Beam propagation camera and method for light beam analysis

Abstract

A beam propagation camera has at least one beam-splitting optical arrangement (240) configured to split a beam, which is incident on the beam-splitting optical arrangement along an optical axis (OA) of the beam propagation camera, into a multiplicity of sub-beams, and a sensor arrangement (250) configured to detect the sub-beams. The beam-splitting optical arrangement has a diffractive structure (241) configured such that at least two of the sub-beams are spatially separated from one another on the sensor arrangement and have respective foci longitudinally offset from one another along the optical axis.

Claims

1. A beam propagation camera, having an optical axis and comprising: at least one beam-splitting optical arrangement configured to split an optical beam, which is incident on the beam-splitting optical arrangement along the optical axis, into a multiplicity of sub-beams; and a sensor arrangement configured to detect the sub-beams; wherein the beam-splitting optical arrangement comprises a diffractive structure; and wherein the diffractive structure is configured such that at least two of the sub-beams are separated spatially from one another on the sensor arrangement and have respective foci offset from one another longitudinally along the optical axis.

2. The beam propagation camera as claimed in claim 1, wherein the diffractive structure is arranged decentered in relation to the optical axis.

3. The beam propagation camera as claimed in claim 1 wherein the beam-splitting optical arrangement further comprises a refractive optical element.

4. The beam propagation camera as claimed in claim 3, wherein the diffractive structure has a focal length f.sub.1 and the refractive optical element has a focal length f.sub.0, wherein the ratio f.sub.1/f.sub.0 is at least 2.

5. The beam propagation camera as claimed in claim 3, wherein the refractive optical element and the diffractive structure have a monolithic configuration.

6. The beam propagation camera as claimed in claim 3, wherein the refractive optical element is a plano-convex lens element.

7. The beam propagation camera as claimed in claim 3, wherein the refractive optical element is arranged such that a focal plane of the refractive optical element corresponds to a pupil plane in a path of the optical beam.

8. The beam propagation camera as claimed in claim 1, wherein the diffractive structure is configured as a phase diffractive optical element.

9. The beam propagation camera as claimed in claim 1, wherein the diffractive structure is configured as a transmission diffractive optical element.

10. The beam propagation camera as claimed in claim 1, wherein the diffractive structure is configured as a Fresnel lens element or as a Fresnel zone plate.

11. The beam propagation camera as claimed in claim 1, wherein the diffractive structure has an increasing diffraction efficiency with an increasing order of diffraction.

12. The beam propagation camera as claimed in claim 11, wherein a decrease in intensity accompanying an increasing defocusing of the sub-beams generated by the beam splitting on the sensor arrangement is at least partly compensated by the diffraction efficiency increasing with the increasing order of diffraction.

13. The beam propagation camera as claimed in claim 1, wherein the diffractive structure is operated in transmission.

14. The beam propagation camera as claimed in claim 1, wherein the diffractive structure is operated in reflection.

15. The beam propagation camera as claimed in claim 1, further comprising a first analysis unit configured to analyze the beam prior to reflection at an object and a second analysis unit configured to analyze the beam after reflection at the object.

16. The beam propagation camera as claimed in claim 15, wherein the object is a flying object in a laser plasma source.

17. The beam propagation camera as claimed in claim 1, further comprising an apodization filter.

18. The beam propagation camera as claimed in claim 17, wherein the sensor arrangement comprises the apodization filter.

19. The beam propagation camera as claimed in claim 17, wherein the apodization filter comprises a gray filter arranged in a pupil plane in a path of the optical beam.

20. A method for light beam analysis, comprising: beam-splitting a beam propagating along an optical axis into a multiplicity of sub-beams; measuring respective spot sizes generated by the sub-beams on a sensor arrangement; and calculating at least one of divergence (θ), focal position (z.sub.0) and waist size (w.sub.0) beam parameters from the measured spot sizes; wherein said beam splitting comprises using a diffractive structure to spatially separate at least two of the sub-beams from one another on the sensor arrangement through a focus offset in a longitudinal direction of the optical axis.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) In the figures:

(2) FIG. 1 shows a schematic illustration for explaining a basic design possible both for realizing a droplet position determination and for determining the focal position of the laser beams, to be updated accordingly, in a laser plasma source;

(3) FIGS. 2A-2C show schematic illustrations for explaining the design and the mode of operation of a beam-splitting optical arrangement employed within the scope of the present invention;

(4) FIG. 3 shows a schematic illustration for explaining the principle of beam fanning, which takes place in accordance with the present invention;

(5) FIGS. 4A-4B show schematic illustrations for explaining an exemplary (measurement) beam path in a beam propagation camera according to the invention;

(6) FIGS. 5A-5D show diagrams for explaining exemplary configurations of a diffractive structure present in a beam propagation camera according to the invention; and

(7) FIG. 6 shows a schematic illustration of the design of an EUV light source in accordance with the prior art.

DETAILED DESCRIPTION

(8) FIG. 1 shows a schematic illustration of a basic design possible in a laser plasma source (such as e.g. the one in FIG. 6), both for determining the droplet position and the focal position of the laser beams to be updated accordingly, wherein both laser beams in the “forward direction” (prior to incidence on the respective target droplet) and laser beams in the “backward direction” (i.e. the inferred radiation reflected back from the respective target droplet) are evaluated.

(9) In accordance with FIG. 1, some of the incident laser beam with a Gaussian profile is decoupled at a first partly transmissive mirror 105 and analyzed by a first analysis unit 110. The part of the incident laser beam passing through the partly transmissive mirror 105 and a further partly transmissive mirror 125 reaches a metallic target droplet 130 (e.g. a tin droplet) by way of a focusing optical unit 128, with some of the laser beam being reflected back at said target droplet and returning to the partly transmissive mirror 125 collimated via the focusing optical unit 128. At the partly transmissive mirror 125, some of the laser beam is decoupled in turn to a second analysis unit 120. Moreover, beam traps (not plotted in FIG. 1) for capturing the respectively unused portion of the radiation incident on the partly transmissive mirror 105 or 125 may be provided.

(10) A schematic beam path for analyzing the laser beam in the “backward direction” is depicted in FIG. 4A, with the field planes being denoted by “F” and the pupil planes being denoted by “P” in each case. “130” denotes the metallic target droplet in FIG. 4A, “405” is an afocal telescope group and “250” denotes the sensor arrangement. A shift in the position of the target droplet 130 has as a consequence a change in the image obtained on the sensor arrangement 250.

(11) Thus, the analysis of the laser beams both in the “forward direction” (laser beam prior to incidence on the respective target droplet 130, denoted as “forward beam” below) and in the “backward direction” (laser beam after the reflection at the respective target droplet 130, denoted as “backward beam” below) allows a statement to be made about the relative setting of the laser beam and target droplet 130 in relation to one another, wherein—with reference again being made to FIG. 1—the setting or focal position of the laser beam can be deduced from the result obtained with the first analysis unit 110 and the droplet position can be deduced from the result obtained with the second analysis unit 120.

(12) In principle, depending on the case, different conventions are possible and conventional for the beam dimension measure and the divergence measure. In the field of laser technology for example, the moments

(13) $\begin{array}{cc}{w}_x^2\left(z\right)=\frac{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right){\left(x-\stackrel{\_}{x}\left(z\right)\right)}^2}{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right)},w_y^2\left(z\right)=\frac{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right){\left(y-\stackrel{\_}{y}\left(z\right)\right)}^2}{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right)}& \left(1\right)\\ \mathrm{with}& \\ \stackrel{\_}{x}\left(z\right)=\frac{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right)x}{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right)},\stackrel{\_}{y}\left(z\right)=\frac{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right)y}{{\int }_{-\infty }^{+\infty }\mathrm{dxdy}I\left(x,y;z\right)}& \left(2\right)\end{array}$
often serve as a basis of a beam dimension definition in accordance with
w(z)=√{square root over (w.sub.x.sup.2(z)+w.sub.y.sup.2(z))}  (3)
or

(14) $\begin{array}{cc}w\left(z\right)=\sqrt[4]{w_x^2\left(z\right)w_y^2\text{(z)}}& \left(4\right)\end{array}$

(15) Here, I(x, y; z) denotes the light intensity for the selected sectional plane.

(16) If a Gaussian beam is based on a diameter of 5*σ(wherein σ denotes the standard deviation or width of the normal distribution in accordance with the conventional terminology and emerges from the second moment), the distance of the relevant spot on the sensor arrangement preferably has a value of at least 5*σ.

(17) When analyzing the forward beam and the backward beam in the basic setup of FIG. 1, it should be noted that only the forward beam should be considered in an idealized manner as a “Gaussian beam”, for which, in the region of the image-side focus, the following applies to a good approximation for the beam dimension w as a function of the propagation coordinate z
w(z)=√{square root over (w.sub.0.sup.2+θ.sup.2(z−z.sub.0).sup.2)}  (5)
where w.sub.0 denotes the waist size, θ denotes the divergence and z.sub.0 denotes the waist position (focal position).

(18) Below, problems are initially discussed, which, for example in the case of analyzing the backward beam in the second analysis unit 120, emerge from the fact that the beam to be examined is not an ideal Gaussian beam but rather a comparatively sharply cut off beam (also referred to as “top hat” beam below). In the case of such a sharply cutoff beam, an Airy light distribution emerges in the focus (far field) and in the aberration-free ideal case:

(19) $\begin{array}{cc}I\left(r=\sqrt{x^2+y^2},z=z_0\right)=\pi {{P\left(\frac{1}{L_c}\right)}^2\left[\frac{2J_1\left(2\pi \frac{r}{L_c}\right)}{2\pi \frac{r}{L_c}}\right]}^2& \left(6\right)\\ \mathrm{where}{L}_c=\frac{\lambda }{\mathrm{NA}}& \end{array}$
denotes the characteristic length, P denotes the entire power transmitted through the system and J.sub.1(x) denotes the first order Bessel function. However, the moments in accordance with equation (5) are not defined due to the asymptotic decrease

(20) $I\left(r,z=z_0\right)\propto \frac{1}{r^2}$
in this light distribution. The problem of also evaluating the backward beam with a “hard cutoff” resulting herefrom can be overcome by a suitable “artificial” apodization: In a first embodiment, this can be carried out by virtue of a suitable mask being realized “electronically” in the plane of the sensor arrangement, said mask apodizing the intensity curves in accordance with the replacement
I(x,y;z).fwdarw.I(x,y;z)A(x−x,y−y)  (7)
by a suitably selected apodization function (wherein this apodization can be denoted “soft” to the extent that discontinuities only occur in the higher derivatives of the apodization curve). By way of example, the function

(21) $\begin{array}{cc}{A}_R\left(x,y\right)=\frac{1}{2}\left(1+\cos \left(\pi \frac{\sqrt{x^2+y^2}}{R}\right)\right)& \left(8\right)\end{array}$
with the cutoff radius R in the range 5L.sub.c<R<10L.sub.c, which is only discontinuous from the second derivatives, is suitable to this end.

(22) In a second embodiment, an apodization (which is “soft” within the above meaning)
u(x,y;z.sub.NF)θ(x.sup.2+y.sup.2≤R.sub.NA).fwdarw.u(x,y;z.sub.NF)A.sub.R.sub.NA(x,y)  (9)
can be realized by introducing a structured gray filter with a corresponding profile into the near field or into a pupil plane. Here, u(x,y;z) denotes the beam amplitude (which determines the intensity by way of I(x,y;z)=|u(x,y;z)|.sup.2) and R.sub.NA denotes the aperture radius (defining the opening or numerical aperture NA).

(23) FIG. 3 shows a schematic illustration for explaining the principle of the beam fanning, which takes place in accordance with the invention.

(24) Here, a light beam (e.g. a sample beam decoupled from electromagnetic radiation to be analyzed) is split or replicated in various sub-beams or used beams, wherein, firstly, a longitudinal focus offset in the propagation direction and, secondly, transversal splitting (for enabling a simultaneous evaluation in a sensor arrangement) are obtained for these used beams. As can be seen from the isofocal line (denoted by IFC) sketched out in FIG. 3, the focus is different for each one of the individual sub-beams. The sensor arrangement 250 placed into the beam path results in different spot images, wherein the size is smallest in the middle or at the perfect focus and increases to the edge. An analysis of the image recorded by the sensor arrangement, in which the size of the spot image is established as a function of the index (e.g. from −3 to 3), therefore enables the determination of the focal position.

(25) A beam-splitting optical arrangement 240 explained in more detail below serves to realize both the longitudinal focal offset and the transverse splitting of the sub-beams.

(26) In the exemplary embodiment, the beam-splitting optical arrangement 240 has a diffractive structure 241 and a refractive optical element (refractive lens element) 242, which have a monolithic embodiment here and together form a multi-focal optical element, as indicated schematically in FIG. 2A.

(27) In a specific exemplary embodiment, the refractive optical element 242 can be a plano-convex lens element, wherein the diffractive structure 241 can be formed on the plane surface of this plano-convex lens element. In a further embodiment, the refractive optical element 242 (e.g. plano-convex lens element) can also be attached to a separate diffractive optical element (DOE) by way of an index-matched lacquer. In accordance with these refinements, an element with a low optomechanical complexity (in respect of holder, adjustment mechanism etc.) is realized in each case, by which the beam-spitting according to the invention can be obtained.

(28) However, the invention is not restricted to the integration of diffractive structure and refractive optical element or, in particular, to the described monolithic refinement. Hence, diffractive structure and refractive optical element or lens element can also have a separate configuration and a (preferably small) distance from one another in further embodiments.

(29) A sensor arrangement 250 is situated in the pupil plane (Fourier plane) of the optical beam path, and the focal plane of the refractive optical element 242 is likewise situated in a pupil plane (Fourier plane) of the beam path.

(30) In principle, in accordance with the occurring orders of diffraction, a diffractive lens element has positive and negative focal lengths in accordance with

(31) $\begin{array}{cc}{f}_{\mathrm{diff}}=\frac{f_1}{k},k=0,\pm 1,\pm 2,\mathrm{.Math.}& \left(10\right)\end{array}$

(32) Here, f.sub.1 denotes the focal length of the first positive order of diffraction and k denotes the beam index or the order of diffraction. Here, the intensity of the respective focus depends directly on the embodiment and approximation form of the underlying (approximately parabolic) phase profile. In combination with a refractive lens element with a focal length of f.sub.0, a multi-focal optical system emerges with a plurality of used focal lengths f.sub.k, k=0,±1, . . . , k.sub.max, wherein the following applies approximately if the distance between the diffractive structure and the refractive lens element is neglected:

(33) $\begin{array}{cc}{f}_k\approx \frac{f_0{f}_1}{f_1+k{f}_0}& \left(11\right)\end{array}$

(34) This relation is elucidated in FIG. 2B for f.sub.1>>f.sub.0.

(35) As indicated in FIG. 4B, a lateral split emerges as a consequence of a break in symmetry, which is introduced in a targeted manner, or of a decentration of the beam-splitting optical arrangement 240. This decentration is depicted schematically in FIG. 2C and it is achieved in the exemplary embodiment by virtue of the refractive optical element 242 being arranged symmetrically with respect to the optical axis OA and the diffractive lens element 241 being arranged decentered by a path d.sub.x or displaced perpendicular to the optical axis OA.

(36) Below, the evaluation of the measured beam sizes for establishing the sought-after beam parameters (divergence θ, focal position z.sub.0 and waist size w.sub.0) is explained. Even though both the longitudinal focal offset obtained by the diffractive structure according to the invention and the lateral offset of the sub-beams caused by the break in symmetry are to be taken into account here, these are initially ignored—merely for the purposes of a better understanding—below, i.e. an evaluation in the case of a conventional beam analysis without the longitudinal focal offset according to the invention and without the lateral offset of the sub-beams is described first.

(37) When evaluating a conventional beam analysis without the longitudinal focal offset according to the invention and without the lateral offset of the sub-beams, the measured beam dimensions w(z) can initially be squared, whereupon the beam data in the focus can be established by way of a fit on the basis of the equation
w.sup.2(z)=A+Bz+Cz.sup.2  (12)
(i.e. a second order polynomial describing a parabola) by virtue of the parameters or “fitting coefficients” A, B and C being determined in accordance with equation (12). According to equation (5), the following relation exists between the fitting coefficients and the beam parameters:
A=w.sub.0.sup.2+θ.sup.2z.sub.0.sup.2,B=−2θ.sup.2z.sub.0,C=θ.sup.2  (13)

(38) Hence, the sought-after beam parameters (divergence θ, focal position z.sub.0 and waist size w.sub.0) emerge in a simple manner from the fitting coefficients in accordance with

(39) $\begin{array}{cc}\theta =\sqrt{C},z_0=-\frac{B}{2C},w_0=\sqrt{A-\frac{B^2}{4C}}& \left(14\right)\end{array}$

(40) According to the rules of the Fourier representation, the waist size w.sub.0 and divergence θ are coupled by way of the relation
w.sub.0θ=c  (15)
where c denotes a constant, which depends on the beam properties and the selected conventions for the beam dimension and diversions measure. The following applies for an ideal Gaussian fundamental mode and the moment-based beam measures:

(41) $\begin{array}{cc}c=\frac{\lambda }{\pi }& \left(16\right)\end{array}$
where λ denotes the light wavelength. By contrast, the modified form

(42) 0$\begin{array}{cc}c=M^2\frac{\lambda }{\pi }& \left(17\right)\end{array}$
applies for an aberrated Gaussian beam, with the propagation-invariant beam parameter product M.sup.2≥1 as fundamental quality measure. By comparing the variable w.sub.0θ from equation (15) with the variable λ/π from the equation (16), it is therefore possible to determine how closely the analyzed beam corresponds to an ideal Gaussian beam or whether it is, for example, a comparatively strongly aberrating beam.

(43) According to the explanation above of the evaluation in the case of a conventional beam analysis, which was merely provided for introductory purposes and for improved understandability, the following describes how this evaluation can be undertaken for the beam analysis according to the invention, i.e., in particular, taking into account the longitudinal focal offset and the lateral offset of the sub-beams obtained by the break in symmetry.

(44) The effect of the break in symmetry can be described in the paraxial beam transfer matrix formalism, by virtue of the conversion being made to homogeneous coordinates when describing the beam in accordance with

(45) $\begin{array}{cc}\underset{\_}{\mathrm{ray}}=\left(\begin{array}{c}x\\ u\\ l\end{array}\right),{\underset{\_}{\mathrm{ray}}}^{\prime }=\left(\begin{array}{c}{x}^{\prime }\\ u^{\prime }\\ l\end{array}\right)& \left(18\right)\end{array}$
where the variables with apostrophes (x′, u′) represent the object space and the variables without apostrophes (x, u) represent the image space. The additional third dimension (with a “one entry”) renders it possible likewise to represent translations and tilts in the form of transfer matrices in the extended formalism.

(46) The Fourier representation proceeding from the object-side focal plane of the refractive lens element is conveyed by the transfer matrix

(47) $\begin{array}{cc}\underset{\underset{\_}{\_}}{M}\left(k\right)=\underset{\underset{\underset{\mathrm{element}\mathrm{to}\mathrm{the}\mathrm{screen}}{\mathrm{Propagation}\mathrm{from}\mathrm{the}\mathrm{lens}}}{︸}}{\left(\begin{array}{ccc}1& f_0+z& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}\underset{\underset{\mathrm{Effective}\mathrm{focu}s\text{-}\mathrm{splitting}\mathrm{of}\mathrm{the}\mathrm{multi}\text{-}\mathrm{focus}\mathrm{lens}\mathrm{element}}{︸}}{\underset{\underset{\mathrm{Decentered}\mathrm{diffractive}\mathrm{lens}\mathrm{elemen}t}{︸}}{\left(\begin{array}{ccc}1& 0& -d_x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ -\frac{k}{f_1}& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& d_x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}\underset{\underset{\mathrm{Refractive}\mathrm{lens}\mathrm{element}}{︸}}{\left(\begin{array}{ccc}1& 0& 0\\ -\frac{1}{f_0}& 1& 0\\ 0& 0& 1\end{array}\right)}}\underset{\underset{\underset{\mathrm{plane}\mathrm{to}\mathrm{the}\mathrm{lens}\mathrm{element}}{\mathrm{Propagation}\mathrm{from}t\mathrm{he}\mathrm{Fourier}}}{︸}}{\left(\begin{array}{ccc}1& f_0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}=\left(\begin{array}{ccc}-z\left(\frac{1}{f_0}+\frac{k}{f_1}\right)-k\frac{f_0}{f_1}& f_0-\frac{f_0}{f_1}k\left(z+f_0\right)& -\frac{d_x}{f_1}k\left(z+f_0\right)\\ -\left(\frac{1}{f_0}+\frac{k}{f_1}\right)& -\frac{f_0}{f_1}k& -\frac{d_x}{f_1}k\\ 0& 0& 1\end{array}\right)& \left(19\right)\end{array}$
in the extended formalism. The following desired transverse beam splittings can be read off in a simple manner from the matrix elements as a result of the decentration d.sub.x:

(48) $\begin{array}{cc}\frac{\partial u}{\partial d_x}=\frac{\partial }{\partial d_x}{M}_{23}=-\frac{k}{f_1}& \left(20\right)\\ \frac{\partial u}{\partial d_x}=\frac{\partial }{\partial d_x}{M}_{13}=-\frac{k}{f_1}\left(f_0+z\right)=\frac{\partial u}{\partial d_x}\left(f_0+z\right)& \left(21\right)\end{array}$

(49) These two equations describe the beam fanning, proportional to the decentration, of a collimated beam, as is elucidated in FIG. 3 (in which the longitudinal focal splitting can also be identified).

(50) The detailed quantitative imaging properties of a complete afocal measurement beam path with a diffractive multi-focal lens element with internal decentration, as shown schematically by FIG. 4b, are disclosed by the transfer matrix of the overall system (from the object-side coherent point source to the sensor arrangement 250). If, like in FIG. 4b, the focal length of the object-side Fourier optical unit is denoted by f′, the focal length of the refractive optical element 242 is denoted by f.sub.0 and the focal length of the diffractive structure 241 is denoted by f.sub.1, the following composition emerges:

(51) $\begin{array}{cc}\underset{\underset{\_}{\_}}{M}\left(k\right)=\underset{\underset{\underset{\mathrm{element}\mathrm{to}\mathrm{the}\mathrm{screen}}{\mathrm{Propagation}\mathrm{from}t\mathrm{he}\mathrm{lens}}}{︸}}{\left(\begin{array}{ccc}1& f_0+z& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}\underset{\underset{\mathrm{Effective}\mathrm{sp}\mathrm{litting}\mathrm{of}\mathrm{the}\mathrm{mult}i\text{-}\mathrm{focus}\mathrm{lens}\mathrm{element}}{︸}}{\underset{\underset{\mathrm{Decentered}\mathrm{diffractive}\mathrm{lens}\mathrm{elemen}t}{︸}}{\left(\begin{array}{ccc}1& 0& -d_x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ \frac{-k}{f_1}& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& d_x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}\underset{\underset{\begin{array}{c}\mathrm{Refractive}\mathrm{lens}\\ \mathrm{element}\end{array}}{︸}}{\left(\begin{array}{ccc}1& 0& 0\\ \frac{-1}{f_0}& 1& 0\\ 0& 0& 1\end{array}\right)}}\times \mathrm{.Math.}\mathrm{.Math.}\times \underset{\underset{\underset{\mathrm{the}\mathrm{fourier}\mathrm{plane}\mathrm{to}\mathrm{the}\mathrm{lens}\mathrm{element}}{\mathrm{Propagation}\mathrm{from}}}{︸}}{\left(\begin{array}{ccc}1& f_0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}\underset{\underset{\mathrm{Telescope}}{︸}}{\left(\begin{array}{ccc}\mathrm{mag}& 0& 0\\ 0& \frac{1}{\mathrm{mag}}& 0\\ 0& 0& 1\end{array}\right)}\underset{\underset{\mathrm{Ojective}\mathrm{lens}\mathrm{element}\left(\mathrm{from}\mathrm{the}\mathrm{object}\mathrm{to}\mathrm{the}\mathrm{Fourier}\mathrm{plane}\right)}{︸}}{\left(\begin{array}{ccc}1& f^{\prime }& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ \frac{-1}{f^{\prime }}& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& f^{\prime }-z^{\prime }& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}& \left(22\right)\end{array}$

(52) After multiplying out, the relevant transfer matrix elements are:

(53) $\begin{array}{cc}{M}_{11}=\frac{\partial x}{\partial x^{\prime }}=f_0\frac{\left(f_0+z\right)k-f_1}{\mathrm{mag}{f}_1{f}^{\prime }}& \left(23a\right)\\ M_{12}=\frac{\partial x}{\partial u^{\prime }}=\frac{1}{\mathrm{mag}{f}_0{f}_1{f}^{\prime }}[z^{\prime }{f}_0^2\left(f_1-f_{0}k\right)+\mathrm{.Math.}\hspace{1em}\mathrm{.Math.}-z\left({\mathrm{mag}}^2{f}^{\mathrm{\prime 2}}\left(f_{0}k-f_1\right)-f_0^2{z}^{\prime }k\right)-{\mathrm{mag}}^2{f}^{\mathrm{\prime 2}}{f}_0^2]& \left(23b\right)\\ M_{21}=\frac{\partial x}{\partial x^{\prime }}=\frac{-1}{F}=\frac{f_{0}k}{\mathrm{mag}{f}_1{f}^{\prime }}& \left(23c\right)\\ M_{22}=\frac{\partial x}{\partial u^{\prime }}=-\mathrm{mag}{f}^{\prime }\left(\frac{1}{f_0}+\frac{k}{f_1}\right)-\frac{f_0{z}^{\prime }k}{\mathrm{mag}{f}_1{f}^{\prime }}& \left(23d\right)\\ \frac{M_{13}}{d_x}=\frac{\partial x}{\partial d_x}=-\frac{k}{f_1}\left(f_0+z\right)& \left(23e\right)\\ \frac{M_{23}}{d_x}=\frac{\partial x}{\partial d_x}=-\frac{k}{f_1}& \left(23f\right)\end{array}$

(54) From the condition M.sub.12 0, the image-side longitudinal focal positions emerge as

(55) $\begin{array}{cc}{z}_k\left(z_0\right)=\frac{-f_0^{2}k+z_0\left(f_1+f_{0}k\right)}{f_1\left(f_0+z_0\right)k}& \left(24\right)\end{array}$

(56) Here, z.sub.0=Mag.sub.0.sup.2 z′ denotes the focal position of the image-side chief ray and

(57) ${\mathrm{Mag}}_0=\frac{f_0}{\mathrm{mag}{f}^{\prime }}$
denotes the far field imaging scale thereof. By inserting equation (24) into equations (23a)-(23f), the beam-specific imaging properties of the system are obtained in the respective focus as

(58) $\begin{array}{cc}{M}_{11}\left(k;z_k\left(z_0\right)\right)=-{\mathrm{Mag}}_0=\frac{f_1}{f_1+f_{0}k+z_{0}k}=-{\mathrm{Mag}}_k\left(z_0\right)& \left(25a\right)\\ M_{12}\left(k;z_k\left(z_0\right)\right)=0& \left(25b\right)\\ M_{21}\left(k;z_k\left(z_0\right)\right)=\frac{-1}{F_k}={\mathrm{Mag}}_0\frac{k}{f_1}⟹F_k=-\frac{f_1}{{\mathrm{Mag}}_{0}k}& \left(25c\right)\\ M_{22}\left(k;z_k\left(z_0\right)\right)=-\frac{1}{{\mathrm{Mag}}_0}\frac{f_1+\left(f_0+z_0\right)k}{f_1}=\frac{f_1}{M_{11}\left(k;z_k\left(z_0\right)\right)}& \left(25d\right)\end{array}$

(59) The size imaging scale Mag.sub.k(z.sub.0)=−M.sub.11(k;z.sub.k(z.sub.0)) is identical to the reciprocal angle imaging scale in accordance with M.sub.22 (k;z.sub.k(z.sub.0))M.sub.11(k;z.sub.k(z.sub.0))=1 and in this case depends both on the reflection index and on the defocus z.sub.0 in accordance with

(60) $\begin{array}{cc}{\mathrm{Mag}}_k\left(z_0\right)={\mathrm{Mag}}_0\frac{f_1}{f_1+\left(f_0+z_0\right)k}& \left(26\right)\end{array}$

(61) The non-vanishing property of the term M.sub.21(k; z.sub.k (z.sub.0)) corresponds to a non-vanishing refractive power (reciprocal focal length F.sub.k) for the overall system and means beam-dependent telecentricity for the higher orders of diffraction.

(62) If the relations (25a)-(25d) are inserted into the equation (5) describing the focal curve and if the value z=0 is selected for the position of the sensor arrangement 250, the following is obtained when taking into account the imaging scales in accordance with equation (26):
w.sub.k=w.sub.k(z=0)=√{square root over (w.sub.k,0.sup.2+θ(−z.sub.k(z.sub.0)).sup.2)}  (27)
where

(63) 0$\begin{array}{cc}{w}_{k,0}=\frac{{\mathrm{Mag}}_k\left(z_0\right)}{{\mathrm{Mag}}_0}{w}_0\mathrm{and}& \left(28\right)\\ {\theta }_k=\frac{{\mathrm{Mag}}_0}{{\mathrm{Mag}}_k\left(z_0\right)}{\theta }_0& \left(29\right)\end{array}$
denote the waist sizes and the divergence angles of the used beams in relation to the chief ray and the substitution w.sub.0=w.sub.0,0 applies. By solving, transposing and using equation (26), the following emerges as a conditional equation for the image-side far-field parameters θ.sub.0, w.sub.0 and z.sub.0

(64) $\begin{array}{cc}{w}_k^2={w_0^2\left(\frac{f_1}{f_1+\left(f_0+z_0\right)k}\right)}^2+{{\theta }_0^2\left(\frac{f_0^{2}k-z_0\left(f_1-f_{0}k\right)}{f_1}\right)}^2& \left(30\right)\end{array}$

(65) From this, the beam parameters are no longer determinable by a simple parabolic fit like in the case of equations (12)-(14). A possible scheme for determining the far-field parameters is obtained by rewriting the equation (30) in the style of equations (12)-(14) under the definition of the parameter pattern set

(66) $\begin{array}{cc}{A}_1=w_0^2.Math.m_1\left(k,z_0\right)={\left(\frac{f_1}{f_1+\left(f_0+z_0\right)k}\right)}^2& \left(31a\right)\\ A_2={\theta }_0^2{z}_0^2.Math.m_2\left(k\right)={\left(\frac{f_1-f_{0}k}{f_1}\right)}^2& \left(31b\right)\\ A_3=-2{\theta }_0^2{z}_0.Math.m_3\left(k\right)=\frac{f_0^{2}k}{f_1}\frac{\left(f_1-f_{0}k\right)}{f_1}& \left(31c\right)\\ A_4={\theta }_0^2.Math.m_4\left(k\right)={\left(\frac{f_{0}k}{f_1}\right)}^2& \left(31d\right)\end{array}$
into the form
w.sub.k.sup.2=A.sub.1m.sub.1(k,z.sub.0)+A.sub.2m.sub.2(k)+A.sub.3m.sub.3(k)+A.sub.4m.sub.4(k)  (32)

(67) By a linear fit of the patterns m.sub.1 (k, z.sub.0), m.sub.2(k), m.sub.3(k) and m.sub.4 (k) to the measured spot sizes, the parameters A.sub.1 to A.sub.4 are obtained, from which the far-field parameters are determined in a largely analogous manner to equation (14) by way of

(68) $\begin{array}{cc}{\theta }_0=\sqrt{A_4},z_0=-\frac{A_3}{2A_4},w_0=\sqrt{A_1}.& \left(33\right)\end{array}$

(69) Solving the equations (32) is more complex than the equations (14) obtained previously for the conventional beam evaluation as a result of the explicit dependence of the pattern m.sub.1 (k, z.sub.0) on z.sub.0. In accordance with one embodiment, this can be accounted for by selecting an iterative procedure as described in the following. In a first iteration step, m.sub.1 (k, z.sub.0) is replaced by m.sub.1(k,z.sub.0.sup.(0)=0) and a first estimate for z.sub.0.sup.(1) is obtained by the fit in accordance with equation (28) and the calculation in accordance with equations (32). In the next step, the improved pattern m.sub.1 (k, z.sub.0.sup.(1)=0) is calculated and the improved estimate z.sub.0.sup.(2) is obtained. The method is continued until a termination criterion placed on the iteration is satisfied and the parameters no longer change from iteration to iteration within the scope of the admissible boundaries.

(70) Below, possible designs of the diffractive structure 241 present in the beam-splitting optical arrangement 240 according to the invention are discussed.

(71) In principle, the beam-splitting optical arrangement 240 can be designed in two steps, wherein the base parameters (focal length f.sub.0 of the refractive optical element 242, focal length f.sub.1 of the diffractive structure 241 and decentration d.sub.x) are set in a first step and the specific step profile of the diffractive structure 241 is predetermined in a second step while optimizing the diffraction efficiencies for the individual sub-beams or orders of diffraction.

(72) The following emerges from equation (23) for the longitudinal focus offset between the marginal rays with the indices k=±k.sub.max and the chief ray:

(73) $\begin{array}{cc}.Math.z_{k_{\max }}\left(0\right)-z_0\left(0\right).Math.=\frac{f_0^2{k}_{\max }}{f_1\pm f_0{k}_{\max }}\stackrel{!}{=}{\kappa }_1\mathrm{DoF}& \left(34\right)\end{array}$
where the depth of field range of the far field DoF should be covered to a portion κ.sub.1 (typical value κ.sub.1=1). The lateral split between the marginal rays and the chief ray is

(74) $\begin{array}{cc}.Math.{\stackrel{\_}{x}}_{k_{\max }}-{\stackrel{\_}{x}}_0.Math.=\frac{f_0}{f_1}{k}_{\max }{d}_x\stackrel{!}{=}{\kappa }_2\frac{L_{\mathrm{sensor}}}{2}& \left(35\right)\end{array}$
and it should comprise half the given sensor length L.sub.sensor to a portion κ.sub.2 (typical value κ.sub.2=3/4). The depth of field range is defined as λ/NA.sup.2 for beams with a hard cutoff (e.g. top-hat beams) and as M.sup.2.Math.λ/π.Math.θ.sup.2 for Gaussian beams.

(75) From the relations (34) and (35), the following design rules emerge directly for the focal length f.sub.1 of the diffractive structure 241:

(76) $\begin{array}{cc}{f}_1=\frac{k_{\max }{f}_0}{{\kappa }_1\mathrm{DoF}}\left(f_0\mp \mathrm{DoF}\right)\underset{f_0>>\mathrm{DoF}}{\approx }\frac{k_{\max }{f}_0^2}{{\kappa }_1\mathrm{DoF}}& \left(36\right)\end{array}$
and for the decentration d.sub.x thereof:

(77) $\begin{array}{cc}{d}_x=\frac{{\kappa }_2}{2}\frac{L_{\mathrm{sensor}}}{k_{\max }}\frac{f_1}{f_0}& \left(37\right)\end{array}$

(78) By setting the two parameters f.sub.1 and d.sub.x, the phase function Φ(x, y) which should be realized by the diffractive structure 241 in the first order of diffraction is as follows:

(79) $\begin{array}{cc}\Phi \left(x,y\right)=\frac{2\pi }{\lambda }\left[\sqrt{{\left(x-d_x\right)}^2+y^2+f_1^2}-f_1\right]& \left(38\right)\end{array}$

(80) This function emerges from the phase difference of a spherical wave emanating from the location (d.sub.x,0, f.sub.1) and a plane wave with a propagation vector parallel to the z-axis considered at the position z=0, where the wavelength is denoted by λ.

(81) A suitable approximation of this phase function can be carried out by the following two operations:

(82) Initially, the phase is brought to the uniqueness range [0,2π] by the modulo operation in accordance with

(83) $\begin{array}{cc}{\Phi }_{\mathrm{mod}}\left(x,y\right)=\mathrm{mod}\left(\frac{2\pi }{\lambda }\left[\sqrt{{\left(x-d_x\right)}^2+y^2+f_1^2}-f_1\right],2\pi \right)& \left(39\right)\end{array}$

(84) Subsequently, the phase corrected thus is suitably transformed in this base range by a map U(w) with a definition range 0≤w≤1 in accordance with

(85) 0$\begin{array}{cc}{\Phi }_{\mathrm{DOE}}\left(x,y\right)=U\left(\frac{{\Phi }_{\mathrm{mod}}\left(x,y\right)}{2\pi }\right)& \left(40\right)\end{array}$

(86) The function U(w) describes the complex transmission function over the unit cell, normalized to a length of one, of a regular periodic grating. The diffraction efficiencies η.sub.k=|c.sub.k|.sup.2 for the orders of diffraction of such a grating, which are characterized by the index k=0, ±1, ±2, . . . , are determined by the Fourier coefficients

(87) $\begin{array}{cc}{c}_k=\underset{0}{\overset{1}{\int }}\mathrm{dx}\exp \left(-2\pi i\mathrm{kx}\right)U\left(x\right).& \left(41\right)\end{array}$

(88) A further the design object consists of adapting the diffraction efficiencies to the measurement application by the suitable selection of the complex transmission function U(w). In order to avoid light losses, a pure phase element with U(w)=exp(iϕ(w)) is considered without loss of generality below, which phase element, from a manufacturing technology point of view, can be realized by virtue of a thickness profile t(x, y) being introduced into the surface of a glass body, e.g. by etching. Here, the correspondence between thickness function phase is

(89) $\begin{array}{cc}t\left(w\right)=\frac{\varphi \left(w\right)}{2\pi }\frac{\lambda }{n_g-1}& \left(42\right)\end{array}$
where n.sub.g denotes the refractive index of the substrate material.

(90) The diffractive structure 241 present in the beam-splitting optical arrangement 240 according to the invention can be realized as an (e.g. binary) phase DOE. The class of binary phase DOEs (with two different phase values) which is easiest to manufacture has only one step per elemental cell. Only the phase shift Δϕ and the step position (given by the duty factor dc=w.sub.step) are available as design degrees of freedom. With the next higher class of binary DOEs with two levels per unit cell and the four design degrees of freedom of phase shift Δϕ, level widths b.sub.1 and b.sub.2, and distance d.sub.12 between the two levels, it is possible to set an optimized profile of the orders of diffraction up to and including the third order. The phase of the electromagnetic radiation diffracted at the phase DOE corresponds to the step heights of the etched steps.

(91) FIGS. 5A-5D depicts phase curves and the corresponding diffraction efficiencies for two different optimization objects:

(92) In FIGS. 5A,5B, the balance (as uniform as possible) of the used orders of diffraction −3≤k.sub.max≤+3 was optimized, with FIG. 5A showing an elemental cell of the grating. As depicted in FIG. 5B, in the phase DOE depicted here, a diffraction efficiency in the range of 0.11-0.12 is obtained in each case for the used orders of diffraction, whereas the remaining (unused) orders of diffraction are loaded with little energy. In the specific exemplary embodiment, this refinement has as a consequence that approximately 80% of the energy radiated-in lies within these used orders of diffraction, when use is made of a total of seven sub-beams or used orders of diffraction (comprising the (−3)rd to (+3)rd order of diffraction), and it can be used for the measurement. In particular, the beam-spitting optical arrangement 240 according to the invention thus renders it possible to generate diffraction efficiencies enabling a highly efficient, largely uniform distribution of the energy over the used orders of diffraction using a comparatively simple DOA (diffractive optical absorber) basic design (namely a binary or two-stage phase DOE), and so this refinement is advantageous, particularly in the case of weak light conditions.

(93) In further embodiments, as depicted in FIGS. 5C,5D, it is also possible to deviate in a targeted manner from the uniform distribution of the energy over the used orders of diffraction described above, wherein, for example in accordance with FIG. 5D, a substantially V-like curve of the diffraction efficiencies can be realized (by a corresponding grating design in accordance with FIG. 5C), in which the higher orders of diffraction are provided with more energy. As a result of this, it is possible to take account of the fact that the spots generated on the sensor arrangement 250 become wider toward the outside or with increasing distance from the ideal focal position, i.e. the corresponding regions become darker. By a curve of the diffraction efficiencies as shown in FIG. 5D, it is possible to at least partly compensate this effect of the decrease in intensity accompanying the increasing defocusing of the sub-beams generated by splitting the electromagnetic radiation to be analyzed in the sensor arrangement, as result of which the used region of the sensor arrangement can be enlarged. Therefore, an optimization was carried out in FIGS. 5C,5D in respect of an increase in the diffraction efficiencies, which is as linear as possible, with the magnitude of the order of diffraction in order to at least partly compensate the drop in intensity due to the beam widening away from the focus.

(94) In further embodiments, instead of a phase DOE, the diffractive structure according to the invention can also be realized by a transmission DOE or an (absorbing) grayscale DOA or by any other DOE systems, e.g. multi-stage DOEs, etc.

(95) Ultimately, a restriction of the concept according to the invention of using a diffractive structure for realizing a multi-focal beam-splitting optical arrangement or beam propagation camera is given by the restriction of the smallest achievable strip distance during the DOE production. The smallest-possible strip distance, denoted here as critical dimension cd, is the following for the embodied DOE design with two levels having the same height:
cd=Δr.sub.minmin(b.sub.1,b.sub.2,d.sub.12,1−b.sub.1−b.sub.2−d.sub.12)≈0.1Δmin  (43)
Here,

(96) $\begin{array}{cc}\Delta r_{\min }\approx \frac{f_1\lambda }{2r_{\max }}=\frac{f_1\lambda }{2\left(d_x+D_{\mathrm{aperture}}\right)}& \left(44\right)\end{array}$
denotes the smallest ring spacing of the zone plate with a focal length f.sub.1 at the maximum used radius of the zone plate. The latter is given by r.sub.max=d.sub.x+D.sub.aperture, where D.sub.aperture denotes the diameter of the aperture and d.sub.x denotes the desired decentration.

(97) Even though the invention has been described on the basis of specific embodiments, numerous variations and alternative embodiments are evident to the person skilled in the art, e.g. through combination and/or exchange of features of individual embodiments. Accordingly, such variations and alternative embodiments are concomitantly encompassed by the present invention, and the scope of the invention is restricted only within the meaning of the appended patent claims and equivalents thereof.