Nonparaxial model for the focusing of highnumerical-aperture

Transcription

1 Q. Cao and J. Jahns Vol. 0, No. 6/June 003/J. Opt. Soc. Am. A 1005 Nonparaxial model for the focusing of highnumerical-aperture photon sieves Qing Cao and Jürgen Jahns Optische Nachrichtentechnik, FernUniversität Hagen, Universitätsstrasse 7/PRG, Hagen, Germany Received November 1, 00; revised manuscript received February 13, 003; accepted February 0, 003 Recently, a paraxially individual far-field model was presented for the focusing and imaging analysis of pinhole photon sieves. By use of a local Taylor expansion of the integrated function of the Rayleigh Sommerfeld diffraction formula, the small-size property of the individual pinholes, and the linear superposition principle, we extend this model to the nonparaxial case of high-numerical-aperture photon sieves. Some related problems, such as the validity range of this nonparaxial model and the selection conditions for the individual pinholes, are also discussed in detail. 003 Optical Society of America OCIS codes: 0.560, , , , , INTRODUCTION The focusing and imaging of soft x rays and extreme ultraviolet (EUV) radiation have many applications in physical and life sciences, such as in high-resolution microscopy, spectroscopy, lithography, and in telescopes. Unfortunately, the traditional refractive lens cannot be used for this purpose because all solids are strongly absorbing in these spectral regions. The Fresnel zone plate can be used for this purpose. 1 However, the largest spatial resolution of a traditional Fresnel zone plate is about the order of the width of the outermost zone, 4 and is accordingly limited in technology by the smallest structure (0 40 nm) that can be fabricated by lithography. 5,6 To get higher spatial resolution, Kipp et al. 7 recently suggested a new kind of diffractive optical element called a photon sieve, which consists of a large number of pinholes properly distributed over the Fresnel zones, for the focusing and imaging of soft x rays. This kind of diffractive optical element potentially can also be used as an EUV telescope for a solar orbiter 8 or in other image-forming applications. It is well known that, for a traditional Fresnel zone plate, each open ring has a net positive contribution to the field value at the desired focal point. Similarly for a photon sieve, each pinhole has a net positive contribution to the field value at the desired focal point. This kind of collective behavior can greatly enhance the intensity at the desired focal point and lead to focusing. However, it should be emphasized that the success (we refer to the increase in spatial resolution and the suppression of sidelobes) of photon sieves does not come from this point but from the following two points. The first is the use of larger total size for the whole element, though this point is not clearly mentioned in the initial work. 7 As a check, from Fig. 4 of Ref. 7 one can see that the total diameter of the photon sieve is much larger than that of the traditional Fresnel zone plate. According to the Rayleigh resolution formula, the use of larger optical elements has the potential to produce higher spatial resolution. The second point is the use of smooth filtering for the population of pinholes. This kind of filtering can effectively suppress higher-order oscillations (i.e., sidelobes) around the main focal point. In the initial work on the photon sieve, 7 the authors employed the traditional Fresnel zone plate theory with numerical calculation of Fresnel Kirchhoff diffraction integrals to analyze and design the photon sieves. More recently, we presented the paraxially individual far-field model 9 for the focusing and imaging analysis of photon sieves. This model is derived from the paraxial Fresnel diffraction integral and is valid for low-n.a. photon sieves, such as the optical prototype of the photon sieve. 7 It is well known that the Fresnel diffraction integral ignores the fourth- and higher-order phase terms in the spherical phase factor. This treatment is valid in almost all cases for the visible spectral range. 10 However, as we show below, it is no longer valid for high-n.a. photon sieves working in the soft x ray region, because in this spectral region the wave number k is very large. As a conquence, the paraxially individual farfield model is also no longer valid for high-n.a. photon sieves working in the soft x ray region. In this paper, we shall extend the individual far-field model to the nonparaxial case of high-n.a. photon sieves and discuss some related problems. The basis for this generalization is the nonparaxial expression for the far fields of the individual pinholes and the linear superposition principle. Within the framework of scalar field diffraction, 11 this nonparaxial model is applicable to an arbitrary illuminative beam with an arbitrary complex amplitude distribution at the photon sieve plane. For simplicity, we shall limit our analysis to the performance of the individual pinholes and not discuss the statistical aspects of the photon sieves.. NONPARAXIAL MODEL FOR HIGH- NUMERICAL-APERTURE PHOTON SIEVES As shown in Fig. 1, a general photon sieve consisting of a great number of pinholes whose locations and radii are properly chosen is located at the xy plane, the desired focal point is located at the point (X 0, Y 0), and the /003/ $ Optical Society of America

2 1006 J. Opt. Soc. Am. A/ Vol. 0, No. 6/ June 003 Q. Cao and J. Jahns Fig. 1. Schematic view of a photon sieve illuminated by an arbitrary scalar field. distance between the xy plane and the XY plane is q. We denote by V(x, y) A(x, y)exp jkl(x, y) the complex amplitude distribution of the illuminative beam at the photon sieve plane, where k / is the wave number, is the wavelength, j ( 1) 1/ is the imaginary unit, L(x, y) is the eikonal, and A(x, y) is the real amplitude. According to the linear superposition principle, the total diffracted field U(X, Y) at the focal plane is the simple sum of those individual diffracted fields from different pinholes. That is to say, U(X, Y) N n 1 U n (X, Y), where U n (X, Y) is the individual diffracted field from the nth pinhole and N is the total number of pinholes. Consider the nth pinhole whose central location and radius are denoted by (x x n, y y n ) and a n, respectively. From the generalized Rayleigh Sommerfeld diffraction integral formula, 10 1 one can write the corresponding diffracted field U n (X, Y) as U n X, Y 1 V n x, y S n exp jkr 01 jk 1 r 01 dxdy, (1) where r 01 q (X x) (Y y) 1/, S n is the integral region of the nth pinhole, and V n (x, y) V(x, y) inside the pinhole. Equation (1) is considered to be exact within the framework of scalar field diffraction. 11 When q, Eq. (1) reduces to the more familiar but less general form of the Rayleigh Sommerfeld diffraction formula U n X, Y 1 q V n x, y exp jkr 01 dxdy, () S n r 01 where we have ignored the constant factor j. Equation () is the starting point of this paper because the condition q is always satisfied for an arbitrary photon sieve. It is well known that in the paraxial Fresnel approximation, the r 01 in the denominator in Eq. () is replaced by q and the r 01 in the exponent is replaced by q (X x) (Y y) /(q). This approximation is valid for almost all cases in the visible spectral region. 10 However, it is no longer valid for the high-n.a. focusing systems working in the soft x ray region because of the huge wave number k. In this case, a small change attributed to the discarded fourth-order term (X x) (Y y) /(8q 3 )inr 01 may lead to significant error in the phase kr 01 and subsequently lead to an incorrect result. r 01 q r 01 To understand this property better, let us now consider a high-n.a. photon sieve with D 00 m and q 500 m illuminated normally by a plane wave with wavelength.4 nm, where D is the total diameter of the photon sieve. For this example, the corresponding N.A. is 0. [the N.A. of a photon sieve can be given approximately by N.A. D/(q)]. It should be emphasized that, for a photon sieve or a Fresnel zone plate working in the soft x ray region, an N.A. value of 0.05 can be regarded as high and an N.A. value of 0. can be regarded as rather high. For the individual diffracted field at the desired focal point from a pinhole located at the edge of the photon sieve, the relations X 0, Y 0, x y (D/) hold. By use of these relations and the values of D, q and, one can find that the discarded fourth-order phase term is k(d/) 4 /(8q 3 ) kq(n.a.) 4 / This value shows explicitly that, for those high-n.a. photon sieves working in the soft x ray region, the fourth-order phase term is no longer negligible and the nonparaxial treatment is necessary. Let us now go back to Eq. (). To get an accurate expression for the diffracted field U n (X, Y), we first express r 01 as r 01 q X Y X x Y y x y 1/, (3) where X X x n, Y Y y n, x x x n, and y y y n. The changes of x and y are very small in the integral region S n because the pinhole is very small. By use of this property and the Taylor expansion, one can obtain r 01 q R 1/ r X x Y y q R 1/, (4) where R X Y and r x y. It is worth mentioning that a similar treatment for r 01 was used in Ref. 1 for the off-axis diffraction of a single circular aperture whose center is located at the origin (x 0, y 0). We use the approximation of Eq. (4) for the r 01 in the exponent of Eq. (). For the r 01 in the denominator of Eq. (), the first term on the right-hand side of Eq. (4) is already good enough. Substituting these two approximations into Eq. (), one can get U n X, Y q H exp jkh V n x, y S n exp jk r X x Y y H dxdy, where H (q R ) 1/. Because the change of the field inside the pinhole is very small, the complex amplitude distribution inside the pinhole can be expressed as the local plane wave V n x, y A n exp jkl n (5) exp jk g n x x n h n y y n, (6)

3 Q. Cao and J. Jahns Vol. 0, No. 6/June 003/J. Opt. Soc. Am. A 1007 where A n A(x n, y n ), L n L(x n, y n ), g n ( L/ x) xn,y n, and h n ( L/ y) xn,y n. Obviously, this approximation is always satisfied because the pinholes are very small compared with the range over which the complex amplitude distribution of the illuminative beam has significant change. It is well known that a similar treatment has been used in the famous Shack Hartmann wave-front sensor. 13 Substituting Eq. (6) into Eq. (5), one can get U n X, Y A nq H exp jk L n H S n exp jk r X x Y y H dx dy, where X X g n H and Y Y h n H. The integral area S n is now explicitly given by x y a n. After a derivation that is similar to that in Appendix A of Ref. 9, we further express the field distribution U n (X, Y) in the following form (7) U n X, Y exp jk L n H F X, Y, (8) F X, Y ka a nq n exp H jk H r 0 J 0 k H r r dr, (9) where (X Y ) 1/. One may notice that, in expressive form, Eq. (9) here is very similar to Eq. (5) of Ref. 9. However, one should keep in mind that the parameter H here is not a constant but a function of X and Y. Also, the parameter here has different content from that which appeared in Eq. (5) of Ref. 9. We now investigate the asymptotic behavior of F(X, Y) for large H. Obviously, when ka n /(H) 1, the relation exp jkr /(H) 1 jkr /(H) holds. 14 Substituting this approximation into Eq. (9) and employing a derivation like that in Appendix B of Ref. 9, one can obtain the first two terms F 0 (X, Y) and jf 1 (X, Y): ka n a n q F 0 X, Y Jinc ka n H H, (10) k A n a 4 n q F 1 X, Y 4H 3 3J 0 ka n H J ka n H J 4 ka n H. (11) where Jinc(.) J 1 (.)/(.) is the Jinc function and J n (.) is the nth-order Bessel function of the first kind. If the pinhole is small enough, the term jf 1 (X, Y) is negligible. In this case, the first term F 0 (X, Y) is sufficient for describing the integral F(X, Y) given by Eq. (9). We refer to this approximation as the nonparaxially individual farfield model. Combining Eq. (10) with Eq. (8), one can obtain U n X, Y ka n a n q H exp jk L n H Jinc ka n H. (1) Equation (1), which is very similar to the Fraunhofer diffraction of a circular aperture in the paraxial approximation, is the nonparaxial far-field distribution of the nth pinhole. From Eq. (1) one can find that the phase of U n (X, Y) is uniquely determined by k(l n H) because the function F 0 (X, Y) is always real. The added phase kh is exactly the same as that determined by the optical path length from the center of the pinhole to the observed point (X, Y) at the focal plane. This consistency implies that the added phase kh comes from a purely geometrical effect. On the other hand, the factor F 0 (X, Y) given by Eq. (10) shows the diffraction effect of the pinhole. The real function F 0 (X, Y) can be positive or negative, depending on the concrete parameters. It is interesting that the phase factor exp jk(l n H) changes very rapidly while the factor F 0 (X, Y) changes very slowly with the coordinate (X, Y). As a consequence, the individual diffracted field U n (X, Y) is a fast-oscillating function modulated by a slowly varying envelope. The fastoscillating factor is related to the spatial resolution, and the slowly varying envelope is related to the efficiency. According to the linear superposition principle, the total diffracted field U(X, Y) at the focal plane is the simple sum of those individual nonparaxial far fields, i.e., U(X, Y) N n 1 U n (X, Y). Similar to the paraxial case, 9 the total diffracted field U(X, Y) is not in the farfield region, though each individual diffracted field is already in its own far-field region. It is worth mentioning that a similar idea was recently used to distinguish between Fraunhofer diffraction and Fresnel diffraction. 15 In the work of Ref. 15, the individual diffracted fields of the two pinholes of Young s experiment are always in the far-field region because the pinholes are very small; however, the total diffracted field may be still in the Fresnel diffraction region if the distance between the aperture plane and the observation plane is not large enough. Let us now focus on the field value at the desired focal point (X 0, Y 0). From Eq. (1), one can immediately obtain U n (0, 0) at the desired focal point (X 0, Y 0): U n 0, 0 kqa n a n exp jk L n Jinc ka n R n, (13) where (q r n ) 1/, R n (x n g n ) ( y n h n ) 1/, and r n x n y n. To get effective focusing, those individual diffracted fields should have the same phase (rigorously speaking, argument) at the desired focal point. The argument of the real function Jinc(ka n R n / ) can be zero or, because this real function can be positive or negative. By taking this property into account, one can briefly state the selection condition as follows: k L n m const., Jinc ka n R n 0, (14)

4 1008 J. Opt. Soc. Am. A/ Vol. 0, No. 6/ June 003 Q. Cao and J. Jahns k L n m 1 const., Jinc ka n R n 0. (15) The constants in Eqs. (14) and (15) are the same. Equations (14) and (15) are both valid for choosing the central positions and the radii of those pinholes of a high-n.a. photon sieve. The first relation in Eq. (14) [or Eq. (15)] is used to determine the central positions of those pinholes, and the second in Eq. (14) [or Eq. (15)] is used to determine the radii of those pinholes. Within the framework of scalar field diffraction, Eqs. (14) and (15) are applicable to arbitrary illumination with arbitrary complex amplitude at the photon sieve plane. Of course, when the photon sieve is illuminated by a spherical wave of point source, Eqs. (14) and (15) correspond to the cases of the white zones and the black zones used in Ref. 7, if one chooses the common constant to be k( p q), where p is the distance between the point source and the photon sieve plane. One can easily prove that the relation q r n /(q) holds for a low-n.a. photon sieve. Simply substituting this approximation into the first relation of Eq. (14) [or Eq. (15)], replacing the in the second relation of Eq. (14) [or Eq. (15)] by q, and letting the constant be kq, one can find that, just as we expect, Eqs. (14) and (15) reduce to Eqs. (10) and (11) of Ref. 9 in the paraxial approximation. To understand the nonparaxially individual far-field model better, let us now investigate the special case of spherical wave illumination from a point source. In this case, the eikonal is given by L(x, y) ( p r ) 1/, where r x y. Substituting this relation into Eq. (13), one can get U n 0, 0 kqa n a n exp jk P n Jinc ka n f n r n, (16) where P n ( p r n ) 1/ and f 1 n P 1 n Q 1 n. If the center of the pinhole is located at a white zone for which the constant in the first condition of Eq. (14) is k( p q), then the relation exp jk(p n ) exp jk( p q) holds. In this case, the contribution to the focusing from the nth pinhole is simply proportional to a n Jinc(ka n r n /f n )/Q n because the common phase factor exp jk( p q) has no influence on the final intensity at the focal point. When Jinc(ka n r n /f n ) 0, the pinhole has a positive contribution to the focusing; when Jinc(ka n r n /f n ) 0, the pinhole has a negative contribution to the focusing. The changes from a positive (negative) contribution to a negative (positive) contribution happen when Jinc(ka n r n /f n ) 0. According to the definition of a Fresnel zone, we know that the difference between the optical path length P n from the point source through the center of the pinhole to the focal point and the optical path length p q from the point source directly to the focal point is an integral number of wavelengths. That is to say, P n ( p q) m. This condition is exactly equivalent to the relation exp jk(p n ) exp jk( p q) mentioned above. Similarly, by use of a local linear approximation that assumes the central position of the investigated pinhole is located at the center of the white zone, one can calculate the optical path length T u from the point source through the upper edge of the white zone to the focal point according to T u p (r n w/) 1/ q (r n w/) 1/, where w is the width of the local zone. For the same reason, one can calculate the optical path length T b from the point source through the lower edge of the white zone to the focal point according to T b p (r n w/) 1/ q (r n w/) 1/. According to the definition of a Fresnel zone, one knows that the difference of these two optical path lengths is half a wavelength. That is to say, T u T b /. Because the values r n w w /4 are much smaller than P n and Q n, one can use the following local Taylor expansions: T u P n r n w/( f n ) w /(8f n ) and T b P n r n w/( f n ) w /(8f n ). Substituting these expansions into the relation T u T b /, one can get w f n. (17) kr n Strictly speaking, the center of the pinhole is not located exactly at the center of the white zone. The real distance w u from the center of the pinhole to the upper edge of the white zone is slightly smaller than w/; and the real distance w b from the center of the pinhole to the lower edge of the white zone is slightly larger than w/. Fortunately, the two slight differences of w u w/ and of w b w/ are complementary to each other, because w u w/ 0 but w b w/ 0. As a consequence, the difference between the real width w u w b and the w value given by Eq. (17) is much smaller than either of w/ w u and w b w/. In other words, Eq. (17) is much more accurate than the local linear approximation itself, though the former is derived from the latter. By use of Eq. (17) in Eq. (16), one obtains the following relation for a pinhole centrally located at a white zone U n 0, 0 1 d w J d, (18) w where d a n is the diameter of the pinhole. To our surprise, Eq. (18) here has the same form as Eq. (16) of Ref. 9, even though the latter is derived from the paraxial Fresnel diffraction integral. This coincidence explains why Eq. (16) of Ref. 9 is in excellent agreement with the calculation result labeled Total in Fig. of Ref. 7, though the high-n.a. photon sieves are also considered in Ref. 7 (see Fig. 3 there). The detailed comparison between Eq. (18) here and the corresponding numerical calculations of Ref. 7 has actually been done in Ref. 9 because of the above-mentioned coincidence. The results are in excellent agreement VALIDITY RANGE The validity of the nonparaxially individual far-field model depends on two approximations. One is Eq. (4), the other is exp jkr /(H) 1. We first discuss the latter. When this approximation holds, Eq. (9) can be simplified to Eq. (10). As we said above, Eq. (9) here is very similar to Eq. (5) of Ref. 9. Also, Eq. (10) here is very similar to Eq. (6) of Ref. 9. In fact, they have the same

5 Q. Cao and J. Jahns Vol. 0, No. 6/June 003/J. Opt. Soc. Am. A 1009 mathematical correspondences. That is to say, the validity of Eq. (10) here for Eq. (9) here has the same condition as the validity of Eq. (6) of Ref. 9 for Eq. (5) of Ref. 9. It has been shown that Eq. (6) of Ref. 9 is highly accurate for Eq. (5) of Ref. 9 when N f a n /( q) 0.05, where N f is the Fresnel number of the pinhole. Correspondingly, one can deduce that Eq. (10) is highly accurate for Eq. (9) if the condition a n /( H) 0.05 is satisfied. Obviously, this condition is satisfied when N f 0.05, because H q. We now discuss the approximation of Eq. (4). According to the Taylor series expansion, this approximation is valid for the phase factor when 4 k(r X x Y y ) /(8H 3 ) 1, where 4 is the lowest-order phase term of those discarded. For the focusing and imaging of a photon sieve, the area of interest is actually the neighborhood of the focal point, because the focal spot is very small. For this area, the relations H q, X x n, and Y y n hold. Then, taking the relations x r cos, y r sin, X x n r n cos, Y y n r n sin, and r a n into account, one can get X x Y y r n r cos( ) r n a n, where and are two related angles in the corresponding polar coordinates. By taking all those analyses into account, one can further get 4 k(a n r n a n ) /(8q 3 ). If r n is approximately of the order of a n, the relation k(a n r n a n ) /(8q 3 ) 1 must be well satisfied, because a n is very small. When r n a n, the approximation k(a n r n a n ) /(8q 3 ) kr n a n /(q 3 ) holds because, in this case, the term a n is negligible compared with the term r n a n. The quantity kr n a n /(q 3 ) increases with the increase of r n for an arbitrary given a n and reaches the maximum (N.A.) N f when r n reaches its maximum D/. Therefore, one can deduce that Eq. (4) is highly valid when 4 (N.A.) N f 1. Briefly speaking, the nonparaxially far-field model is valid when the sufficient conditions N f 0.05 and (N.A.) N f 1 are both satisfied. We emphasize that the first condition is more important than the second. The second is automatically satisfied when the first is satisfied, if the N.A. value is not extremely high. For example, the value of (N.A.) N f is as small as 0.00 even for a high N.A. value of 0., when N f The validity condition N f 0.05 can be rewritten in the form d q/(5 ) 1/. To have an idea of the order of the value q/(5 ) 1/, we consider the underlying Fresnel zone plate for the simple case of plane wave illumination. It is still suitable to employ the paraxial approximation to estimate the widths of the Fresnel zones near the center of the underlying zone plate. In the paraxial approximation, one can derive that the width w m of the mth white (or black) zone is w m q/(8m) 1/, where m is a small positive integer. From the expression of w m one can find that the value q/(5 ) 1/ is about the width of the second white zone corresponding to m. Therefore, roughly speaking, the nonparaxially individual far-field model is highly valid when the largest diameter of the pinholes is not larger than the width of the second white (or black) zone of the underlying zone plate when the photon sieve is illuminated by a spherical wave of point source and p q. If the diameter of the pinholes is larger than q/(5 ) 1/, then the quasi-far-field correction term jf 1 (X, Y) is needed. According to the analysis of Ref. 9, the paraxially quasi-far-field correction term is highly accurate when N f 0.. By use of an analysis similar to that for the far-field term F 0 (X, Y), one can deduce that the quasi-far-field correction term is valid when a n /( H) 0.. Because H q, one can further use the more sufficient condition N f 0.. This sufficient condition can also be expressed as d q/(1.5 ) 1/. The value of q/(1.5 ) 1/ is about 1.43 times the width of the first white (or black) zone of the underlying zone plate when the photon sieve is illuminated by a plane wave. The sum of the far-field term F 0 (X, Y) and the quasi-far-field correction term jf 1 (X, Y) is suitable for describing the individual diffracted fields from such very large pinholes. Similar to the validity condition N f 0.05 for the far-field term, the validity condition N f 0. is still stricter than the condition (N.A.) N f 1if the N.A. value is not extremely high. For example, the value of (N.A.) N f is as small as even for a high N.A. value of 0., when N f 0.. To have an intuitive impression of the validity of the nonparaxial model, we now directly test it for the pinholes of a typical high-n.a. photon sieve with D 00 m and q 500 m, illuminated normally by a plane wave with.4 nm. For this example, the N.A. value is as high as 0.. Employing simply the knowledge of the Fresnel zone plate, one can find that the underlying zone plate has 415 white zones and 416 black zones (excluding the white half-zone corresponding to m 0), and the width of the outermost white (or black) zone is 6.1 nm. There- Fig.. Comparison between the far-field term F 0 (R ) and U n (R )exp( jkh), where U n (R ) is the exact diffracted field distribution at the focal plane. The solid line is the real part of U n (R )exp( jkh), and the asterisks plot the far-field term F 0 (R ). The field values have been normalized according to F 0 (R 0) 1. See text for the concrete parameters. (a) For a pinhole corresponding to N f ; the negligible imaginary part of U n (R )exp( jkh) is completely indistinguishable from zero in this case. (b) For a pinhole corresponding to N f 0.06; the dashed curve is the small imaginary part of U n (R )exp( jkh).

6 1010 J. Opt. Soc. Am. A/ Vol. 0, No. 6/ June 003 Q. Cao and J. Jahns Fig. 3. Comparison between the sum F 0 (R ) jf 1 (R ) ofthe far-field term and the quasi-far-field term and U n (R )exp( jkh) for a very large pinhole, where U n (R ) is the exact diffracted field distribution at the focal plane. See text for the concrete parameters. The field values have been normalized according to F 0 (R 0) 1. (a) The solid line is the real part of U n (R )exp( jkh) and the asterisks plot the far-field term F 0 (R ). (b) The solid line is the imaginary part of U n (R )exp( jkh), while the asterisks plot the quasi-far-field correction term F 1 (R ). at the focal plane is circularly symmetric about the point (X x n, Y y n ) at the focal plane. As a consequence, U n (X, Y) is uniquely dependent on the parameter R and therefore can be expressed as U n (R ). This property allows us to use simple one-dimensional plots to show the diffracted field distributions. In the concrete comparison, we do not directly compare the nonparaxially individual far-field model of Eq. (1) with the numerical calculation U n (R ) of Eq. (). Instead, we compare the farfield term F 0 (R ) with U n (R )exp( jkh), because this treatment can avoid the appearance of the fast-oscillating factor exp( jkh). We first consider a pinhole with diameter d nm (about 5.5 times the width of the local zone) centrally located at a position satisfying r n m. For this pinhole, the N f value is As shown in Fig. (a), the far-field term F 0 (R ) is in excellent agreement with the exact numerical result U n (R )exp( jkh). To show how fast the far-field term F 0 (R ) begins to fail, we consider a pinhole with diameter d 303 nm (about 1.56 times the width, 194 nm, of the local zone) centrally located at a position satisfying r n m. For this pinhole, the N f value is As shown in Fig. (b), the difference between F 0 (R ) and U n (R )exp( jkh) [i.e., the imaginary part of U n (R )exp( jkh)] is observable. This observable difference implies that the quasi-far-field correction term may be needed in this case. We then test the diffracted field distribution of a very large pinhole with diameter d 585 nm (about 1.5 times the width, 390 nm, of the first white zone) centrally located at a position satisfying r n m inside the first white zone. For this pinhole, the N f value is 0.4 (little larger than the value 0.). As we have analyzed above, for such a very large pinhole, the quasi-far-field correction term is definitely needed. The comparison between the sum F 0 (R ) jf 1 (R ) of the far-field term and the quasi-far-field correction term and U n (R )exp( jkh) is shown in Fig. 3. Good agreement is observed in each case. Out of curiosity, we finally consider an extremely large pinhole with diameter d 678 nm (about 1.74 times the width of the first white zone) centrally located at a position satisfying r n m. For this pinhole, the N f value is 0.3. From Fig. 4, one can see that the difference between F 0 (R ) jf 1 (R ) and U n (R )exp( jkh) is small. This small difference shows that the sum F 0 (R ) jf 1 (R ) is still basically valid even after the inequality N f 0. is broken down. Fig. 4. Comparison between the sum F 0 (R ) jf 1 (R ) ofthe far-field term and the quasi-far-field term and U n (R )exp( jkh) for an extremely large pinhole. All notations are the same as those in Fig. 3. See text for the concrete parameters. fore, the ultimate spatial resolution, which is 1. times the width of the outermost zone of the underlying zone plate, 7 can reach as high as 7 nm. We use the Rayleigh Sommerfeld diffraction formula of Eq. () to calculate the exact diffracted field distribution U n (X, Y) at the focal plane. Because the illuminative beam is a normal plane wave, the individual diffracted field U n (X, Y) 4. CONCLUSIONS AND DISCUSSION We have extended the individual far-field model to the nonparaxial case of a high-n.a. photon sieve working in the soft x ray region. Within the framework of scalar field diffraction, 11 this nonparaxial model is applicable to an arbitrary illuminative beam with arbitrary complex amplitude distribution at the photon sieve plane. The selection conditions for the positions and the radii of the pinholes of the sieve are explicitly given by Eqs. (14) and (15). We have also discussed the validity problem. In particular, we have shown that the sufficient conditions for the validity of this generalized model can be analytically described by N f 0.05 and (N.A.) N f 1. The

7 Q. Cao and J. Jahns Vol. 0, No. 6/June 003/J. Opt. Soc. Am. A 1011 quasi-far-field correction term has also been presented for describing the diffracted field distributions of those very large pinholes whose Fresnel number N f is larger than The nonparaxially individual far-field model, as well as the quasi-far-field correction term, can be used for the focusing and imaging analysis and the fast simulation of those high-n.a. photon sieves working in the soft x ray region for which the paraxial Fresnel diffraction formula is invalid. The analyses presented in this paper are based on the Rayleigh Sommerfeld diffraction formula, 10 1 which is the integral-form solution of the scalar Helmholtz wave equation. This formula is valid provided that the longitudinal field component is negligible. Strictly speaking, 16 the purely linear polarization field with only one electric (or magnetic) field component [say, E U(X, Y)e x, where e x is the unit vector in the x direction] cannot exist in the real world because it can never satisfy the condition E 0. There must exist an associated longitudinal field component W(X, Y)e z, where e z is the unit vector in the z direction. The total longitudinal field component can be given approximately by [see Eq. (3) of Ref. 19 or Eqs. (6) and (9) of Ref. 17] W X, Y j U X, Y. (19) k X Substituting the relation U(X, Y) N n 1 U n (X, Y) into Eq. (19), one can get N W X, Y n 1 W n X, Y, (0) W n X, Y j U n X, Y, (1) k X where W n (X, Y) is the individual longitudinal field associated with the individual transverse field U n (X, Y), which can be expressed as exp jk(l n H) F(X, Y). The factor exp jk(l n H) changes much faster than the factor F(X, Y). As a consequence, one can obtain approximately U n (X, Y)/ X jk( H/ X)U n (X, Y). Substituting this approximation and the relation H/ X (X x n )/H into Eq. (1), one can obtain W n X, Y X x n H U n X, Y. () In particular, the field value W n (0, 0) at the focal point is given by W n (0, 0) x n U n (0, 0)/. Unlike the situation when all the individual transverse fields have positive contributions to U(0, 0), some individual longitudinal fields have positive contributions to W(0, 0) but others have negative contributions to W(0, 0), because x n can be positive or negative. As a consequence, the total longitudinal field W(0, 0) is far smaller than U(0, 0). Similarly, W(X, Y) is far smaller than U(X, Y) in the neighbor of the focal point because (X x n ) can also be positive or negative. Therefore, to state it simply, the longitudinal field component can be ignored and the scalar diffraction theory is highly valid. 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