New apodizing functions for Fourier spectrometry

eral results. This check is particularly important since our numerical results show substantial deviation from those presented by Yeh8 even for elliptic cross sections of small eccentricity. For example, when x = 1. 90, m = f, and b/a = 1. 05 (kgq2 /4 = O. 1 and ko q coshto = 2. 0 in Yeh's notation), we find a back scattering cross section (ako /4) of 0. 56 as opposed to 0. 7 for Yeh. However, for these parameters the abovementioned approximation scheme gives an independent check on our expansion coefficients and confirms the accuracy of our calculation to within 1%. The agreement obtained in all limiting cases confirms the validity of the results presented here. V. CONCLUSION Relatively simple expressions have been developed for the machine computation of electromagnetic scattering from dielectric rods of arbitrary cross section. As an example of the application of these formulas, results for the scattering cross section of rods with rectangular and elliptic cross section have been presented. Graphs are obtained showing cross section versus size, eccentricity, and index of refraction. In particular, once results for a given polarization and angle of incidence are obtained, the formulas give results for the other polarization and arbitrary angle of incidence with little additional effort. Although numerical results obtained for elliptic rods disagree with one set previously published in the literature, our independent approximation methods support the results presented here. *This work was performed as a National Research Council Postdoctoral Associate. Present address: Sperry Rand Re- search Center, Sudbury, Mass. 01776. 'L. Eyges, Ann. Phys. (NY) 81, 567 (1973). 2 L. Eyges, Ann. Phys, (NY) 90, 266 (1975). 3 P.C. Waterman, Proc. IEEE 53, 805 (1965); J. Acoust. Soc. 4 Am. 45 1417 (1969); Alta Freq. 38 (Speciale) 348 (1969). Waterman's work exploits the surface integral equations that arise by applying Green's theorem in that problem. In the present work volume integral equations, that hold at all points in space, are used. 5C. Yeh, J. Math, Phys. 4, 65 (1963). 6 C. Yeh, J. Opt. Soc. Am, 55, 309 (1965). 7 C. C. Strickler, IRE Trans. Ant. Prop. AP-5, 267 (1957). 8J. H. Richmond, IEEE Trans. Ant. Prop. AP-13, 334 (1965). 9 K. Mie and J. Van Bladel, IEEE Trans. Ant. Prop. AP-l1, 52 (1963), 'OD. S. Jones, The Theory of Electromagnetism (Macmillan, New York, 1964), p. 335. The difference between Jones's formula and ours is due to the difference in his MKS units and our CGS-Gaussian units. t "See, for example, L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, Reading, Mass., 1972). 2 i p. Morse and H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York, 1953), p. 1322. 13 H. C. van de Hulst, Light Scattering by Small Particles (John Wiley, New York, 1957). New apodizing functions for Fourier spectrometry Robert H. Norton and Reinhard Beer Space Sciences Division, Jet Propulsion Laboratory, Pasadena, California 91103 (Received 6 October 1975) A new class of apodizing functions suitable for Fourier spectrometry (and similar applications) is introduced. From this class, three specific functions are discussed in detail, and the resulting instrumental line shapes are compared to numerous others proposed for the same purpose. Practitioners of Fourier spectrometry and, indeed, most individuals employing numerical Fourier transforms commonly employ apodization to improve the appearance of their outputs. All users are aware that such a damping of secondary maxima entails an inevitable smearing of the output. To the Fourier spectrometrist, the smearing manifests itself as a loss of spectral resolution, a loss that strikes him particularly hard because of the effort entailed in achieving a satisfactory spectral resolution in the first place. There is, of course, no real need to apodize at all: The unapodized, point-by-point output of a Fourier spectrometer system contains all the information derivable from the interferogram. Nonetheless, spectral analysts more familiar with the "smooth" output of scanning spectrometers dislike such "unsmooth" output intensely and demand that it be smoothed and interpolated. The apodi2 zation function is commonly used for this purpose, 1' but with the foregoing in mind, Fourier spectrometrists 259 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976 are more conscious than most of the need to find and use the most suitable functions. It has become clear that there is no such thing as an "optimum" apodizing function. This statement will, hopefully, be clarified later in the present paper wherein we demonstrate that, at best, some functions can be regarded as more useful than others, In the process, we shall arrive at the conjecture that, while there is no single optimum function, there may well exist a class of such functions, the nature of which is presently unknown. For terminology, we employ the name apodizing function for the function multiplying the interferogram amplitudes and instrumental line shape (ILS) for the Fourier transform of the apodizing function. I Furthermore, we use the identities U= I/L, where 1 denotes path difference and L maximum path difference A. Nelson and L. Eyges 259 and a- 2,ruL, where a is the wave number of units of reciprocal path difference (usually cm-'). We also employ the convention3 sincx= sinx/x. 100 THE PRACTICE OF APODIZATION In 1967, Fellgett 4 said ". . .the orthogonal properties of the sinc function... are easily destroyed by apodization. This is why I believe that apodization should be done only by experts. " Notwithstanding such structures, apodization is undertaken daily by experts and nonexperts alike. The fact of the destruction of the orthogonality of the sinc function is indisputable. In the present context, the consequence is that the output points following the Fourier transformation process no longer are statistically independent. Calculation of parameters such as signal-to-noise ratio must therefore proceed with caution. -c 10= - 0- V) 0 'A a 10' I- :5 0 In a decade of use of Fourier spectrometry in astronomy and atmospheric physics we have continually sought to find better apodizing functions because the acquisition of such interferograms is sufficiently time-consuming and costly that it behooves us to make the best possible use of the data at hand. The most extensive investigation of apodizing functions is that of Filler, 2 who employed combinations of trignometric functions to generate a number of useful apodizations. Connes 5 has used algebraic functions with similar success and, emboldened by this, we have now investigated algebraic and trigonometric-algebraic functions. In all, some 3000 functions were tested. In order to distinguish useful functions, we have employed a criterion scheme devised by Filler. He introduced three criteria for the ILS. (i) The half-width relative to the sinc function (i. e., the unapodized case). We denote this by W/W 0 . (ii) The absolute size of the largest secondary maximum (not necessarily the first) relative to sinc. We denote this by Ih/ho I. (iii) The convergence rate of the secondary maxima. This criterion is important mainly when, as is our practice, spectra are stored in unapodized form and the chosen apodization is applied by convolution at the time of production of an output. The fewer points that are needed to define the instrumental function, the less costly will be the convolution. To these criteria, we add a fourth: (iv) The degree of destruction of the statistical independence of the apodized spectral points. The more closely the zero crossings of the apodized ILS match those of the sinc function, the less damage is done to the independence of the data. The basic selection is undertaken using criteria (i) and (ii). The outcome is best displayed on a plot of Ih/ho I against W/W 0 , which we call a "Filler Diagram." Such a plot is presented in Fig. 1. Roughly 1150 func260 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976 NJ 0 2 10 1.4 1.6 1.8 NORMALIZED HALF-WIDTH [W/Wo] FIG. 1. Filler diagram for about 1150 apodizing functions. The V-shaped patterns are from various families of functions, unidentified for clarity. Attention is drawn to the concentration of the plotted points into the upper right half of the diagram. tions are plotted, the V-shaped patterns being individual families of functions in which a single parameter has been varied. The functions are plotted without identification to avoid confusion, but each of them (or at least the family) has been proposed for use as an ILS function at one time or another. Some specific cases will be discussed later. For the moment, we wish only to draw attention to the most arresting feature of the plot: There appears to be a boundary, to the left of, and below, which no function penetrates. Families of functions approach, and then retreat from, this boundary. None ever crosses it. The equation of the boundary appears to be quadratic: logo I h/h01I 1. 939 - 1. 401(W/W0 ) - 0. 597(W/Wo)2 . Note that it is perfectly possible to derive functions that result in a half-width less than sinc (W/Wo<1). This produces the well known phenomenon of "super-resolution" but we find that all such functions give enhanced secondary maxima (I h1/ho I > 1) and still fall above and to the right of the apparent boundary. We therefore offer the conjecture that such a boundary does, indeed, exist and issue a challenge to the mathematically minded to solve the following problems: (i) to prove (or disprove) that such a boundary exists; (ii) to find an expression describing the boundary. The one given here is purely an empirical fit; and (iii) to use the expresR. H. Norton and R. Beer 260 sion (or otherwise) to derive ILS functions that fall on, or arbitrarily close to, the boundary given that the apodizing function must fall to zero at maximum path difference. Of course, it would be preferable if the resultant ILS functions also converged rapidly and had zero crossings coincident with sinc. If the conjecture is true and problems (ii) and (iii) are soluble, the outcome will be the best possible ILS functions that can exist for a given Ihl and W. However, the existence of a boundary will also imply that there is no such thing as an optimum function because an infinite number of functions could fall on the boundary. Consequently, the experimenter will still be required to decide on his own choice of desirable damping and half-width on the basis of his particular needs and prejudices. It is also worth drawing attention to the fact that two widely used apodizing functions-"triangular" apodization and the Hamming functionr-result in ILS functions that fall well to the right of the boundary and are far from "best" functions from any standpoint. APODIZING FUNCTIONS Filler? came to the conclusion that a set of apodizing functions Da given by Da(U)=cos(irU/2)+a cos(37rU/2) 0' a• 1 could scarcely be improved upon. Indeed, we ourselves used D., functions for several years in that same belief. However, since that time we have discovered several functions with better performance. One of these is a variant of another of Filler's functions-one that he termed Ec: C2 = 0. 22498 C3 = 0. 07719 and gives W/W 0 = 1.228, Ih/ho I = 0.33 (an absolute secondary maximum height of 0. 071 for a central lobe of height - 1). We are uninformed as to the route that led Connes to these values; we believe, however, that the set of functions presented in Table I is superior. The functions F1, F2, and F3 are displayed in Fig. 2. It will be noted that they decay smoothly to about U= 0. 9, flatten out and even increase slightly, and are truncated at U= 1. 0. FO, of course, takes the value 1 for 0' U' 1. 0. All the functions are zero for U> 1. 0. The specific values in Table I were chosen to give a set of ILS functions that fell as close as possible to the boundary, converged reasonably rapidly and did as little damage as possible to the statistical independence of the neighboring spectral points. It was also found that there was sufficient flexibility in our requirement for a "weak, " a "medium, " and a "strong" apodization to permit us to choose coefficients such that the resultant values of W/W 0 became 1.2, 1.4, and 1.6 for ILS functions Il, 12, and 13, respectively. The degree to which we have been successful is illustrated in Figs. 3, 4, and 5 where the 3 ILS functions are plotted individually with I0 (= sinc [27raL]) to show the damping achieved for the particular broadenings. Because the apodizing functions FN(U) are expressed as power series, the terms are separable and the ILS are readily derived as n IN(Cc) = E Eo(U)= 1+(1+ a)cos(vU)+ a cos(2wrU) 0' a - 1 The addition of a constant term p gives a set of functions we call Py: where Q0= sinca, P1 :(u)= i+p+ (1+ a) cos(1rU)+ acos(27U), - 1' a' , O'p' 3(sinca - cosa) Q1= a2 1 which has quite desirable properties. For example, setting a = - 0. 0325 and p = 0. 3 gives a function for which W/W 0 = 1. 41 and I h/hoI = 0.064, significantly better than for either D,, or Er, alone. In fact, none of Filler's functions fall close to the boundary for any value of a. Pg comes much closer but is not recommended because the convergence is rather slow. The next section treats our preferred functions. THE PREFERRED FUNCTIONS - 15[(1 - 3/a2 ) sinca + (3/a2 ) cosa] Q2 = - i=O In general, n does not exceed 4 because we have found no particular improvement in going beyond that value. The form is similar to a function devised by Connes5 which had the coefficients C 0 = 0. 23977, C 2 =0.45806, 261 105[(l - 15/a 2 ) cosa + 3(2 - 5/a a4 2 ) sinc a] 945[(1 - 45/a 2 + 105/a 4 ) sinca + 5/a 2 (2 - 21/a 2 ) cosa] a4 a = 2,f raL. Note that, since C3 0 for all the functions, Q3 plays no role in the evaluation but is included for completeness. The expressions for the Qi are accurate but tend to n FN(U)=ZCi(l-U2)i; ECi-=1, N=0, 1, 2, 3. i=O a2 Q3 = The preferred set of functions is simply algebraic, of the general form n N=0, 1, 2, 3 CMi i=O J.Opt. Soc. Am., Vol. 66, No. 3, March 1976 TABLE I. Coefficients of the preferred apodizing functions. Apodizing function number C0 C1 C2 C3 C4 Comments 0 1 0 0 0 0 No apodization 1 0. 5480 -0. 0833 0. 5353 0 0 "Weak" apodization 2 0.26 - 0. 154838 0.894838 0 0 "Medium" apodization 3 0. 09 0. 3225 "Strong" apodization Coefficients 0 0. 5875 0 R. H. Norton and R. Beer 261 1. 3 0.i 0 I 0. O.-.4 X I I2 -1 0. -O.: 1/L EL] PATHDIFFERENCE NORMALIZED be numerically unstable for values of a less than about 0. 7. For small arguments, it is better to use the expansions a4 a2 a4 az 1+ Q2 =1 T10 280 a2 a4 a 8 a8 as 15 120 -1 330 560 a8 a - 33 264 - 3 459 456 4 +0-4 Q11a 3 18 Q4 8 a 5040 +362 880 + 120 a a a 792 - 61 776 + 7 413 120 a4 1144 4/L The convergence of the secondary maxima (and their values) is shown in Table IIL Note that after some gyrations in the first few secondary maxima, the later maxima decline monotonically. Since the oscillations are, in all cases, about zero, the error associated with the truncation of the ILS when it is used to convolve and interpolate an unapodized spectrum is always less then the next secondary maximum, presuming that the truncation occurs at a zero crossing. The Filler diagram for these ILS functions is shown in Fig. 6, together with the line representing the boundary suggested by Fig. 1. It will be seen that all three functions fall close to the boundary and therefore appear to be suitable functions. The small deviations from the boundary should not be construed as implying that they cannot be improved upon since the boundary itself is purely empirical and somewhat ill defined; in fact, we find them to be perfectly adequate for our pur- ++- Q.=1a 2 22 3/L FIG. 4. Instrumental line shape resulting from function 2 compared to sinc (function 0). FIG. 2. The preferred apodizing functions. Q"': 1 -6 .3/L.-.. 2/ 2/L [cm-' if L in cm] FREQUENCY a a a 102 960 14 002 560 which are accurate to < 10-9 for a < 0. 7. 1.0 1. 0.0 X 0.. 0.6- 0.4 -3 0. Intuetlleshp FIG 3. 0. rslIn fo fuc Io 1 0.2 I1 -0.2 -A. 2 1/L 2/L 3/L 4/L [cm'l If L in cm] FREQUENCY FIG. 3. Instrumental line shape resulting from function l compared to sinc (function 0). 262 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976 1/L 2/L 3/L 4/L [cm if Lincm] FREQUENCY FIG. 5. Instrumental line shape resulting from function 3 compared to sinc (function 0). R. H. Norton and R. Beer 262 TABLE II. Half-width, height, and position of the first 10 secondary maxima of the instrumental line shapes resulting from the preferred apodizations. a ILS function Relative half-width Position of halfamplitude number IV/W0 point 0 1. 0000 0.30168 - 0.21723 0.71523 0.12837 1.22956 - 0. 09133 1.73549 0. 07091 2.23873 - 0.05797 2. 74078 0. 04903 3.24220 - 0. 04248 3. 74324 0.03747 4. 24404 - 0. 03353 4.74467 0.03033 5.24518 1 1.2000 0.36202 - 0. 05804 0.81683 0.05349 1.24845 - 0. 04389 1.73810 0. 03598 2.23767 - 0. 03020 2.73874 0. 02592 3.23993 - 0. 02266 3.74099 0. 02012 4.24189 - 0. 01807 4.74264 0. 01640 5.24327 2 1.4000 0.42235 - 0. 01414 0. 99105 0. 00805 1.32982 - 0. 01346 1.74455 0. 01364 2.23138 - 0.01248 2. 73048 0. 01119 3.23175 - 0. 01003 3.73334 0.00905 4.23483 - 0. 00822 4. 73615 - 0. 00752 5.23730 3 1. 6000 0.48268 - 0. 00286 1. 23237 -0. 00044 1. 47514 - 0. 00303 1. 79028 0. 00373 2. 26494 - 0. 00369 2. 75555 0.00346 3. 25127 - 0.00319 3. 74918 0.00293 4. 24813 - 0. 00270 4. 74760 0. 00250 5. 24735 Height (upper) and position Gower) of secondary maxima h, h2 alleights are relative to a central peak h, h, STATISTICAL DEPENDENCE Since apodizing functions used in Fourier spectrometry broaden the central lobe, there is necessarily an interaction between any given spectral point and its -lo I I I - X * h7 h8 h9 h8o nearest neighbors. An unapodized spectrum sampled perfectly 7 has all the spectral points falling on the zero crossings of the ILS of every other point in the spectrum. There is no interaction and all spectral points are statistically independent. However, the ILS is usually also employed as an interpolating function, and the total number of output spectral points increased by a factor of 5-10. In this case there will be an interaction because the interpolated points are not independent and, in fact, unapodized spectra interpolated with a sinc function frequently take on a most bizarre aspect. It will be noted from Figs. 3, 4, and 5 that the zero crossings well removed from the origin are essentially coincident with the zeros of the sinc function. This is a substantial advantage of these functions if the output spectrum is not to be interpolated. If they are to be used for interpolation, the small amplitude and rapid convergence of the secondary maxima is helpful. Ii - hh 1; Positions are in units of 1/L. poses and substantially better than other functions we have used in the past. 10° h5 10- csc Actually, the zero crossings do not quite coincide with the zeros of sinc but, in every case, by cr= 2/L the crossings are displaced from sinc by less than 0. 02 (in units of 1/L) and the discrepancy becomes smaller with increasing frequency. I2 en ;Z 2 0 vI3 CONCLUSION C) We believe that the apodizing functions and the resultant instrumental line shapes presented here offer measurable advantages over any previously published and that they are of such a nature as to be readily applicable to many fields, not just Fourier spectrometry. 0 I 0 2 10o-3 1.0 I I 1.2 I I 1.4 I 1.6 I I I 1.8 I 2.0 I 2.2 NORMALIZED HALF-WIDTH [W/Wo] FIG. 6. Filler diagram for the instrumental line shapes resulting from the preferred apodizations. The line represents the boundary seen in Fig. 1, 263 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976 We should like to thank Dr. A. J. Brooke of the University of Texas at Austin who performed most of the computations for Fig. 1 and who originated the Pa function described earlier. This paper presents one phase of the work carried out at the Jet Propulsion Laboratory, California Institute of Technology under National Aeronautics and Space Administration Contract NAS 7-100. R. H. Norton and R. Beer 263 'J. Connes, AFCRL-71-0019, pp. 83-115 (1971). Air Force Cambridge Research Laboratories Special Reports No. 114. 2 A. H. Filler, J. Opt. Soc. Am. 54, 762 (1964). 3 A. E. Siegman, Appl. Opt. 13, 705 (1974). 4P. Fellgett in printed comments on R. Beer and A. H. Cayford, J. Phys. (Paris) 28, C2-33 (1967). 5 J. Connes (private comn unication). R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), p. 98. T Perfect sampling implies that the occupied spectral interval exactly fills the alias. 6 Anisotropic intrinsic photoluminescence in NaT* W. L. Emkey Allentown Campus of the Pennsylvania State University, Fogelsville, Pennsylvania 18051 P. V. Meyers and W. J. Van Sciver ' Lehigh University, Bethlehem. Pennsylvania 18015 (Received 27 September 1975) The intrinsic photoluminescence in NaI is observed to have an emission intensity that is direction dependent. The experimental results are explained by calculating the interference between the direct luminescence and that which has undergone internal reflection. The result is a two-parameter theory. The two parameters are the absorption coefficient, a, and the thickness of a nonluminescing "dead" surface layer, d. Typical values are a - 107 m-' and 1.0 X 10-8 m < d < 3.0 X l0-8 m. I. INTRODUCTION Single crystals of NaI luminesce when illuminated at LNT in the region of the fundamental absorption.' The resulting luminescence is a single emission centered at 4. 2 eV and is associated with the radiative recombination of an electron with a self-trapped hole. The excitation spectrum for this intrinsic luminescence' is seen in Fig. 1, where the excitation spectra for an irradiation in both a (110) and a near (100) direction are given. Two observations from Fig. 1 are noted. (1) Even though essentially all the light is absorbed (a -107 mi1), both spectra exhibit structure throughout the excitation energies, an effect which is typical of highly absorbent luminescent materials. 1-7 (2) The two spectra differ in both structure and intensity. Since these spectra were recorded with the positions of the exciting light and the detector fixed at 900 with respect to each other, two possibilities for this difference are considered: (a) an emission intensity which depends upon the angle of incidence of the exciting light, and (b) an emission intensity which depends upon the relative angular position of the detector with respect to the illuminated surface. incident angle of the exciting light and/or the angular position of the detector with respect to the illuminated surface of the crystal. A schematic of the experimental arrangement can be seen in Fig. 2. Cylindrical samples of pure NaI were obtained from the Harshaw Chemical Company. Cylindrical samples were used so that luminescence coming from the center of the cleaved face would be normal to the exiting surface, thus reducing the effects of multiple internal reflections and refraction of the emission at the exiting surface. Samples were cleaved, mounted, and placed in a special optical Dewar in a time span of 3 min. These operations were all performed in a dry box with a relative humidity less than 2%. Liquid nitrogen was introduced into the cold finger of the Dewar and measurements were taken after the sample reached its lowest temper- 80 o 0 z U- Reported here is the angular dependence of the intrinsic luminescence of NaL. The luminescence is found to be anisotropic. The analysis of this anisotropy is based upon interference between the luminescent source and its image. The model is then extended to describe the structure of the excitation spectrum. 2 40- 2 dl 8 50 II. EXPERIMENTAL PROCEDURE An optical Dewar was constructed so that the luminescence could be monitored as a function of both the 264 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976 7.5 7.0 6.5 6.0 5.5 ENERGY (eV) FIG. 1. Excitation spectra for the intrinsic luminescence of Nal at 80'K (from Ref. 1). Copyright © 1976 by the Optical Society of America 264