Tables of the Modulation Transfer Function of aDefocused Perfect Lens

Leo Levi and Richard H. Austing

Tables are presented giving the MTF of an aberrationless defocused lens with circular aperture for objectsilluminated with incoherent monochromatic light. MTF values are given to six decimal places for re-duced spatial frequencies v = 0(0.01)1 and defocusing A = 0(0.1)0.4, 0.5(0.5)5, 10(5)50, except that 5/Ais an upper limit on v,. Here A is measured in units of Rayleigh's ,4X tolerance on defocusing. The inte-gral of the MTF and (MTF)2 are also given for A < 5. The numerical integration was performed bySimpson's rule over most of the range and by series expansion over the rest. An error analysis is appended.Tables of the line spread function for the focused case are also given.

The appended tables (I) give the modulation transferfunction of the aberrationless lens with circular aperturein the presence of defocusing. The results apply toobjects illuminated incoherently with monochromaticlight.

The spatial frequency () is given in reduced units,which are related to the spatial frequency () by:

V = ('v/2 sino) = 'Fv = (Xov/2A), (1)

where X', Xo are the wavelengths of the light in theimage space and a vacuum, respectively, 0 is the half-angle of the cone subtended by the exit pupil at theimage point, and F,A are the lens f/number andnumerical aperture, respectively, as measured in theimage space.

Defocusing (A) is measured in units of Rayleigh's(X/4) tolerance on defocusing, which is ('/2 sin2 0).Thus,

A = (2 sin'O/X') = (2A'/nXo) = (1/2F2X')5, (2)

where is the defocusing in the units of length in whichX is measured.

The required TF has been shown' to have theform:

T(v, ) = f - (- eiax dxdy, (3)W JJ -(1- y2)i + yr

where a = 27rvA.

L. Levi is with the City College of the City University of NewYork, New York, New York, 10031; R. H. Austing is with theComputer Science Center, University of Maryland, College Park,Maryland 20740.

Received 25 October 1967.

By integrating first over x and expanding the result-ing y integrand in a series of Bessel functions, Hopkins'showed that this MTF can be written in series form:

T(vrA) = cosavp {fJ(a)- E ( sin (2no)[J2 - (a)7ra n= i 2n

-J 2 n+1 (a)]}

- sinavp E sin[(2n + 1)1] [J2 n(a)7ra n =0 2n + 1

- J 2 n + 2(a)], (4)

where A - cos v.To avoid the inconvenience of evaluating the Bessel

functions, the order of integration in Eq. (3) may bereversed, resulting in

T(vrA) = (1 - x2

)2 cos[a(x - vr)]dx,7ra Ir

after a simple transformation.The appended tables were evaluated by integrating

Eq. (5) by means of Simpson's rule with intervals of1/1600, except in the neighborhood of the upper limit.There the cosine was expanded in a Taylor series andintegrated term by term.

The numerical data were obtained on an IBM 7094computer operated at double precision. This methodof evaluating the integral should yield an accuracy ofbetter than 2 X 0-v for a < 10r (see Appendix).These tables give T(ir,A) for the following ranges of vand A:

vr 0(0.01)1, A < 5

= 0(0.01)5/A, A > 5,

A = 0(0.1)0.4, 0.5(0.5)5, 10(5)50.

May 1968 / Vol. 7, No. 5 / APPLIED OPTICS 967

(5)

Table a. MTF of Perfect Lens, DefocusedA Rayleigh Units -2AF2 X

= 0 0.1 0.2 0.3 0.4Vr

0.0.010.020.C30.040.050.o0.07o.oe0.os0.1 C0.110.120.130.140.150.160.170. le0.1S0.200.210.220.230.240.250.260.270.280.240.3C0.310.320.330.340.350.360.370.3e0.340.4C0.410.420.430.440.450.460.470.480.440.5C0.510.520.530.540.550.560.570.50.590.600.610.620.630.640.650.660.670.680.6S0.7C0.710.720.730.740.750.760 770.780.740.800,.810.820.830.840.850.860.870.88

0.9C0.910.930.940.950.960.97

0.99

I 00CocoC09872680.9745370.9618090.9490840.93636509 42 36 510.910946

.89e2500.88 55630.8728P.90.86C2270.8475790.8349460.8223300. 8097 330.7971540.7845960. 77 20610. 7595480.7470600. 7345980. 7221640. 709758.0.6973820.6850380.6727260.66C4490.6482080.6360030.6238380.6117 120.5996280.5875870.5755900.5636390. 55 17 360.5398830.57580790.51 6 3 290. 5046 320.4929 900.4814C60.4698 810.45 84 160. 44 70 140.4356760.4244C43 .4132 000.4020650.3910020. 38CO 130.3690990.35e2620.3475050.3368300. 32 62 390. 3157 330. 30 53170.2949900.2847570. 2746 190.2645790. 2546400. 2448050.2350750.2254540.2159450. 2065 510. 1972750. 1881200. 1790910. 1701890. 16 14200. 1527870.1442940. 1359450. 1277450. 1196990. 1 18 110.1 040880.0965340. 0891 570.0819620.0749560.06 81470.0615440.0551560.0489930.0430650.0373860.0319700.0268340. 02 19970.0174830.0133200.0095460. 0062090.0033850.001149

1.000000o.9872630.9745180.961768 I0 .9490 150.9362610.9235080.910757 I

0 .8980 120.8852730.8725440.8598250.8471190.8344270.82 17520.8090940.7964570.7838410. 7712480.7586800.7461 380.7336240.72114C0.7086880.6962680.6838830.6715 330.6592210.6469480. 6347150.6225250.6103 780.5982760.5862200. 5742130.5622540.5503470.5384930.5266920.5149460.5032 580.4916280 .4800590.4685510 .4571060.4457260.4344 130.42 31680.4119920.4008890.3898580.3789020.3680240.3572240.3465050.3358680. 32 53 160.3148510.3044740.294 1880.2839950.2738980.2638990.2539990.2442030.2345120.2249290.2154 580.2061000. 1968 590.1877380.1787410.1698710.16 11320. 15 25270. 1440610. 13 57370.12 75610.1195370. 1116710 .10 39 660.0964300.0890680.08 18870.0748940.0680970.0615040.05 51240.0489680.0430460.0373720.03 19610.0268280.0219930.0174800.013 3 190.0095450.0062090.0033850.001199

1.30'.000:.987249C.974463.961648c.948809C.935950C 921 076C.91C 191C.897299c. 884404C.871510C .858621C.845740C. 83'871C. 821018o.807 182C .794367C. 781576C. 76813C.756078C.74'376C. 73"708C. 718077C.7054840.692933C.680425C .667962c.655546C.64 179C.630863C.618598C.606388C .594232C. 587134C.570094C.558113C.546 194C.534 337C.522544C.51"815C .499153C.48 5 59C .476032C.464576C .4 53 191C.441a78C.43C 638C .419474C.40e385C.397373C. 3864390.375 585C.364812C.354 122C. 343515C.337994C.322559C. 3122 120. 3CI19560.291 7931C.281719C.271743C.261864C.252084C.242405C.2 3? 8300 .223361C .2 14 000C.2047 50C.1956140. 186595C.177695C. 16-919C. 1602690. 151749C .143364C.135 116C.1270120. 119054C.111249C. 10'601C.096 117c.o8e8o20.081664C.074709C.067946C.061382C.055028c.04e893C.0429900.037331C .031931C.026808C.0219800.0 174730.01' 315C.00Q543c.006208C.003 385C.001199

INTEGRAL CF MTF0.424413 0.423776 0.421868

INTEGRAL CF TF-SQ.0.272421 0.271771

1.OCOOOO I.OOOOO0.987225 0.9871910.974371 0.97424z0.961448 0.9611680.948466 0.9479850.935432 0.9347070.922356 0.9213500.9C9247 0.9079260.896111 0.8944490.882956 0.8809310.869789 0.8673830.856617 0.8538160.843446 0.8402410.830283 0.8266680.817132 0.8131050.8C4001 0.7995610.790892 0.7860450.777813 0.7725640.764766 0.7591240.751756 0.7457340.738788 0.7323970.725865 0.7191210.712990 0.7059100.7CO167 0.6927690.6e7399 0.6797030.674689 0.6667140.662040 0.6538080.649453 0.6409860.636931 0.6282530.624476 0.6156100.612091 0.6030590.599776 0.5906030.587534 0.5782440.575366 0.5659820.563274 0.55382u0.551258 0.5417570.539320 0.5297950.527460 0.5179340.515680 0.5U61750.5C3981 0.4945180.492363 0.4829630.480827 0.4715100.469374 0.460158

0.458003 0.4489090.446717 0.4377600.435515 0.4267130.424398 0.4157660.413366 0.4049180.402420 0.3941700.391560 0.3835210.380787 0.3729700.370101 0.3625160.359503 0.352159

0.348993 0.3418990.338572 0.3317340.328241 0.3216650.317999 0.3116910.3c7849 0.3018120.297791 0.2920270.287825 0.2823370.277954 0.2727400.268177 0.2632390.258496 0.2538320.248913 0.2445200.239428 0.2353040.230044 0.2261840.220763 0.2171610.211585 0.2082370.202514 0.1994130.193552 0.1906900.1847C0 0.1820700.175962 0.1735550.167340 0.1651480. 1588390. 1504600. 1422070.1340860. 1260990. 1182510. 1105480 . 1029950.0955970.0883610.0812930.0744010.0676940.0611790 .0 548680.0487690.0428960 .0372620.0318830.0267750.0219590 .0174600 .0133080.009540O.OC62070.0033850.00 1199

Each column gives the MTF for one value of A to sixdecimal places. Such tables have been publishedpreviously, but these have been far less extensive,detailed, and precise.* For all values of A not ex-ceeding 5, the integral of the MTF and the (MTF)2 aregiven at the bottom of each column.

The line spread function L(x) can be found from theoptical transfer function T(v) by means of the Fouriertransform. When the optical transfer function is real,the line spread function is symmetrical and takes theform:

L(|x|) = f T(v) cos2,rvx dv f T(v)dv. (6)

In the case of a perfect lens with a circular aperture,we have in terms of the reduced frequency

L(xr) = 2 [cos-l - vr'l - Jo,2"r.d -7r o37r)

where the reduced displacement

Xr = (2 sinO/X')x = x/FX' = (2A/Xo)x.

(7)

(8)

For convenience we also use, in the following, t = 2 7rX,as a shorthand notation.

Equation (7) can be shown4 to yield

3 r/2L(x,) = _ sin2u sin(Q cosu)du = (3r/2 2)Hi(), (9)

where H1() is the Struve function, which has the follow-ing series representation 5

(10)i n= f (-) [2! 12' !7r = 1 2n +l L(2n)!j

From this, the series expansion for the line spreadfunction Eq. (9) is readily found to be

L(x,) = 3[ 1.3 - + - . (11)12-31-5 12.33.52-7

For large values of x, the following asymptotic ex-pansion may be used:

L(xr) -;\1 + e -1LE! 42{ n 2n - I 2nn!2

- (-)~(sint + cost) 1 (-1

(4 1)2(4 3)2... [4 (4n- 1)2]

(2n)!

+ ( () ( sin -cst)Z (-1)n

(4 - 12)(4 _ 32)(4 - 52). .. [4 -(4n + 1)2]} (12

(2n + 1)!

0. 1568510. 1486670. 1405990. 1326520. 1248290. 1171340. 1095730. 1021500. 0948720. 0877450.0807760.0 739720.0673430. 0608970.0546440.0485960. 0427650.0371660.0318150. 0267290.0219300.0174430.0132980.0095350.0062060.0033840.001199

0.418708 0.414324

C.269835 0.266651 0.262285

* Barakat and Houston 2 cover vP = 0(0.05) 1, A = 0.8(0.8)4.8,to five decimal places; Steel 3 covers v = 0.(0.1)1, A = 0(2/7r)

28/7r to four decimal places.

968 APPLIED OPTICS / Vol. 7, No. 5 / May 1968

Table lb. MTF of Perfect Lens, Defocused A Rayleigh Units -2AF2X

A 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0VI

0 1.000009 0.986746 0.977698 0.957807C0.942226C0.926037 0.909331 0.890186C0.874672 0.856876 0.838865 C.82"683C0.802415C0.784105 0.765808 0.74756,2C0.729420 0.711423 0.69359,3 0.67596S C.65-57'5 0.64144'1 0.6245913 C.60804,

C .591 80110. 57589'0. 560 321C. 549 10,

I C. 5312 3,i0.515 72110.-50 57(C .48777!C. 474 35CC. 46127ii

10.448562

C. 436 19!IC. 4241811C.412501

I0. 401 154IC .39013C

I 0.379422C.36902C0.358914

0. 344 095 C0.3395510.3302730. 3212490.3 12467C.3039 18C.2955890. 28 74680.279 545C.2718090.264 247C.2568480. 24 96020.24 7498C.235 5250. 2286 720.2219290.215287

C. 208 736C. 2012 66C. 1958690. 1895370. 1832610. 1770340. 1708500. 164 701C0. 1585 830. 1 54890. 146417C. 140361-0. 1343200. 12 82910. 1222740. 116267

0. 1102 730. 104292C.098 3290.092 3860. 0864 700. 0805 870.074745C.068 955C. 0632 270. 0575 730.052009C. 046 5510.041 2170. 0 360280.03 1007C. 026 1800.02 1577C.0172 32C.0131840.0094820. 006 1860.003 3790.001 198

10 1.000000 1.0000C'1 0.986195 0.98531'5 0.970396 0.9671119 0.952826 0.945818 0.933704 0.921859 0.913241 0.895521 0.891640 0.867264 0.869096 0.837445 0.845792 0.806437 0.821901 0.774540 0.797585 0.742087 0.772992 0.709348 0.748260 0.676568 0.723516 0.643979 0.698872 0.611776 0.674431 0.580124 0.650285 0.549201 0.626514 0.519102 0.603189 0.489969 0.580371 0.461849 0.558111 0.434819 0.536453 0.408938 0.515432 0.384236 0.495076 0.360723 0.4754n7 0.338429 0.456440 0.317331 0.438184 0.29742.

0.420645 0.27869.0.403823 0.26111.L0.387715 0.24464~0.372313 0.22926

10.357607 0.21492:10.343585 0.20158,10.330232 0.18920,10.317531 0.17773'10.3105463 0.16714,0.294009 0.1573710.283147 0.1483910.272857 0.14014;I0.263116 0.1325810.253900 0.12569]I0.245188 0.1194010.236956 0.113691,i0.229181 0.1085210.221839 0.10385k*0.214908 0.099651*0.208365 0.0958950.2C2187 0.0925330.196352 0.08955,50.190839 0.0869210.185626 0.08460S0.180691 0.0825930.176015 0.0808440.171576 0.0793550.167355 0.0780880.163332 0.0770260.159489 0.0761500.155806 0.07544C0.152265 0.0748760.148848 0.0744390.145538 0.0741110.142317 0.0738730.139169 0.0737070.136077 0.0735950.133026 0.0735190.130000 0.0734610.126983 0.0734040.123962 0.0733270.120923 0.0732150.117852 0.0730470.114737 0.0728080.111566 0.0724770.108328 0.0720380.1C5014 0.0714730.101613 0.0707660.098120 0.0698990.094526 0.0688580.090827 0.0676290.087020 0.0661990.083102 0.0645570.079074 0.0626930.074938 0.0606030.070697 0.0582810.066359 0.0557280.061933 0.0529460.057430 0.0499420.052865 0.0467270.048255 0.0433180.043620 0.0397360.038986 0.0.360060.034380 0.0321610.029832. 0.02824U0.025379 .0.0242860.021061 0.0203520.016922 0.0164950.013016 0.0127840.009403 0.0092930.006156 0.0061150.003372 0.0033620.001197 0.0U1196

10 1.000000 1.000600 1.0(0000 1.000000 1.000000 1.000000-1 0.984290 0.982982 0.981437 0.979657 0.977642 0.97539313 0.963063 U.958044 0.952134 0.945345 0.937690 0.929183'9 0.936996 0.926215 0.913580 0.899144 0.882966 0.865113.7 0.906773 U.888560 0.867351 0.843295 0.816566 0.787353'1 0.873076 0.846153 0.815045 0.780093 0.741672 0.7001930 0.836568 0.800039 0.758231 0.711780 0.661382 0.6077866 0.797884 0.751220 0.698397 0.640472 0.578585 0.5139340 0.757621 0.700631 0.636920 0.568094 0.495855 0.4219444 0.716334 0.649128 0.575038 0.496340 0.415391 0.3345359 0.674527 u.597480 0.513836 0.426646 0.338984 0.2538046 0.632655 U.546360 0.454235 0.360182 0.268008 0.1812305 0.591121 0.496345 0.396994 0.297853 0.203437 0.1177153 0.550274 0.447920 0.342711 0.2407317 0.145876 0.0636480 0.510413 U.401476 0.291835 0.1I880U0 0.095604 0.0189869 0.471787 u.357318 0.244675 0.141124 0.052621-0.0166590 0.434599 0.315674 0.201420 0.099739 0.016705-0.0439359 0.399009 U.276697 0.162147 0.063747-0.012542-0.0636682 0.365136 0.240477 0.126846 0.032935-0.035642-0.o767903 0.333064 U.207048 0.U95427 0.007O32-0.053200-0.0842759 0.302845 U.176398 0.067742-0.014414-0.065864-0.0870928 0.274503 U.148474 0.043596-0.031717-.0y4295-.*

0861674 0.248037 u. 123190 0.022761-0.045333-0.079142-.0.0823518 0.223425 u..10'0437 0.0u4987-0.055692-0.081023-.0.0764096 0.200630 U.080085-0.009989-o.03215-.o.080508-.0.0690032 0.179599 o,.06199l-o.0a2436-.06838-.7

8 17 -0o.0606966 0.160266 (..046003-0.032618-.0r13500-.07437-0.o0519516 0.142561 U.031965-0.04793-.072626920.09479-.0.0431375 0.126403 o,.019720-0.047206-0.072650-.0.063970-0.0345399 0.111711 0J.0091 12-0.052089-0.071508-0.05865-.0.0263684C.098397-(o.000009-.05656-.0.069515-.0.051993.0.0187693 0.086377-.0077900.0s8183.3.o.06688a.-0.045940-~0.01183440.075564-o,.014367-.59610-.0.63812-.0.040046-0.00561340.065874--o.1972-.633660.60442-0.o034418-0.0001208 0.057223-G,.024425-.0604223-o.056909-0.0029132 0.0046561 0.049533-0.028136-O.U59997-.0.053319-.0024238 0.0087470.042729-U.031110-o0.059166-0.049755-0.019764 0.0121960.036737-o).033437-0.058025-0o.046286-.0.015724 0.0150550 .O31491-u.035204-0.056653-.0.042962-..0012118 D.017381I 0.026927-U.036483-0.055120-0.039820-0.008936 0.0192310.022985-o,.037344-0.03482-.0.03687.0.006162 0.0206620.01961 1-U.037843-0.U51785-0.034180-.0.003775 0.0217260.016754-oJ.0380350.050067-.o.031709-.0.001749 0.0224760.014369-0O.037963-o.048358-0.029477-0.000061 0.022954

I0.012411-o,.037666-0.046681-0.027483 0.001317 0.023203I 0.010843-0,.0371793-0.045054..0.025722 0.002410 0.023256i0.009629-U.036528-0.043488-.o.024186 0.003241 0.023145'0.008739-oJ.035738-0.041993-0o.022865 0.003835 0.022893O0.00

8142-u.034828-0.040571-0.021748 0.004211 0.022523

0.0078114-U.033813-0.039223-.0.020821 0.004390 0.0220500.007730-U.032705-0.037947-0.020072 0.034388 0.021488*0.007870-o.031515-0.036740.-o.019487 0.004222 0.0208440.008216-(,.030248-0.035594-.0.019050 0.003905 0.020127*0.008750-0.028910-0.034502-0.01a747 0.003451 0.0193400.009457-o,.0275U3-o.033454.-0.018562 0.002871 0.018483O.010

322-u.026029-0.032439-0.018481 0.Q02177 0.0175580.011333-u.1024487-0.031446-0.018486 0.001378 0.0165630.012477-0.022877-0.030461-0.018562 0.000486 0.0154960.013744-U.0211936-0.029471-0.018691-0.000490 0.014354

0.015121-u.719443-.o.028463-.018856-0.o001537 0.0131360.016598-0.017615-1.27422-o.019038-.0.002645 0.0118390.018164-0.015710-0.026334-.019218-.0003799 0.0104640.019808-U.013727-0.025184-0o.019376-.0004985 0.0090110.021517-u.011664-0.023958-.0O

1 9 49 1 -. 00 6 1 8 6 0.0074830.023281-o.c009521-0.022643-0.01943...ooO 7383 0.0058890.025087-0.007299-0.021225-.01959-.0.008555 0.0042360.026919-0.005ool-0.019695-.0.019367-.0009678 0.0025390.028764-0.002632-o.018041-0.1997-.o010728 0.0008150.030605-0. 000197-0.016257-.018675-0.o01 1676-0.0009120.032426 0,.002294-0.014336-0.018fl83-.0.012

494 -. 00O26 160.034208 0.004831-0.012277-O.017303-0.01 3150-0.0042640.035931 o.007398-0.010082-0.0163 17-0.01 3614-0.0058200.037574 0.0,09980-0.007754-o.015114-0.013855-.00072430.039116 u,.012555-0.005304-0.013685-.01O

3844 -. 004890.040534 0.0151C1-0.002746-0.012028-.0.013556-0.0095130.041804 u.'01

7591-0.0u010l-0.0310144-0.012968-0.010269

0.042902 0.019997 0.002608-0.008n44-0.012067-.0.0107160.043802 0.022286 0.005348-0.005745-.0.010845-.0.0108130.044482 C-.C24425 0.008084-0.003275-.0935-.0.0105320.044917 u,.026378 0.0)10773-0.000)669.0.00)7461-0.0098510.045086 u0C281C8 0.013370 0.0020.28-0.005341-0.0087640.044967 0.029578 0.015823 0.004762-0.002984-0.0072810.044544 0.030751 0.018078 0.007471-0.000448-0.0054330.043800 0.031594 0.020081 0.010085 0.002197-0.0032690.042725 0.032074 0.021777 0.012529 0.004870-0.0008620.041315 0.032165.0.023112 0.014726 0.007476 0.0016960.039568 0.031846 0.024039 0.016596 0.009912 0.0042930.U37490 0.031125 0.024515 0.018067 0.012075 0.0068020.035097 0.029938 0.024511 0.019072 0.013860 0.0090900.032408 01.028352 0.024008 0.019557 0.015174 0.0110210.029456 0.026366 0.023003 0.019487 0.015938 0.0124730.026280 U0.24014 0.321511 0.018846 0.016097 0.0133400.022930 0.021344 0.019568 0.017647 0.015625 0.0135500.019465 0.018418 0.017232 0.015929 0.014535 0.0130750.015958 0.015317 0.014583 0.013767 0.012880 0.0119350.012489 0.012135 0.011725 0.011265 0.010759 0.0102120.009153 U.008984 0.008788 0.008564 0.008316 0.0080450.006062 0.005998 0.005923 0.005837 0.005741 0.0056350.003349 0.003333 0.0u3314 0.003292 0.003268 0.0032400.001195 0.001194 0.001192 0.001190 0.001188 0.011185

INTEGRAL CF MTF0.424413 0.408756 0.3652320.303132 0.234806 0.172131 0.123455 0.091965 0.075871 0.070103 0.068712

INTEGRAL CF MTF-SQ.0.272421 0.256824 .217339 0.170480 0.130809 0.103720 0.086660 0.074942 0065810 0.058534 0.052847

May 1968 / Vol. 7, No. 5 APPLIED OPTICS 969

000.00 00.00.00.00.00.00.00.1'0.1I0.1;

0.110.120.10. '0.110.1'0.210.220.2;0.210.2;0.2!0.210 210.2EO.2~0. 3C0. 310.300 330.340 350.360. 370.3890 340.400.410.420.430.440.450.460.470.480.440. SC0 510. 520. 530.540.550 560.570. 580. 590 6C0.610.62

0.640 650.660.670.680 640.7C0.7 10.720.730.740 750.760.770.780.740. 8IC0.810.820.830.840 850.860.870.8 80. 840 9C0 910.920.930. 940. 950. 960. 970. 980. 94

1 0.9872682 0.9745373 0.9618094 0.9490845 0.9363656 0.9236517 0.9109468 0.8982504 0.885563C 0.8728891 0.8602272 0.8475793 0.834946;0.8223305 0.8097331 0.7971541 0.7845961 0.77206110.759548C0.7470601 0.7345981 0.72216410.709758i0.69738120.6850380.6727260.66044910.6482080.6360030.6238380.6117120.599628

I0.587587

0. 5755S900.5636390.5517360. 53 98830.52 80790.5163290.5046320. 4929 900.48 14C60.4698810.458416 0.447014 0.435676 0.424404 I0.413200 0.402065 0.391002 0. 38C0013 0.369099 C0.35826210.34750510.3368300.326239 0.3 157 330.3053170.2949900.284757 C0.2746190.264579C0.254640C0.2448C5 C0.235075 C0.225454 C0.215945 C0. 20 6551C0.197275 C0.1881-20 C0.179091 0.170189 00.161420 00.152787 C0.144294 00.135945 00.127745 00.119699 00.111811 00.104088 00.096534 00.089157 00.081962 00.074956 00.068147 00.061544 00.055156 00.048993 00.043065 00.037386 00.031970 00.026834 00.021997 00-.017483 00.013320 00.009546 00.006209 00.003385 00.001199 0

1 .000000.987 140.974070.9608 00.9473 60 .93 3770. 9200 50. 9062 30. 892 3 10 .8783 30. 86 4290. 8 50220 .8 361 30. 82 20 30. 80794,0. 7938 710.779840. 76 58 50.75 1910. 7 3 80 310.7242 320. 7105 120.69687~0.68333!0. 6698 9;0.6565 5!0.6433210.63021]0.6172110.6043310 .59 15 8(0. 57895SC0.5664410. 55407]0.54 18240.5297060. 51 77 10 .50)58500.49413C0.482 5300. 4710 580 .45 97120.4484930. 43 73980.42 64260 .415 5750.4048450.3942320.3837350.3733 520.36 30820. 3 5 292 1).342 869

. 33 29 242. 32 3083). 31 334 5).303708).294 170).2847302. 2753 861.2661 371.2569831. 2479211.2 389 511.2 30073).22 12 861. 212591~.2039 86I. 19 54 74'.187053.17 87 26.170494'162 358.154 3211. 146 384'. 1385 51.130825.12 3209.11 5708.108 3271010 71

.09 3946

.086958

.080 14.07 342 3.066 893.0605 34.0543 57.04 8374.0425 98.037043.03 1728.0266 70.02 1 892.0 7420.01 32 86'.0095 30.006203.00 3384.00 1198

Table c. MTF of Perfect Lens, Defocused A Rayleigh Units 2AF2X

A= 10 15 20 25 30 35 40 45 50

V,

0. 1.OOC000 1.000000 1.000000 I.000000 1.000000 1.000000 1.000000 1. 000T 1.0010000.01 0.94C338 0.883782 C.801408 0.717748 0.615971 0.507639 U.397449 0.29963 .189363

0.02 0.801513 0.614762 0.401707 0.197349 0.U314280.077392-.124343-0.117889-0.0760010.03 0.613337 0.302560 C.04n433-0.102387-0.118066-0.053566 .023082 0.3o0947 0 .047757

0.04 0.409469 0049385-C.115944-0.03190 0.017658 0.061238 .c255C2-0.026C'37-0.033811

0.05 0.22C129-0.089777-C.086291 0.031972 0.052541-0.011160-0.034844 0.0o3464 0.025900

O.Ce 0.067081-0.116306 0.006558 0.054253-0.017842-o.U26093 .020785 0.011709-0.018270

0.07-3.038520-0.070751 C.057192-0.001455-0.028445 0.022461 .002676-3.0i6734 0.311647

0.08-0.095660-0.004660 .041u15-0.034583 0.014614 0.006444-U.015678 0.0139R3-0.004308

0.04-0.111138 0.042977-C.003930-0.012786 0.020028-0.017830 .O2425-u.Ou4607-0.0017

08

0.1C-0.096374 0.056284-0.031630 0.019078-0.008844 0.002945 u.cn1997-0.0U4535 0.006462

0.11-0.064218 0.040155-C.025794 0.020215-0.015198 0.012892-O.C'10396 3.0u9157

0.12-0.026395 0.010533-c.000882-0.001792 0.004496-0.005011 u.005881

0.13 0.008109-0.016105 .019316-0.016229 0.013328-0.008765

0.14 0.033705-0.029614 0.021651-0.009668 0.001051 0.005242

0.15 0.048071-0.028071 0.009049 O.OC5892-0.009899

0.16 0.051519-0.015827-C.004625 0.012954-0.OQ4990

0.17 0.046111 0.000098-C.014892 O.OC6824

0.1E 0.C34776 0.013349-C.01'567-0.0040550.1S 0.02C572 0.020194-C.003346-0.0093880.2C 0.006169 0.019930 .006329-0.0060700.21-0.006443 0.014179 C.0114760.22-0.016041 0.005698 .0106050.23-0.022114-0.002742 C.0053600.24-0.024725-0.009106-C.001 1610.25-0.024329-0.01239C-C.0061530.26-o.c21605-0.0125450.27-0.017303-0.0102010.2e-0.012141-0.0063150.24-0.006733-0.0018760.3C-0.001552 0.0022980.31 0.003071 0.0056590.32 0.006944 0.0079340.33 0.004986 0.0090850.34 0.0122cO0.35 3.0136530.36 0.0144440.37 0.0146930.3e 0.0145200.3S 0.0140410.4C 0.0133560.41 0.0125520.42 0.0116980.43 0.0108470.44 0.01C0400.45 0.0093030.46 0.0086560.47 0.0081080.48 0.0076620.4S 0.0073190.5C 0.0070730.510.520.530.540.55

Tables of this function have been given previously.6

In Tables Ia and b we give considerably more exten-sive and precise tables, computed from Eqs. (11) and(12), respectively, and correct to six decimal places.*

The last column in Table Ha gives the second differ-ence (central, for the last entry in the row). This isuseful in accurate interpolation.

Appendix. Error Estimates

A. Part of Integral Evaluated by Simpson's Rule

The error in Simpson's rule integration is less thanE < nhffmaxIv(x)/180, where n is the number of pointsused in applying Simpson's rule, hi is the interval be-tween these points, and fmaxlV is the maximum valueattained by the fourth derivative of the integrand inthe range of integration.

* Incidentally, for t appreciably greater than 8, the partialsums in Eq. (11) take on large negative values before they con-verge to a small positive value. For example, with t = 20(x,= 3.18) the partial sum becomes -4 X 104 before it begins to in-crease. It was therefore necessary to use double precision to

maintain six-place accuracy in the latter part of Table Ia.

Referring to Eq. (5), we find that the fourth deriva-tive of the integrand will have a term 15 cos a(x -v)

(1 - x2)- K 15(2)- (1 - x)-, all other termsbeing of higher order in (1 - x2). For very smallvalues of (1 - x), all other terms will be negligible,provided a << (1 - x2 ) -1. Writing e for 1 - x, we findfmaxIV(X) < 1.4 e-1. Taking n = 1600, h = 1/1600,and e = 0.005, we find, from Eq. (6), that the error isless than 1.4 X 10-7.

B. Part of Integral Evaluated by Series Expansion

The part of the integral evaluated by series expansionis

AT J f (1x- t cos[a(x - V)]dx.7r1-

On substituting u = 1- x and expanding, this becomes

AT = f (u)- (2 - u)[cosa(1 - v) cosau7r +

+ sina(1 - ) sinau]du.

970 APPLIED OPTICS / Vol. 7, No. 5 / May 1968

Table Ila. Line Spread Function of Per- x 7

fect Lens0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

0.00 .0000CCC.01 .999731]0.02 .99894E0.03 .9976340.04 .9957910.05 .993434C.06 .990564

0.07 .9871710.08 .98327]C.09 .978875

0.100. 110.120. 130. 140. 150. 160. 170.18

0. 19

973976.96858 5

.962 71 1.956 360.949541.942264.934538.926373.917 78 1.90877 1

0.20 .8993570.21 .8895500.22 .8793640.23 .8688110.24 .8579050.25 .8466600.26 .8350900.27 .8232110.28 .8110370.29 .798584

0. 300,. 310.32

0. 340.350 .36C.37C.380.39

The actual displacement x is given by x =FX'x,, where F is the effective f/number inthe image space, X' is the wavelength in theimage space, and x, is the value given inTable II. Beyond x = 50, the following ap-proximation will yield results accurate toabout two units in the sixth decimal place:L(x) = 3/(43.2x2)= 0.076/X2.

785867.772902.759704.746291.732679. 718 883. 704921.690 809.676 564.662202

0.40 .6477400.41 .6331950.42 .6185830.43 .6039200.44 .589223C.45 .5745086.46 559791C.47 .5450860.4-8.530411C.49 .515780

0. 500. 510.520 .530. 540 .5 56. 560 .570.5 80. 59

.50120 8.48 67 10.472299.45 7990.44379 7.429732.415 809.402039

.388435

.3 75008

0.60 .3617700.61 .3487300.62 .3358980.63 .3232860.64 .3109000.65 .2981500.66 .2868440.67 .275189.1.68 .2637930.69 .252660

0.70 .2417990.71 .2312130.72 .2209070.73 .21 08860.74 .20 1153C.75 .117110.76 .1825630.77 .173711C.78 .1651570.79 .15 6901

0.800.810 .8 20 .830. 840.8 50. 860. 870.8 80).89

0. 900. 910.920.9 30. 940. 950. 960.9 70. 980. 99

148943.141285.133 92 5.126 86 2.120096.11 362 3. 10 7442.10154 9.09594 3.0906 18

.08 55 71.080 798.07629 3.072052.0680 70.064 340.060858.057615S.054607

.05 1826

.999997.999682.998840.997473.99558499 3174990248

.986808.982859.978408

.973459

.968020.96209 7.955699.948834.941 512.9 33741.925533.916898.907848

.898 394.888 548.878325.867736.8 56795.8 455 17.83 3916.82 2007.809804.797 324

.78458 177 15 92.75 8373.74493 97 3 130 7

.71 7494

.70 3 516.6893 90.6 7513 3.660760

.64 6289.63 1 736.61 711960 2452

.5877525730365 5 8319.54 36 17.528946.514 320

.499755.48 5264.47086 3.456565442384.428333.41442 5.40067 1.387084.373676

360457.347437334627.322037.309674.297548.285667.274038.262667.2 51562

240728.2 30169.219 89 2.209899.20 019 5.190 7 83

.181664.1 7284 2.164 318.15 6091

.14 8164.140 536.13 320 5.12 617 2.11 943 5. 11299 2.106840. 1030976

.095397

.090101

.08 508 1

.080335.075857.0 7164 3.067686.06 398 1.060523.057304.054 319.05 1560

999 98~.99962].998 72]99730].99 536].99290 1.98992]98643e98243].97 7931

.972933:9674 49.96 147 8.95503 3.948 12 2.94075 5932940.924689

.9 160 12

.906920

.897427.88 7543.877282866657.85 5682.844 37 1.832 739

.820 80080856979606 1

783293.770280757039743584729934.71610 4.70 211 0

.687 7970.673 700.659 3 17

.644837

.63027 7615654.60098 35 862 81

.57 156 4.5 56848.542 149.52 748 1.512 86 1

.498 302

.48 3820

.469429

.45 5 14 2

.440973

.426936

.4 1304 2.399305

.385 735

.372345

.359 14 5

.34 6146.333358

3 20790.30 845 1

27889

.26 154 5

.2 50466

.239659

.229 12 9

.2 18 880.20 8916.19 924118985 8

I18 0769I171976.16 348 115 528 5

* 14 738 8.139 789.132489.12 548 518 7 78

1 1236 310 6240100C405

09485508 9586

0845940798750754240 7123 506730406362406019005699505403305 1296

.999~176999555.99860999713 7.99 5144.992630.98960 1.986 059.982 009.977458

972410.966873.96085 5954363947406.93999 3.932 13492 3840.91 512 1905989

.896455.88653 3.876236.865575.8 54566.84322 2a8315 59

.819 58 9.807330794796

.78200 3

.768966.755702.74222 8728559.71471 1

.70070 2686549.672267.657873

.64338 5.628 1 8.61418 9.5995 145848105700D92

.555377

.540680

. 5260 17

.5 1140 2

.496850

.482377

.467995

.4 5 37 19

.43956 3

.425540.4 1166 1.397940.384388.37111 6

.357836

.344858

.33209 1319546.307230.29 5 152.28 332 12 717 4 3.260425.249373

238594.22 809 1.2 17870.207935.1982 89.18893 517987 6

.17111 3

.16264 8

.15448 2

146614.13904 6.1317 7 5.12480 1

.11812 3

.I11173 8

.10 5644

.09 9836

.0194 316089075

084110.0 794 18.074993.0 7083 1.06692 5.063270.059860.056688.053749.0 5103 5

-5-5-5

-5

-5-5-4-5-4

-4-4-4-4-4-4-4-4-4-3

-3-3-3-3-3-3-2

-2-2

-2-2-1-1-1-1-1-1-1-0

-0-0-0

-000000

3

222

22

2

2

2

2

22222

22222

2222

2222

2

049760 2.049512

May 1968 / Vol. 7, No. 5 APPLIED OPTICS 971

.99995(

.99948 '.99848 1.99696].9949 1.992 3 5989263.9 856.7].98 1 57]e.976971-

.97 1876.9 6629 3.96022 7953688.9466851939227931 1324,.922987

.9 14226.9050 53

.89548 1.885520.8 75 186.864490.853447.8420 70830375

.8 18 3 76.80608 9.793 528

.780 710.76 7650754364.740 869.72 7 182.71 331 7699293

.68 5 126

.670 832

.65642 8

.64 193 1.627357.6 12 72 3598045

.5833 38

.568620

.553906.5392 12.52455 3.509943

.495399

.4809 34.466 562.452 298.438 15 5.424 14 5.41028 2.396577.383042.369690

.356 529343572.330827

. 3 18 30 3

.306011.29 3958.282 1 52.270599.259308.24828 3

.23 7 53 1.22 7056.2 16864206958.1 9 734 1.1880 16.17898 7.1702 5416 1818.15 368 2

.14 5844*13 830 513 1064.124 120.1 17471.1 1 116.1050 50.099273093779

088566

0836290789640740,650 7042 9066548062918059532056384~053468050776

3.999934.:999408.998356.996780994683.99 2066".988933.985289.98 1139976488

97 1 342.965708.959594.953009.94 5960.938457.930 510.92 2 130.91 332 7.904 114

.894 502

.884504.8 74 132.86 3401.8 52324.8409 15.82 9189.81 7160.804 84 5.792258

.7794 14.766 33 1.75 3024.739509.725803.711922.69 7883683702

.669 396

.65498 3

.64047 7625896.61 12 57.596575.58 1867.56 7149.552436.537744.52 3089.508486

.493949

.479492.46 5131.450878436748422752408904.39 5216.38 1699.368365

.355224342287.329564.3 17064.304795292766.280985.269458.25 8193.24719 5

.2 3647 1

.226024

.21 5860

.205983.196 39518 7100.78100.169 397.163 991.152 885

.14 5077

.13 7568

.130 3 57

.12 3442.116823.1I10496.104460.0981711.09q3 245

.088060

.08 31500 7 8 512.0 74 140.0 702 9.066 174.062568.059207.056082.053 18 8.050 518

.99990'.999 32].99822;.996 59'.99444'991 1771.98859;.98489,98069].9 75991

.970 80C.96 5 1 1 .95895].952 32494523C.9 37682.92 969 1.92 126891242A

.903 170

.893 514.883483.873075.862 309.8 51 19 8.839756.82 7999.81 5941

.803598

.790984

.7 78117

.76 5010751681.7 3814 7.72442 2.710 5259.696471682277

.66 7960.653536

639022.6244 35.609 790.59 5 10 5.580395.565677.550965.5362 77.52 1626.507029

.492 4994780 52.46 3701.449459.435342.42 1360,407528.393856.380357.36 7042

.35392134100 5.328 3043 1 582630 35 1291 57 72 7982 126832025 708 124611 1

2 354 14224995214 86020 501119 545218618 71 772 1616 854 3160 16 71 52090

144 31213 683 312 965 212 276 71 16 17 71098 7910 3 87209 815 10 92 714087557

082b740780630 73 7 17069632065802062 22 10588840557820 5291 1050264

5.999871.:999240998083996402.994201.991480

1 .988245F.984499.9802485.975498

I .970254I .964524.9 5831 5.9 51636I.944495

.936902

.928868

.920403

.911 51 7I.902223

I.892533

*.882459.872014.8612 138 50068

.838595

.826807

.8 14 720

.802 349

.789709

.7768 16

.763687

.750337

.736783.72 3040.709126.695057.680850.666 522.652088

.637566

.622 972

.608 323

.593635

.578923

.564205

.549495. 534810.5 20164.505573

. 49105 1

.4 766 12

.462 271

.448042

.433937

.419970

.40615 3

.392498

.3790 17

.36 572 1

.352620

.339725

.327046

.314 591

.302370

.290390

.278659.26 7184.255972.245028

.234359

.223969

.2 13862

.204042.194 513I18 52 76

.1 76 33 5

.16 7692

.15 9346

.1 512 99

.1435 51

.1 361021 2 89 5012 2 095

.115 53 4109266.1032 87.097595.092 186.087056

0 82 20 1. 07 7616.073297.069238.065433.06 1 877.058563.055485.052637.0 500 11

.99983;'.999 14].9979399620.99395;.9911 8].98 789].98409:.97979e.97499!

.969 70]96392'&

.95766].950942.94375].93611C.92 804 1.9 19 533.910606.9012 72

.891 542

.88143 1

.87095SC

.860113

.848935.837430.82 5611.81349 5.80 1096.78843 1

.77 5 514.762362.748990.7 354 17.72165670.7 72 6

.69364 3679423665083650640

636110.62 1510

.6068 56

.592 164.577452.562733.54 802 5.533343.5 1870 2.5041 17

.489603.4 75 17 3.460843.446626.4325344 18 581404 780.391 142

.364402

338448-3257903 13 35 8301 1612892052775002660 5125486 5243949

2333072229452 1286 720307619 3 576184 3691 754 58166,344158 52 81 5051 1

142 79313 537 312 825 11 2142 51 14 894108 65 5102 70 50 970410 91660086558

0728800688460650660615 3 505824505 5 190052364

1 9997871 .999050

.997789996t.04993698990874987537

1 .983689.979338.974488

.969146.9633209571016950244

.943C,1 2

.927209

.9186 59

.909690900 316

.890548

.88J3998698828 5901 1847799.836262.824413.812 26 8799842

.7 871 50

.774209

.7 6 134

.747642

.734049.720270.706324.692227.6779)94.663643.649 190

.634653

.62004 7605388.590694

.57598056 1262

.54t,5 56

.5 17241.502662

.488156.473 736.4594 16.445 2 1.43 1132.417 194.40 3409.389788

.376343.36 3085

350024.3 37172.324537.312128.299954.2 88023.276343.264920.253762.242873

232259.22 192 5.2118 75.202 1 13.192 642.18346 5.7458 3165999.157 7 13.149726

.14 23 7134648

.12 7555

.120 759

.114 25 7

.108047102 12 6

_96491.09113 8.08606 3

.0 812 63.076732.072465.066457.064702.061195.05 7929.054897.052094

Table Ila. Line Spread Function of Per- Xr 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

fect Lens (continued) I 100 .049265 .049021 .048779 .048539 .048 301 .048066 .047832 .047600 .041 371 .04 7143 2

1.01 .046918 .046695 .046474 .046255 .046037 .045822 .045609 .045398 .045189 .044982 11.02 .044777 .044574 .044373 .044174 .043977 .043782 .043588 .043397 .043208 .043020 11.03 .042835 .042651 .042469 .042290 .042112 .041936 .041761 .041589 .041419 .041250 11.04 .041083 .040918 .040755 .040594 .040435 .040277 .040121 .039967 .039815 .039664 11.05 .039516 .039369 .039223 .039080 .038938 .038798 .038660 .038523 .038388 .038255 11.06 .038124 .037994 .037866 .037739 .037614 .037491 .037370 .037250 .037131 .037015 1

1.07 .036900 .036786 .036674 .036564 .036455 .036348 .036242 .036138 .036036 .035935 11.08 .035836 .035738 .035641 .035546 .035453 .03 5361 .0352 71 .03 5182 .035094 .035008 1

1.09 .034924 .034841 .034759 .034679 .034600 .034522 .034446 .034372 .034299 .034227 1

1.10 .034156 .034087 .034019 .033953 .033888 .033824 .033762 .033701 .033641 .033582 111.033525 .033469 .033414 .033361 .033309 .033258 .033209 .033160 .033113 .033067 1

1.12 .033022 .032979 .032937 .032895 .032855 .032817 .032779 .032743 .032707 .032673 1113 .032640 .032608 .032578 .032548 .032519 .032492 .032466 .032440 .032416 .032393 1

1.14 0 32 3 71 .032350 0330.211 0293.032276 .032260 .032245 .0 322 31 .032218 0

1.15 .032207 .032196 .0 3 218 6 .0 32 17 7 .0 32 168 .032161 .0 32 155 .032150 .032146 .032142 0

1.16 .032140 .032138 .032137 .032137 .032138 .032140 .032143 .032147 .032151 .032 156 01.17 .032162 .032169 .032 17 7 .03218 6 .032195 .032205 .0 32 216 .032228 .032240 .032254 0

1.1 .032268 .032282 .032298 .032314 .032331 .032349 .032367 .032386 .032406 .032427 01.19 .032448 .032470 .032492 .032516 .032539 .032564 .032589 .032615 .032641 .032668 0

1.20 .032696 .032724 .032753 .032783 .032813 .032843 .032874 .032906 .032939 .032971 01.21 .033005 .033039 .033073 .033108 .033144 .0 3 3180 .0 33 2 17 .033254 .0 3 3291 .033329 0

1.22 .033368 .033407 .033446 .033486 .033526 .033567 .033608 .033650 .033692 .033735 01.23 .033778 .033821 .033865 .033909 .033954 .033998 .034044 .034089 .034135 .034182 01.24 .034229 .034276 .034323 .034371 .034419 .034467 .034516 .034565 .034614 .034664 01.25 .034714 .034764 .034814 .034865 .034916 .034967 .035019 .035071 .035123 .035175 01.26 .035227 .035280 .035333 .035386 .035439 .035493 .035546 .035600 .035654 .035709 01.27 .035763 .035818 .035872 .035927 .035982 .036038 .036093 .036148 .036204 .036260 01.28 .036316 .036372 .036428 .036484 .036540 .036597 .036653 .036709 .036766 .036823 01.29 .036880 .036936 .036993 .037050 .0 3 710 7 .037164 .0 3 722 1 .037278 .037335 .037392 0

1.30 .037449 .037507 .037564 .037621 .037678 .037735 .037792 .037849 .037906 .037963 -01.31 .038020 .038077 .038134 .038191 .038248 .038305 .038362 .038418 .038475 .038531 -01.32 .038588 .038644 .038700 .038757 .038813 .038869 .038924 .038980 .039036 .039091 -01.33 .039147 .039202 .039257 .039312 .039367 .039422 .039477 .039531 .039586 .039640 -01.34 .039694 .039748 .039801 .039855 .039908 .039961 .040014 .040067 .040120 .040172 -01.35 .040224 .040276 .040328 .040379 .040431 .040482 .040533 .040584 .040634 .040684 -01.36 .040734 .040784 .040834 .040883 .040932 .040981 .041030 .041078 .041126 .041174 -01.37 .041221 .041269 .041316 .041362 .041409 .041455 .041501 .041546 .041592 .041637 -01.38 .041681 .041726 .041770 .041814 .041857 .041900 .041943 .041986 .042028 .042070 -01.39 .042112 .042153 .042194 .042235 .042275 .042315 .042354 .042394 .042433 .042471 -0

1.40 .042510 .042547 .042585 .042622 .042659 .042696 .042732 .042767 .042803 .042838 -01.41 .042873 .042907 .042941 .042974 .043008 .043040 .043073 .043105 .043136 .043168 -01.42 .043199 .043229 .043259 .043289 .043318 .043347 .043376 .043404 .043431 .043459 -01.43 .043486 .043512 .043538 .043564 .043589 .043614 .043638 .043663 .043686 .043709 -01.44 .043732 .043754 .043776 .043798 .043819 .043840 .043860 .043880 .043899 .043918 -01.45 .043936 .043954 .043972 .043989 .044006 .044022 .044038 .044054 .044069 .044083 -01.46 .044097 .044111 .044124 .044137 .044149 .044161 .044173 .044184 .044194 .044204 -01.47 .044214 .044223 .044232 .044240 .044248 .044255 .044262 .044269 .044275 .044280 -01.48 .044286 .044290 .044294 .044298 .044302 .044304 .044307 .044309 .044310 .044311 -01.49 .044312 .044312 .044311 .044311 .044309 .044307 .044305 .044303 .044299 .044296 -0

1.50 .044292 .044287 .044282 .044277 .044271 .044265 .044258 .044251 .044243 .044235 -o1.51 .044226 .044217 .044207 .044197 .044187 .044176 .044164 .044153 .044140 .044128 -o1.52 .044114 .044101 .044086 .044072 .044057 .044041 .044025 .044009 .043992 .043975 -01.53 .043957 .043939 .043920 .043901 .043881 .043861 .043841 .043820 .043799 .043777 -01.54 .043755 .043732 .043709 .043685 .043661 .043637 .043612 .043586 .043561 .043534 -0

1.55 .043508 .043481 .043453 .043425 .043397 .043368 .043339 .043309 .043279 .043248 -01.56 .043217 .043186 .043154 .043122 .043089 .043056 .043022 .042988 .042954 .042919 -01.57 .042884 .042848 .042812 .042776 .042739 .042702 .042664 .042626 .042588 .042549 -01.58 .042509 .042470 .042430 .042389 .042348 .042307 .042265 .042223 .042181 .042138 -01.59 .042094 .042051 .042007 .041962 .041917 .041872 .041827 .041781 .041734 .041688 -0

1.60 .041641 .041593 .041545 .041497 .041449 .041400 .041350 .041301 .041251 .041200 -01.61 .041150 .041098 .041047 .040995 .040943 .040891 .040838 .040785 .040731 .040677 -01.62 .040623 .040568 .040513 .040458 .040403 .040347 .040290 .040234 .040177 .040120 -01.63 .040062 .040004 .039946 .039888 .039829 .039770 .039710 .039651 .039591 .039530 -0

1.64 .039470 .039409 .039347 .039286 .039224 .039162 .039099 .039037 .038974 .038910 -01.65 .038847 .038783 .038719 .038654 .038590 .038525 .038459 .038394 .038328 .038262 -01.66 .038196 .038129 .038062 .037995 .037928 .037860 .037793 .037724 .037656 .037588 -o1.67 .037519 .037450 .037380 .037311 .037241 .037171 .037101 .037030 .036960 .036889 -01.68 .036818 .036746 .036675 .036603 .036531 .036459 .036387 .036314 .036241 .036168 -0

1.69 .036095 .036022 .035948 .035874 .035800 .035726 .035652 .035577 .035503 .035428 -o

1.70 .035353 .035277 .035202 .035127 .035051 .034975 .034899 .034823 .034746 .034670 -01.71 .034593 .034516 .034439 .034362 .034285 .034207 .034130 .034052 .033974 .033896 -01.72 .033818 .033740 .033662 .033583 .033505 .033426 .033347 .033268 .033189 .033110 -01.73 .03 30 31 .032951 .032872 .032792 .0 3 271 3 .032633 .032553 .032473 .032393 .0 32 313 '-0

1.74 .032233 .032152 .032072 .031992 .031911 .031830 .031750 .031669 .031588 .031507 -01.75 .031426 .031345 .031264 .031183 .031102 .031021 .030939 .030858 .030777 .030695 -01.76 .030614 .030532 .030451 .030369 .030288 .030206 .030124 .030043 .029961 .029879 -01.77 .029798 .029716 .029634 .029552 .029471 .029389 .029307 .029225 .029143 .029062 01.78 .028980 .028898 .028816 .028734 .028653 .028571 .028489 .028407 .028326 .028244 01.79 .028162 .028081 .027999 .027918 .027836 .027755 .027673 .027592 .027510 .027429 0

1.80 .027348 .027266 .027185 .027104 .027023 .026942 .026861 .026780 .026699 .026618 01.81 .026537 .026457 .026376 .026296 .026215 .026135 .026054 .025974 .025894 .025814 01.82 .025734 .025654 .025574 .025495 .025415 .025335 .025256 .025177 .025097 .025018 01.83 .024939 .024860 .024781 .024703 .024624 .024546 .024467 .024389 .024311 .024233 01.84 .024155 .024077 .023999 .023922 .023844 .023767 .023690 .023613 .023536 .023459 0

1.83 .023383 .023306 .023230 .023154 .023078 .023002 .022926 .022850 .022775 .022700 01.86 .022625 .022550 .022475 .022400 .022326 .022251 .022177 .022103 .022029 .021956 01.87 .021882 .021809 .021736 .021663 .021590 .021517 .021445 .021372 .021300 .021228 01.88 .021157 .021085 .021014 .020943 .020872 .020801 .020730 .020660 .020590 .020520 0

1.89 .020450 .020380 .020311 .020242 .020173 .020104 .020035 .019967 .019899 .019831 0

1.90 .019763 .019696 .019628 .019561 .019494 .019428 .019361 .019295 .019229 .019163 01.91 .019097 .019032 .018967 .018902 .018837 .018773 .018709 .018645 .018581 .018518 01.92 .018454 .018391 .018328 .018266 .018204 .018141 .018080 .018018 .017957 .017895 0;1.93 .017835 .017774 .017714 .017653 .017593 .017534 .017474 .017415 .017356 .017298 01.94 .017239 .017181 .017123 .017066 .017008 .016951 .016894 .016838 .016781 .016725 01.95 .016669 .016614 .016558 .016503 .016449 .016394 .016340 .016286 .016232 .016179 0

1.96 .016125 .016072 .016020 .015967 .015915 .015863 .015812 .015760 .015709 .015659 01.97 .015608 .013338 .015308 .015458 .015409 .015360 .015311 .015262 .015214 .015166 01.98 .015118 .015071 .015023 .014976 .014930 .014883 .014837 .014792 .014746 .014701 0

1.99 .014656 .014611 .014567 .014523 .014479 .014435 .014392 .014349 .014306 .014264 0continued

972 APPLIED OPTICS / Vo. 7, No 5 / May 1968

Table Ila. Line Spread Function of Per- XI. 0.000 0.001 0.002 0.003 0.001 0.005 0.005 0.007 0.003 0.0098I L ens continuea)

2.002.012.022.032.042.052.062.072.082.09

.01422.01381.01343.01308.01276.0 1247.01221.01197.01176.01158

2. 0 .011422.11 .01 1292.12 .011182. 13 .01 1102.14 .01104'2.15 .01101.2.16 .010992.17 .01100;2.18 .011022.19 .01106'

2.202.212.222.232.242. 252.262.272.282.29

.01112i.01120:.01 129:.0 11393.0115 1;.011633.0117 7'.011913.012071.0 12223

2.30 .0123932.31 .0125512.32 .01272!2.33 .01289!2.34 .0130642.35 .0132332.36 .0133942.37 .0135622.38 .0137202.39 .013874

2.40 .0140212.41 .0141612.42 .0142942.43 .0144182.44 .0145332.45 .0146372.46 .0147312.47 .0148142.48 .0148862.49 .014945

2.50 .0149922.51 .0150262.52 .0150482.53 .0150562.54 .0150512.55 .0150322.56 .0150002.57 .0149552.58 .0148962.59 .014824

2.60 .0147402.61 .0146422.62 .0145332.63 .0144112.64 .0142772.65 .0141322.66 .0139762.67 .0138102.68 .0136342.69 .013449

2.70 .0132552.71 .0130542.72 .0128442.73 .0126282.74 .0124062.75 .0121782.76 .0119462.77 .0117102.78 .0114702.79 .011228

2.802.812.822.832.842.852.862.872.882.89

2.902.912.922.932.942. 952.962.972.982.99

.010984.0 10739.0 10493.0 10248.0 10004.009 761.009520.009283.009049.0088 19

.0085 94

.008 374

.008 160

.007952

.007 751

.007557

.007371

.007 193

.007023

.006861

2 .014186 .013778 .013409 .013058 .012736 .012441 .012185 .011955 .011743 .01156

6 .011415 .011289 .01 1187 .01 1109 .011042 .01 1017 .010992 .011006 .01103i9.01 107-

3 .01 113'3 .01121;3 .01130:6 .0 1140'2.01152'3 .01165:.011781.01193'L.012083 .01224!

.012401,

.0 12 57'1 .01274;i.01291;,.013081I .01324C

I.01341!.01357E

I .01373e*.01388S

.01403!.0 1417!*.014301.01443C.014543.014647.014 740.0 14 822.014 892.0 149 50

.0 14996

.0 15029

.0 15049

.0 5056

.0 15049

.0 15029.014996.014 950.0 14 890.0 14817

.014731

.0 14 632

.014 521

.0 14398

.014 26 3

.0 14117

.013 960

.0 3793.0 13616.0 13430

.0 13236

.0 1303 3

.0 1282 3

.0 12606

.0 12 38 3

.012 15501 192 3.011686.0 11446.01 1204

.0 10960.0 107 14.0 10469.0 10224.009979.009737.009496.0092 59.00 9026.00 8796

.008 571

.008352

.008 139

.007932

.00 773 1

.007538

.00735 3

.007175

.007006

.006846

0 .0141386 .0137382 .013 3666 .01 302 38 .01270 88 .0124217 .0121623 .01 193 160 0117276 .01 1549

2 .01 13984 .0112720 .01 117 11 .01 10944 .0110400 .0110076 .0109963 .0110050 .01 103 34 .011079

5 .0111422 .01 12203 .0113137 .0114194 .0 115361L.01 16653.01 1034.01 1949.-012102

3 .012261

r.012424.012590012759.012929.013098

3.0132663.0134321 .013594

.0137513.013904

1.0140501.014189.014319I.014442.014554.01465 7

I.014749

.0148 30

.0 14899

.0 14 956

*.015000.0 1503 2.0 150 50.015056.015048.01502 7.014992.014944.014 88 3.014809

.01472 1

.01462 1

.014509

.014385

.014249

.014102

.013944

.013 776

.013598.01341 1

.013216

.013012

.012802

.012 584

.012 361

.012 132

.01 1899

.011662

.01 142 2.01 1179

.010935

.010690

.010444

.010199

.009955

.009713

.009473

.009236

.009002

.008773 .

.008549 .3

.008331 .3

.008118 .

.00791 1

.00 7712 .

.007519 .3

.007335 .

.007158 .

.006990 .

.006830 .

.01409 7.013 69 9.0133 30.0 12990.0 12678.012 39 4.0 1213 8.0 11909.0117.08.0 11533

.011384.01126 1.01 1162.011087.0 1103 5.0 11005.0 1099 6.011007.0 11037.01108 5

.0 11149

.0 11229

.0 11323

.0 11430

.0 11549

.01 1678'.011817.011964.0 12117.0 12 277

.0 12440

.0 12607

.0 12776

.0 12946

.0 13115

.01328 3

.0 13448.0 13610.013 76 7.0 13919

.0 14064

.0 14202

.01433 2

.0 1445 3

.0 1456 5

.0 14667

.014 75 7

.0 1483 7

.0 14905

.0 1496 1

.0 150040 1503 4

.0 1505 1

.0 1505 6

.015046

.015024

.014988

.014939

.014876

.014800

.014712

.01461 1

.01449 7

.01437 2

.01423 5,01408 7.013928013759.01358001339 2

01319601299101278001256201233801210901187601163801 139 8011155

010911010665 t01042001017 5009931009688009449009212008979008 751 .

308527308309308097307891307692307500307316307141306973306814

.01405.0 1366.0 1329.0 1295.0 1264.012 36.0 1211.0 1188.01168.01151

.0 1137

.0 1125.0 1115.0 1108.0 1103.0 1100.0 1099.0 1100.0 1104.0 1109

.01115

.01123

.01133.0 1144.0 1156.0 1169.0 1183.0 1197'.0 12 13:.0 1229:

.012457

.0 1262'

.012 79:

.012961

.0 13 13;

.01329'

.0 1346'

.0 1362].013 78.01393--

.014073

.0 1421!

.0 1434!

.0 1446!

.0 145 7]

.0 1467]

.0 14 766

.014844

.0 14911

.014965

.0 15003

.0 150 36

.0 15053

.0 15055

.0 15045

.015021

.0149 84

.0 1493 3

.0 14 869

.0 14 792

.0 14702.014 600.0 14485.0 14 359.0 14220.0 14071.01391 1.0 13 741.013561.0 13 373

013176012971012759012540012316012086011852011614011374011131

010886010641010395010150009906009664009425009189008956008728

.008 505.008287.008076.007871.007672.00 7482.007299.007 124.006957.006799

6 .0140151 .01 36235 .013260.7 .0129258 .0126197 .0123403 .0120908 .0118679 .0116717 .011501

1 .01 13580 .0112394 .01 11451 .0110751 .0110283 .0110026 .0109979 .0110121 .01 10450 .01 1096

6 .01 11647 .01 12463 .0113431 .0114531 .0115742 .0117051 .0118463 .0119943 .0121493 .012309

1 .0124744 .0126413.012810

3 .0129793 .0131493 .0133164 .01 3481

.013642

.0137983 .013948

3 .014092i.0142293 .0143571 .014476

.014586

.014686

.014774*.014852.014917.014970

.01501 1*.015039*.015053*.01505 5.0 15043.0 150 18.014979.014927.014862.014784

.014693

.014589

.014473

.014345

.014206

.0140 56

.013895

.01 3724.013543.01 3353

.013155

.012950

.012 737

.012518

.012293

.012063

.011828.011590.01 1350.011106

.010862

.010616

.010371

.010126

.009882

.009640

.009401

.009165.008933.008706

.008483

.008266

.008055 .

.007851

.007653

.007463

.007281

.007106 .3

.006941 .3

.006784 3

.013975 .01392.0 13 58.013 22.012 84.0 12 58.0 1231.0 1206.0 1184.01165.0 1148

.01134

.0 1122

.01113

.01106

.0 1102

.0 1100

.0 10 99

.0 1101

.01 105

.0 1110

.01117

.01125

.011 35.0 1146.01158.01171.0 1186..012 00'.0 12 16.0 1232

.0 12 493

.012651.0 12 82'.01299i.01316'..01333:

.0 1 349'

.01365:

.0 1381:

.01396]

.014103

.0 14 24;

.01436

.0 14483

.0 14 59].

.0 1469!

.0 14 782

.01485S

.0 14 923

.014 97 5

.0150 14

.01504 1

.015 054

.0 150 54

.015 04 1

.015015

.014 97 5

.014921

.0 1485 5

.014 77 5

.0 1468 3

.0 14 578

.0 1446 1

.0143 32

.0 14 191

.0 14040

.0 138 78

.013 706

.013 524

.0133 34

.013 13 5

.0 12 929012 71501249601227001204001180 5011566011325011082

01083 7010592010346010101009858009616009377009142008910008683

008461008245008034007830D07634007444D0726330709030692 5306769 .006753 .3306738

1603486 .:01354813.01286119 .012 5614 .0122886 .0120436 .0 1182 53 .01163 56 .0 11471

5 .01 13329 .01 12197 .0111309 .0110644 .0 110210 .0109998 .0109984 .0 110170 .0110542 .011108

1 .0 111795 .01 12654 .01136.44 .0114766 .01 15999 .0 1173 30 .01 18759 .0120245 .0121815 .012342

3 .012 5073 .0126751 .012844S .01 30135 .01 31823 .0133491 .013513F.0136733.01 3828

3 .013978

.014120

.01425 53.014382

3 .0144991.0146073 .0147041.014 7913.014866

1 .0149293.014979

.01 5017

.0 15043

.015055*.015054.015039

*.015011*.014970.014915.014847.0 14 767

.0 14673

.014567

.014448

.014318

.014177

.014024

.013861

.0 1368 8

.013506

.0 13 314

.013115

.012908

.012694.012473.012247.012016.011781.011542.011301.011058

.010813

.010567.010322.010077.009833.009592.009354.009119.008887.008661

.008439 .

.008223 ..008014 ..007810 ..007614 ..007426 .3.007245 .3.007073 .3.006909 .3

01385.0 135 1.01315.0 12 80.0 12 SO.0122 6.0 1202.01180.01161.01145

.0 1132

.0 1120

.01112

.0 110 5

.0110 1.0 10 99.0 1099.0 1102.0 105.011 11

.01118

.0 112 7

.0 113 7

.0 1148

.01161

.011 74

.0 1188.0 1204.01219,.0 12 3 Si

.0 12 52:~

.0 1269:

.0 12 86:

.0 130 33

.0 13 19'

.01336(

.013 52'

.0 1368'

.0 1384]

.0 1 399;

.01413'

.0 142 63

.0 143 9'

.0 14 5 IC

.01461],

.0 147 1]-

.01479!

.0 14 87

.014934

.014984

.0 15021

.015044

.0 1505!5

.0 15053

.0 15037

.0 15008

.0 1496 5

.0 14 909

.0 14 840

.0 14758

.014 663

.0 145 56

.0 144 36

.0 14 305

.0 14 162

.0 14 008

.0 13 844

.0136 70.013487.0132 95

013095012 887012 672012451012224011993011 757011 51801 12 77011033

010788010543010297010052009809009568009330009095008864008638

D08417008202307993007 791307595307407307228307056306893

34 .0138551 013474i6 .013 12330 .01279912 .012 504'2 .01223 70 011997IS .01 17857 .0116006 .011441

0 .0113079 .0111992- 0111159 .01 10538 .0110158 .0109979 .0110010 .01102 39 0110645 .011121

7 01 11954 .0112835- 0113858 .0115002- 01162 56 .011 7609 .0119040D.0120555 .0122128 .012 374

3 .0125401 .012 7081L.0128783 .0130470 .0132165.013382

3 .0135453.0137053.013859.0 14007

4 .0141483 .014281.014406

3.014521.014627

3.0147223.014807.014879.014940.014988

015024*.015046*.015056*.015052.015035*.015004.0 14960.0 14903.0 148 32.0 14749

.0 14653

.0 14 544

.0 14423

.014291

.0 14147

.0 13992

.0 1382 7

.0 13652

.0 13468.0 13275

.0 13074

.012 865

.0 12650

.0 12428

.012201

.0 11970

.0 11734

.011494

.0112 52.011009

.0 10764

.0 105180 10273010028.009785.009544.009306.009072.008842.0086 16

.008396

.008181

.007972

.00 7771

.00 7576

.007389

.0072 10

.007039

.006877

.006724 0

0000000000

0000000000

0000000000

000-0-0-0-0-0-0-0

-o-0

-0-0-0-0-0-0-0

-0-0-0-0-0-0-0-0-o-0

-0-0-0-0-0-0-0-0-0-0

-0-0-0-0-0-0-0-0-0-0

-0000000000

000000000

May 1968 Vol. 7, No. 5 APPLIED OPTICS 973

14

Table llb. Line Spread Function of Perfect Lens

Xr 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2. .014221 .011426 .011128 .012391 .014021 .014992 .014740 .013255 .010984 .008594 505

3. .006709 .005703 .005608 .006153 .006898 .007409 .007405 .006836 .005866 .004787 195

4. .003905 .003422 .003384 .003678 .004C89 .004391 .004424 .004146 .003635 .003048 9'5. .002556 .002284 .002267 .002447 .002702 .002897 .002932 .002775 .002469 .002110 546. .001805 .001635 .001626 .001746 C01917 .002051 .002081 .001984 .001785 .001547 337. .001343 .001228 .001224 .001309 .O,1430 .001527 .001552 .001487 .001350 .001183 228. .001038 .000957 .000955 .001018 .001108 .001181 .001201 .001155 .001056 .000933 159. .000827 .00076R .000767 .000814 OO83 .000939 .00U956 .000923 .000848 .000756 11

10. .000675 .000629 .000629 .000666 .000720 .000765 .000779 .000754 .000696 .000624 811. .000561 .000525 .000525 .000555 .000599 .000635 .000647 .000627 .000582 .000524 612. .000474 .000445 .000446 .000470 .000506 .000535 .000545 .000530 .000493 .000446 513. .000405 .000382 .000383 .000403 .000433 .000457 .000466 .000454 .000423 .003385 414. .000351 .000332 .000332 .005350 .000374 .000395 .000403 .000392 .000367 .000335 315. .000307 .000291 .000291 .000306 .000327 .000345 .000351 .000343 .000322 .000295 316. .000271 .000257 .000258 .0002 70 .000288 .000304 .000309 .000302 .000284 .000261 217. .000240 .000229 .000229 .000240 .000256 .000269 .004274 .000268 .000253 .000233 218. .000215 .000205 .000206 .000215 .000229 .000240 .000245 .000240 .000226 .000209 119. .000193 .000185 .000185 .000194 .000206 .000216 .000220 .000215 .000204 .000188 1

20. .000175 .000167 .000168 .000175 .000186 .000195 .000199 .000195 .000184 .000171 121. .000159 .000152 .000153 .000159 .000169 .000177 .000180 .000177 .000168 .000156 122. .000145 .000139 .000140 .000146 .000154 .000161 .000164 .000161 .000153 .000143 123. .000133 .000128 .000128 .000133 .000141 .000148 .000150 .000148 .000140 .000131 124. .000122 .000118 .000118 .000123 .000130 .000136 .000138 .000136 .000129 .000121 025. .000113 .000109 .000109 .000113 .000120 .000125 .000127 .000125 .000119 .000112 026. .000105 .000101 .000101 .000105 .000111 .000116 .000118 .000116 .000110 .000103 027. .000097 .000094 .000094 .000098 .000103 .000107 .000109 .000107 .000103 .000096 028. .000090 .000087 .000088 .000091 .000096.000100 .000101 .000100 .000095 .000090 029. .000084 .000082 .000082 .000085 .000389 .000093 .000095 .000093 .000089 .000084 0

30. .000079 .000076 .000077 .000079 .000083 .000087 .000088 .000087 .000083 .000378 0

31. .000074 .000072 .000072 .000074 .000078 .000081 .000083 .000082 .000078 .000074 0

32. .000070 .000067 .000068 .000070 .000073 .000076 .000078 .000077 .000073 .000069 033. .000065 .000063 .000064 .000,66 .000069 .000072 .000073 .OOOJ72 .000069 OOOJ65 034. .000062 .000060 .000060 .000062 .000065 .000068 .000069 .000068 .000065 .000062 035. .000058 .000057 .000057 .000059 .000061 .000064 .000065 .000064 .000062 .000058 036. .000055 .000054 .000054 .000056 .000058 .000060 .000061 .000061 .000058 .000055 037. .000052 .000051 .000051 .000053 .000055 .000057 .000058 .000057 .000055 .000052 038. .000050 .000048 .000048 .000050 .000052 .000054 .000055 .000054 .000052 .000050 039. .000047 .000046 .000046 .OOOC47 .000050 .000051 .000052 .000052 .000050 .000047 0

40. .000045 .000044 .000044 .000045 .000047 .000049 .000050 .000049 .000047 .000045 041. .000043 .000042 .000042 .000043 .000045 .000047 .000047 .000047 .000045 .000043 042. .000041 .000040 .000040 .000041 .000043 .000044 .000045 .000045 .000043 .000041 043. .000039 .000038 .000038 .000039 .000041 .000042 .000043 .000042 .000041 .000J39 '3

44. .000037 .000036 .000036 .000037 .000039 .000040 .000041 .000041 .000039 .000037 045. .000036 .000035 .000035 .000036 .000037 .000039 .000039 .000039 .000037 .000036 046. .000034 .000033 .000033 .000034 .000036 .000037 .000038 .000037 .000036 .000034 047. .000033 .000032 .000032 .000033 .000034 .000035 .000036 .000036 .000034 .OOOj33 0

48. .000031 .000031 .000031 .000032 .000033 .000034 .000034 .000034 .000033 .000031 0

49. .000030 .000029 .000029 .00;OC30 .000031 .000033 .000033 .000033 .000032 .000030 0

Expanding cos au and sin au in a Taylor series and C. Numerical Checkintegrating, we find

AT = - [cosa(l7r

.-P) E aA~,nn = +

+ sinal- ) :E a2n +,A2n+l ,n =

As a numerical check, T(0.5, 1/7r) = 0.379515 wasevaluated on a desk computer by means of Eq. (4) andwas found to agree exactly with the correspondingresult obtained on the computer.

Ao = 6 [(4/3)- (e/5) - (/56) - . . .1,

A, = e e[(2/5) - (e/14) - . .. 1,

112 = e '[(2/7) - . . .]

With e = 0.005, the last retained term is less than3 X 10-9!

References1. H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 91

(1955).2. R. Barakat and A. Houston, J. Opt. Soc. Amer. 54, 768

(1964).3. W. H. Steel, Opt. Acta 3, 65 (1956).4. W. H. Steel, Rev. Opt. 31, 334 (1962).5. G. N. Watson, Theory of Bessel Functions (Cambridge Uni-

versity Press, Cambridge, 1962), 2nd ed., pp. 328, 333.6. F. Kottler and F. H. Perrin, J. Opt. Soc. Amer. 56, 377 (1966).

Note that throughout their expression (30), (2z)"'2 should bereplaced by (16z)flO and that the exponent 1/2 was omitted inthe second factor (v/2).

974 APPLIED OPTICS / Vol. 7, No. 5 / May 1968

where