h interval, both the standard and the improved model approximations have the same tangents with the light trajectory. The standard linear approximation (ds) is simply a straight line, i.e., the tangent line to the light trajectory. The relative positions of the real light trajectory and the arc of the approximating circle will depend on their curvatures next to the initial point of the approximation interval. The following formula for the curvature of the approximation circle can be derived from formula (13): circle 1 r circle 2 sin 2. L On the other hand, there is the standard formula for the curvature of the trajectory of light in the atmosphere: light 1 r sin i n h dn dh. Figure 3 presents the graph of the ratio light circle for the initial angle i z 2. It can be seen that for the small h, the curvature of the light trajectory is 1% 2% bigger than the curvature of the approximating circle from the improved model. The curvature ratio value drops to 1 and below when h becomes bigger. It means that for the small h, the light trajectory is located in the area between the arc of the approximating circle ( s) and its chord (L) and very close to the approximating circle. Furthermore, the light trajectory can intersect with the approximating circle next to the end of the approximation interval. The scheme of this location of the curves and Assume that h h h 2. Then, formulas (13) and (16) give the final expressions for the finite air mass element M and the finite trajectory element s : s L r 4r 2 2 h r 2 h cos 2 i h 2 sin h 2r h r h 4r 2 h cos 2 i h 2 sin 2 i, (17) M h s h r 4r 2 2 h r 2 h cos 2 i h 2 sin h 2r h r h 4r 2 h cos 2 i h 2 sin 2 i. (18) 3. Calculation Error Analysis The comparison of the standard model approximation and the improved model approximation for the real Fig. 3. Graph of the curvature ratio of the real light trajectory and the approximating circle of the improved model, where h is the finite height of the layer and light, circle, and light circle are the curvature of light, the curvature of the approximating circle, and the ratio of these curvatures related to the finite point of the finite height interval, respectively APPLIED OPTICS Vol. 46, No October 2007

5 Fig. 4. Scheme of the relative positioning of the real light trajectory, the standard model approximation, and the improved model approximation, where dh h is the height of the layer, ds is the trajectory element of the standard model, s is the finite trajectory element of the improved model, and L is the chord correspondent to s. their intersection is presented in Fig. 4. From Fig. 4, it can be seen that for the same h, the error of the standard model can be calculated as the sum of the differences between the standard model and the improved model and between the improved model and the light trajectory. It also supports the initial expectation that the second-order approximation of the improved model is better than the standard one. Because of the previous conclusion on the location of the light trajectory element between the arc of the approximating circle and its chord, to make the proper calculation results of formulas (17) and (18), the approximation error from formula (13) should be estimated. An assumption that the relative approximation error of s L should not exceed 0.01% leads to 2.8. Because of Eq. (12), this gives the constraint on the finite angular elements and i for the single calculation step: i 2.8. (19) Fig. 5. Graphs of, i, and i and their square root approximations, where h is the finite height of the layer, is the diamond marked curve, i is the triangle marked curve, i is the square marked curve, an approximation for is the diamond shapes, and an approximation for i is the circle shapes. Because, i, and i depend on both h and h, the inequality (19) actually gives some constraints on the h values. For the estimations of the angles values, the worst case when an observer is located on the Earth s surface and the initial angle i z 2 has been chosen. The results of estimations of the different values of and i and their sum i as functions of h, the elevation above the ground in this case, are presented in Fig. 5. The presented graphs show very consistent behavior of (the diamond curve), i (the triangle curve), and their sum (the square curve) when h reduces from 1 km to 50 m. Furthermore, in this domain, both and i can be perfectly approximated by the following functions C h 0.5 with some constants C deg km 0.5 (the series of circle and diamond shapes): h, i h. (20) This is quite enough for the U.S. Standard Atmosphere, 1976 model [21], which has the initial h 50 m. Approximations (20) are also perfectly held when h reduces from 50 to 1 m. Because the error analysis was conducted for the worst case when the initial (zenith) angle i z 2 it can be concluded that by using expressions (17) and (18), the improved model can approximate both the M and the s with any necessary precision for all z. This conclusion shows the advantages of the improved model over both Link and Neuzil s and Kasten s models when the observer is located on the Earth s surface, h U 0, and the initial angle is i z 2. In this domain, formulas (4) and (5) cannot be directly used, and both Link and Neuzil s and Kasten s models have to change variables and use approximations in the integration intervals r 0, r 0 2km [11] and r 0, r km [13] correspondingly. Because the rays traced from the selected observer are horizontal, they have long paths in these height intervals: approximately 175 km for Link and Neuzil s model and approximately 83 km for Kasten s model. In those ranges, the atmosphere has the highest gradient of densities that affects and changes both refraction angles and approximated paths of the rays [8,12,14] and strongly influences the air masses as well. 4. Choice of the Atmospheric Model Approximations In the calculations of atmospheric density profile for the improved model, the linear approximation of densities based on the real values of the U.S. Standard Atmosphere, 1976 tables [21], for the heights up to 86 km of geometrical altitude is used. A possible way to 10 October 2007 Vol. 46, No. 29 APPLIED OPTICS 7095

6 Table 1. Approximation for the U.S. Standard Atmosphere, 1976 Density Profile H gp (m) H gp 0 11, H gp 44, ,000 20, e 11,000 Hgp ,000 32, H gp 201, ,000 47, H gp 57, ,000 51, e 47,000 Hgp ,000 71, H gp 184, ,000 84, H gp 198, improve the calculations is to use the U.S. standard atmosphere approximation formulas as published by the National Oceanic and Atmospheric Administration (NOAA), NASA, and U.S. Air Force (USAF) [22]. The formulas provide temperature, pressure, and density profiles with the heights up to 71 km of geopotential altitude ( km of geometrical altitude). They can be additionally extended to km of geopotential altitude (86 km of geometrical altitude) by using the standard constant temperature lapse rate [23] formula for the higher layer. The NOAA approximations [22] give smaller densities than the values tabulated in the U.S. standard atmosphere, Table 1 presents the set of modified formulas used to approximate the U.S. standard atmosphere, 1976 density profile, in the improved model calculations (H gp (m) geopotential altitude). The conversion from the geometrical altitude H gm to the geopotential one H gp can be done by the following standard expression: H gp r 0H gm, (21) r 0 H gm where r m is the standard value of Earth s radius [21]. For the improved model, the precision of calculations is chosen equal to h 50 m that is equal to the initial resolution of the U.S. Standard Atmosphere, 1976 tables. 5. Air Mass Number Calculation and Comparison Let us assume that an observer is located on the height h 0, and the light goes through the atmosphere from the height h 1 to the observer. Then expressions (17) and (18) give the following formulas for the air mass numbers M h0,h 1 and the length of the light trajectory s h0,h 1 for the height interval h 0, h 1 : s h0,h 1 s h h r h 2 r 2 h 2r h r h 4r h 2 cos 2 i h 2 sin r 2 h cos 2 i h 2 sin 2 i, (22) M h0,h 1 M h h h r h 2 r 2 h 2r h r h 4r h 2 cos 2 i h 2 sin r 2 h cos 2 i h 2 sin 2 i. (23) The results of calculations of the relative air mass numbers M a by expression (23) for the entire atmosphere and the different zenith angles (z) are presented in Table 2. The second and the third columns of Table 2 present results for the improved model with two different density approximations (linear and NOAA) of the U.S. Standard Atmosphere, For comparison, the fourth and fifth columns show Table 2. Comparison of the Relative Air Mass Numbers for the Different Models z Improved Model Linear Approximation Improved Model NOAA Approximation Link [11] U.S., 1966 Kasten [15] ISO, APPLIED OPTICS Vol. 46, No October 2007

7 Link and Neuzil s model results for the U.S. Standard Atmosphere, 1966, and Kasten s model results for the ISO Standard Atmosphere, The U.S. Standard Atmosphere, 1976, is identical to both the U.S. Standard Atmosphere, 1966, for altitudes up to 50 km and the ISO Standard Atmosphere, 1972, for altitudes up to 50 km [15,21,24]. Kasten [13] noticed that the atmosphere between 34 and 84 km (mesosphere) contributes 0.5% or less to the absolute air mass. From this work, it can be also seen that the atmosphere layer above 50 km contributes less than 0.1% to the air mass numbers. So the different atmospheric models should not affect the results of calculations (less than 0.1%), and the presented comparison should give a correct picture. To be consistent in comparison with the latest Kasten s model, the mean radius of Earth is taken as r km, and the average atmospheric refractive index n 1 0, where Kasten [13] showed that this refraction coefficient is exact for the wavelength 700 nm, which divides the energy distribution of the visible solar spectrum on two equal parts. The ISO Standard Atmosphere, 1972, has an initial resolution h 50 m, and in my communication with Young [25], he noticed that in his and Kasten s work [15] the actual values were based on linear interpolation of the densities in 50 m intervals, using the analytical formulae in the Appendix [13]. It can be seen that the improved model with the linear approximation of the density profile from the U.S density values and Kasten s model with ISO, 1972, give very similar results of air mass calculations. The improved model with the modified NOAA approximation formulas of the density profile gives slightly larger air mass values than both the improved model with the linear approximation and Kasten s model. However, these differences are 0.15% (0.05 of the relative air mass) for both of these model results when the zenith angle approaches 90. Even though Link and Neuzil s model gives larger air mass values than the improved model when the zenith angle equals 90, this comparison may not be correct. Even though the U.S. Standard Atmosphere, 1966 has an initial resolution h 50 m, there is no available information what tabulation intervals Link and Neuzil used in their calculations. In their model [11] Link and Neuzil used the atmospheric refractive index with the different calculated for the wavelength 540nm, the peak of the visible solar spectrum. And they may also have used a different value for the mean radius of Earth r km. If, for the sake of comparability, Link and Neuzil s values of the atmospheric refractive index and the mean radius of Earth are substituted into the improved model and the calculations with the maximal altitude 50 km are made, larger values for the relative air mass numbers for the zenith angle equals 90 are obtained: for the linear approximation: M , M ,r km , and M ,r km ; for the modified NOAA approximation: M , M ,r km , and M ,r km The comparison of these improved model results with Link and Neuzil s model results shows that with the proper , the improved model gives larger air mass values than Link and Neuzil s model values. The difference is 0.12% 0.4% of the relative air mass), which comes closer to the comparison difference for Kasten s model. 6. Conclusion The improved air mass numbers model that was developed shows a consistent behavior without any calculation problems when the size of the finite height element h is reduced from 1 km to 50 m and below, which guarantees good estimations in the calculations of the air mass numbers for the entire zenith angle range at the observer. With the properly chosen, the air mass numbers calculated by the improved model are slightly larger than the corresponding values of both Link and Neuzil s and Kasten s models. This is in good agreement with the improved model construction, which gives the second-order approximation for the light trajectory element. The maximal deviation from Link and Neuzil s and Kasten s values occurs when the zenith angle approaches 90. It reaches 0.15% for the NOAA approximation of the atmospheric density profile. In this case, for both Link and Neuzil s and Kasten s models, the comparison difference with the improved model reaches 0.05 of the relative air mass, which can be evaluated as an additional 0.51 km of length of the light trajectory in the homogeneous atmosphere in standard conditions. The paper is dedicated to my father, Nikolay Kronidovich Kivalov. I thank Andrew T. Young for looking through this work and making some valuable comments. His bibliography and his site on the Internet are a great source of information related to research in the field of atmospheric refraction and its applications. They were very helpful for me when I was finishing this paper. I also thank my colleague Laura Zan for her great help in improving the understandability of the paper. References 1. A. T. Young, Cassini s Model, explain/atmos_refr/models/cassini.html. 2. B. Oriani, De Refractionibus Astronomicis Ephemerides Astronomicae Anni 1788: Appendix ad Ephemerides Anni 1788 (Appresso Giuseppe Galeazzi, 1787), pp J. B. Biot, Sur les Réfractions Astronomiques (Additions à la Conn. des Temps de 1839) (De l Académie des Sciences, 1836), pp J. D. Forbes, On the transparency of the atmosphere and the law of extinction of the solar rays in passing through it, Philos. Trans. R. Soc. London 132, (1842). 5. P. Bouguer, Traité d Optique sur la Gradation de la Lumière 10 October 2007 Vol. 46, No. 29 APPLIED OPTICS 7097

8 (Mem. de l Académie Royale des Sciences. H. L. Guerin and L. F. Delatour, 1760). 6. P. S. Laplace, Traité de Mecanique Celeste (Chez J. B. M. Duhrat, 1805), Vol. 4, Chap P. S. Laplace, Celestial Mechanics. Translated with a commentary, by Nathaniel Bowditch (Chelsea Publishing, 1966). 8. A. T. Young, Understanding astronomical refraction, Observatory 126, (2006). 9. A. Bemporad, Sulla teoria d estinzione di Bouguer, Mem. Soc. Astron. Ital. 30, (1901). 10. A. Bemporad, Zur Theorie der Extinktion des Lichtes (Mitteilungen der Grossh. Sternwarte, 1904). 11. F. Link and L. Neuzil, Tables of Light Trajectories in the Terrestrial Atmosphere (Hermann, 1969). 12. U.S. Standard Atmosphere Supplements, 1966 (U.S. Government Printing Office, 1966). 13. F. Kasten, A new table and approximation formula for the relative optical air mass, Arch. Meteorol., Geophys. Bioklimatol., Ser. B 14, (1965). 14. International Organization for Standardization, Standard Atmosphere, International Standard ISO2533 (1972). 15. F. Kasten and A. T. Young, Revised optical air mass tables and approximations formula, Appl. Opt. 28, (1989). 16. A. T. Young, Laplace s extinction theorem (2006), http: //mintaka.sdsu.edu/gf/explain/extinction/laplace.html. 17. L. H. Auer and E. M. Standish, Astronomical refraction: computational method for all zenith angles, Astron. J. 119, (2000). 18. C. Y. Hohenkerk and A. T. Sinclair, The Computation of Angular Atmospheric Refraction at Large Zenith Angles (NAO technical note, Royal Greenwich Observatory, 1985). 19. S. Y. van der Werf, Ray tracing and refraction in the modified US1976 atmosphere, Appl. Opt. 42, (2003). 20. S. Khromov and L. Mamontova, Meteorological Handbook (Gidrometeoizdat, 1963). 21. U.S. Standard Atmosphere, 1976 (U.S. Government Printing Office, 1976). 22. U.S. Standard Atmosphere, 1976, as published by NOAA, NASA, and USAF, atmos/1976%20standard%20atmosphere.htm. 23. Properties of the U.S. Standard Atmosphere 1976 (2005), M. J. Mahoney, A Brief Note on Standard Reference Atmospheres (2005), ReferenceAtmospheres.html. 25. A. T. Young, Department of Astronomy, San Diego State University, 5500 Campanile Drive, San Diego, California , USA (personal communication, 2006) APPLIED OPTICS Vol. 46, No October 2007