METHOD FOR DETERMINING A PHASE OF AN INPUT BEAM BUNDLE

Abstract

A method is presented for determining a phase of an input beam (110, E.sub.in) without a reference ray. In the method, an input beam (110, E.sub.in) having a plurality of input rays is split into a main beam (112, E1) and a reference beam (114, E2) in such a way that each input ray is split into a main ray of the main beam (112, E1) and a comparative ray of the reference beam (114, E2). The main beam (112, E1) is propagated along a first interferometer arm, and the reference beam (114, E2) is propagated along the second interferometer arm. The propagated main beam (112, E1) and the propagated reference beam (114, E2) are superposed to form an interference beam having a plurality of interference rays. The propagation along the first and second interferometer arms is carried out such that at least one interference ray of the interference beam is a superposition of a main ray of the propagated main beam (112, E1) assigned to a first input ray of the input beam (110, E.sub.in), and of a comparative ray of the propagated reference beam (114, E2) assigned to a second input ray of the input beam (110, E.sub.in) different from the first input ray.

Claims

1. A method for determining at least one phase of an input beam using an interferometer system, wherein the interferometer system comprises a detection surface and an optical system including a first interferometer arm and a second interferometer arm, wherein the interferometer system further comprises a beam splitter and a beam combiner, wherein the input beam comprises a plurality of input rays each including a phase, the method comprising: splitting the input beam into a main beam and a reference beam using the beam splitter such that each input ray is split into a main ray of the main beam and a comparative ray of the reference beam, wherein the main ray and the comparative ray are uniquely assigned to each input ray; propagating the main beam along the first interferometer arm and the reference beam along the second interferometer arm; superposing the propagated main beam and propagated reference beam using the beam combiner to form an interference beam including a plurality of interference rays; generating a hologram by: propagating the interference beam onto the detection surface; and measuring at least one interference pattern of the interference beam; determining a propagator mapping of the optical system, wherein the propagator mapping describes a propagation of the propagated main beam into the propagated reference beam; reading out the hologram with at least a portion of a test beam including a test phase to generate a first beam; applying the propagation mapping to at least a portion of the test beam to generate a second beam; comparing the first beam and the second beam; and determining at least a portion of the phase of the input beam from the test phase of the test beam; repeating said reading out the hologram, said applying the propagation mapping, said comparing the first beam and the second beam until the first beam and the second beam are substantially identical except for at least one of local intensity differences or a global phase; said propagating the main beam along the first interferometer arm and the reference beam along the second interferometer arm takes place in such a way that at least a portion of the interference rays of the interference beam is in each case a superposition of a main ray of the propagated main beam assigned to a first input ray of the input beam, and of a comparative ray of the propagated reference beam assigned to a second input ray of the input beam different from the first input ray; and performing said method without using a reference ray.

2. The method of claim 1, wherein said propagating the main beam along the first interferometer arm and the reference beam along the second interferometer arm takes place such that an interference ray of the interference beam comprises a superposition of a main ray of the propagated main beam assigned to a first input ray of the input beam and of a comparative ray of the propagated reference beam assigned to a second input ray of the input beam identical to the first input ray.

3. The method of claim 1, wherein the test beam is arbitrarily selected from a set of beams that become a first beam when applying the propagation mapping.

4. The method of claim 1, wherein at least one of the first interferometer arm or the second interferometer arm comprises a beam rotation element, and wherein said propagating the main beam along the first interferometer arm and the reference beam along the second interferometer arm further comprises: rotating at least one of the main beam or the reference beam about an axis of rotation extending along the propagation direction by a rotational angle using the beam rotation element.

5. The method of claim 1, wherein at least one of the first interferometer arm or the second interferometer arm of the optical system comprises at least one lens, and wherein said propagating the main beam along the first interferometer arm and the reference beam along the second interferometer arm further comprises: at least one of focusing or defocusing at least one of the main beam or the reference beam with the at least one lens.

6. The method of claim 1, wherein said determining the propagator mapping further comprises: determining a first propagator mapping describing the propagation of the input beam through the beam splitter and along the first interferometer arm; determining a second propagator mapping describing a propagation of the input beam through the beam splitter and along the second interferometer arm; inverting the first propagator mapping; and multiplying the inverse of the first propagator mapping with the second propagator mapping in order to determine the propagator mapping.

7. The method of claim 1, wherein the interferometer system further comprises a beam source configured to emit an initial beam propagating along a propagation direction wherein the initial beam comprises a plurality of initial rays, wherein the method further comprises: irradiating at least a portion of an object with the initial beam such that at least a portion of the initial rays of the initial beam are transformed by reflection on object points of an outer surface of the object into the plurality of input rays of the input beam, wherein the phase of each input ray is a phase shift with respect to an unambiguously bijectively assigned initial ray of at least a portion of the initial rays.

8. The method of claim 1, wherein the detection surface is configured to store the hologram, wherein the method further comprises: reading the hologram in order to reconstruct the holographic mapping of the object, wherein the reading includes diffracting at least one of the reference beam or the main beam on the hologram.

9. The method of claim 1, comprising at least one of: selecting a frequency of the test beam which is part of a spectrum of the main beam and determining the hologram at the frequency; determining at least one of an intensity or amplitude of at least one of the propagated main beam the propagated reference beam on the detection surface at the frequency of the test beam; determining a complex interference term of the hologram at the frequency of the test beam; determining a complex phase-matching factor which takes into account possible scales and total phase differences between the determination of the propagator mapping and the complex interference term.

10. The method of claim 9, wherein said determining at least one of an intensity or amplitude of at least one of the propagated main beam or the propagated reference beam and said determining a complex interference term of the hologram are performed using at least one of a phase shifting method or a carrier phase method.

Description

[0122] The method described here is explained in more detail below with reference to exemplary embodiments and the associated figures.

[0123] FIGS. 1a, 1b, 1c, 1d, 1e and 2 show exemplary embodiments of a method described here with reference to schematic representations.

[0124] FIGS. 3 to 5 show simulations and measurement results for an exemplary embodiment of a method described here.

[0125] Identical, similar or identically acting elements are provided with the same reference signs in the figures. The figures and the size relationships of the elements illustrated in the figures to each other are not to be considered as true to scale. Rather, individual elements may be shown exaggeratedly large for better presentability and/or for better understanding.

[0126] On the basis of the schematic representations of FIGS. 1a, 1b, 1c, 1d and 1e, the exemplary embodiments of interferometer systems for exemplary embodiments of a method described here are explained in more detail. FIGS. 1a to 1d respectively show Mach-Zehnder-like interferometer systems, while FIG. 1e shows a Fabry-Prot-like set-up.

[0127] Each of the interferometer systems of FIGS. 1a to 1d comprises a detection surface 124 and an optical system having a beam splitter 140 and a beam combiner 144 for providing a first and a second interferometer arm. An input beam 110 incoming through a diaphragm 120 is split into a main beam 112 and a reference beam 114 by means of the beam splitter 140. The main beam 112 is propagated along the first interferometer arm, and the reference beam 114 is propagated along the second interferometer arm. Each of the two interferometer arms contains optical components of the optical system. The first interferometer arm contains a first deflection element 142, and the second interferometer arm contains a second deflection element 146. The deflection elements 142, 146 may be mirrors, for example. In the interferometer systems according to FIGS. 1a, 1b, 1c and 1d, the propagated main beam 112 and the propagated reference beam 114 are combined into an interference beam by means of the beam combiner 144, which is arranged downstream of the beam splitter 140 in the propagation direction of the main beam 112 or the reference beam 114, and interfere on the detection surface 124. The resulting interference pattern is stored digitally or physically as interferogram 126 or as hologram 126. The exemplary embodiments of FIGS. 1a, 1b, 1c and 1d differ in this respect as follows.

[0128] The optical system of the exemplary embodiment of FIG. 1a comprises an optical element 130 introduced into the second interferometer arm. For example, the optical element 130 may be a beam rotation element and/or a diffractive optical element. By means of the optical element 130, it is possible to rotate and/or change the reference beam 114 propagating along the second interferometer arm in comparison to the main beam 112 such that in the superposition using the beam combiner 144, rays with different phases respectively interfere. Alternatively or additionally, it is possible for the first interferometer arm to contain an optical element (not shown in FIG. 1).

[0129] The optical system of the exemplary embodiment of FIG. 1b comprises a lens 132, which may also be a lens system, introduced into the second interferometer arm. Merely by way of example, a concave lens 132, i.e. a diverging lens, is shown. The reference beam 114 can be focused and/or defocused by means of the lens 132. It is also possible to rotate the reference beam 114 about its own axis by means of the lens (see also FIG. 1c). It is furthermore possible that the phases of the comparative rays of the reference beam 114 can be modulated due to different optical path lengths in the lens 132.

[0130] The optical system of the exemplary embodiment of FIG. 1c comprises a first lens 136 introduced into the first interferometer arm and a second lens 134 introduced into the second interferometer arm. The first lens 136 and/or the second lens 134 may also respectively be a lens system. Merely by way of example, the first lens 136 is shown as a concave lens, i.e. as a diverging lens, and the second lens 134 is shown as a convex lens, i.e. as a converging lens. The first lens 136 and the second lens 134 may be designed differently. For example, the two lenses differ in their curvature and/or their refractive power. Furthermore, FIG. 1c shows by way of example the propagation of an input ray 108 of the input beam 110 arranged away from a central ray of the input beam 110. Within the scope of manufacturing tolerances, the central ray coincides with at least one of the optical axes of the lenses.

[0131] The input ray 108 is split by the beam splitter 140 into a first main ray 116 of the main beam 112 and a first comparative ray 118 of the reference beam 114. The first main ray 116 is propagated along the first interferometer arm, i.e. in particular by the first lens 136, while the first comparative ray 118 is propagated along the second interferometer arm, i.e. in particular by the second lens 134. In this case, the first main ray 116 is refracted away from the optical axis, while the first comparative ray 118 is first refracted toward the optical axis. A focus is in this case formed behind the ray deflection element 146. Subsequently, the propagated main beam 112 and propagated reference beam 114 are superposed by the beam combiner 144 into an interference beam having a first interference ray 152 and a second interference ray 154. The first interference ray 152 and the second interference ray 154 strike the detection surface 124 at different points, particularly away from a central interference ray 150, which reflects the propagation of the central ray of the input beam 110 through the interferometer arms. The first interference ray 152 contains the first main ray 116, and the second interference ray 154 contains the first comparative ray 118. Due to the different propagations within the two interferometer arms, the first main ray 116 and the first comparative ray 118 thus strike the detection surface 124 at different points.

[0132] In one exemplary embodiment, the interferometer system may also comprise only the first lens 136 or the second lens 134. The respectively other lens 134, 136 may however in particular allow an adaptation of the phase fronts of the main beam 112 and the reference beam 114. For example, the respective phase fronts of the main beam 112 and of the reference beam 114 have the same curvature due to the use of two lenses or two lens systems 134, 136.

[0133] The optical system of the exemplary embodiment of FIG. 1d comprises a diffractive optical element 138 introduced into the second interferometer arm. For example, the diffractive optical element is an optical lattice 138. By means of the diffractive optical element 138, the phases of the comparative rays of the reference beam 114 may for example be modulated due to different optical path lengths.

[0134] The interferometer system of FIG. 1e is based on the set-up of a Fabry-Prot interferometer. As before in FIG. 1c, the propagation of an input beam 110 including an input ray 108 of the input beam 110 arranged away from a central ray of the input beam 110 is shown by way of example. The interferometer system comprises a diaphragm 120 through which the input ray 110 comes, a beam combiner 144, a beam splitter 140 arranged downstream of the beam combiner 144 in the propagation direction, and a detection surface 124.

[0135] The beam splitter 140 and the beam combiner 144 are inclined with respect to the propagation direction of the input beam 110 and in particular are not oriented orthogonally to the propagation direction. Furthermore, the beam combiner 144 has a curvature. The beam combiner 144 can be a further beam splitter and/or a deflection element. The beam splitter 140 may be a curved partially transparent mirror. The beam combiner 144 can have a reflection of at least 80% and at most 95%, for example 90%, and a transmission of at least 5% and at most 15%, for example 10%, at a wavelength of the input beam 110. Although this makes it possible for the input beam 110 to lose much intensity, i.e., for example 90% of the intensity, when passing through the beam combiner 144, the loss in a later reflection is, conversely, lower.

[0136] The input beam 110 passes through the beam combiner 144 and is subsequently split by the beam splitter 140 into a main beam 112 and a reference beam 114. In addition, the input ray 108 is split into a first main ray 116 and a first comparative ray 118. The main beam 112 including the first main ray 116 propagates along the original direction of the input beam 110 towards the detection surface 124. The reference beam 114 including the first comparative ray 118 is deflected in the direction of the beam combiner 144 which directs the reference beam 114 through the beam splitter 140 in the direction of the detection surface 124 by means of a further reflection. The main beam 112 and the reference beam 114 interfere as an interference beam comprising a first interference ray 152 and a second interference ray 154. Due to the different propagations of the main beam 112 and of the reference beam 114, the first interference ray 152 and the second interference ray 154 strike the detection surface 124 at different points. The central interference ray 150 again reflects the propagation of the central ray of the input beam 110 through the interferometer arms. The first interference ray 152 contains the first main ray 116, and the second interference ray 154 contains the first comparative ray 118.

[0137] The overlap of the main beam 112 and the reference beam 114 on the detection surface 124 can, for example, be adjusted by means of the inclination of the beam combiner 144 and/or the beam splitter 140 with respect to the propagation direction of the input beam 110. By selecting the reflection and transmission of the beam combiner 144, it is possible to adjust an intensity ratio of the main beam 112 and the reference beam 114 in the interference beam. For example, the intensities of the two beams 112, 114 in the interference beam are identical. It may be advantageous to provide a Fabry-Prot-like interferometer system with a low finesse, since undesired resonance effects, for example in the wavelength, can be avoided in this way.

[0138] It is possible that, due to partial re-reflection of the reference beam 114 on the beam splitter 140, further beams with lower intensity could be produced. Such beams can be taken into account in the method described here. For example, a propagator mapping U which transforms the main beam 112 (E1) into the sum of the reference beam 114 (E2) and said further beam (E2, E2, . . . ), U(E1)=(E2+E2+E2+ . . . ), may be taken into account for this purpose. In this case, E2 in the above formulae would have to be substituted by (E2+E2+E2+ . . . ). In this case, the linearity of the field equations can be utilized, whereby a plurality of beams can be combined to form one beam.

[0139] On the basis of the schematic representation of FIG. 2, an exemplary embodiment of a method described here is explained in more detail. FIG. 2 shows schematically the reconstruction of the phase of the input beam 110 by using a test beam 170. The test beam 170 is split into two parts by means of a further beam splitter 200. The further beam splitter 200 may be a physical beam splitter, such as a prism, or a digital beam splitter, such as a numerical division, for example. By means of a first portion of the test beam 170, a first beam 172 is generated by reading the hologram 126 formed based on the interference beam.

[0140] Furthermore, a propagation mapping is applied to a second portion of the test beam 170 in order to generate a second beam 174. The propagation mapping corresponds to a propagation of the second portion of the test beam 170 through an optical system having a first path length 190, a first further deflection element 202, a second path length 192, a second further deflection element 204 and a third path length 194. The optical system may contain further optical components, such as lenses, prisms and/or small wave plates. The optical system or the propagation mapping corresponds to the optical system of the interferometer system. In other words, the optical system corresponds to a propagation of the main beam 112 through the first interferometer arm and a backpropagation of the propagated main beam 112 through the second interferometer arm. For example, the optical system of the interferometer system corresponding to FIG. 2 exclusively contains beam splitters and deflection elements, for example mirrors.

[0141] The first beam 172 and the second beam 174 are superposed by means of a further, digital and/or physical beam combiner 206 and compared to a reference unit 210. If the first beam 172 and the second beam 174 are substantially identical (with the exception of local intensity differences and/or a global phase), the test beam 170 corresponds to the input beam 110, or the main beam 112, or the reference beam 114.

[0142] Based on the measurement data of FIGS. 3 to 5, an exemplary embodiment of the method for determining a phase is explained in more detail. In the exemplary embodiment, the method is used, by way of example, for an interferometer system in accordance with the exemplary embodiment of FIG. 1a whose optical system is composed of beam splitters and deflection elements, for example mirrors. The first interferometer arm and the second interferometer arm of the interferometer are designed with different lengths and have a first length L1 and a second length L2, respectively. The input beam 110 is a monochromatic plane wave described by the amplitude factor exp(i.Math.k.Math.x) and the fixed frequency . The propagation direction corresponds to the z axis, and the detection surface 124 corresponds to the x, y plane at z=0.

[0143] The input beam 110 is split into a main beam 112 and a reference beam 114. The reference beam 114 is propagated along a second interferometer arm and rotated in the second interferometer arm such that the propagated main beam on the detection surface 124 is rotated by 180 compared to the main beam propagated through a first interferometer arm. A central ray of the reference beam running along the propagation direction serves as an axis of rotation. Furthermore, the propagated reference beam is shifted by a path length dx in comparison to the propagated main beam.

[0144] The rotation corresponds to the transformation (x,y).fwdarw.(x,y). For the propagated comparative ray field, the following then applies:

E2expi(k.sub.x[xdx]k.sub.yy+k.sub.z[L2L1]), and thus

IF.sub.s,=exp (k.sub.x[2x+dx]+k.sub.y2yk.sub.z[L2L1]), where k.sub.z={square root over ((/c).sup.2k.sub.x.sup.2k.sub.y.sup.2)}.

[0145] In the position space, the complex interference term IF corresponds to a point-by-point multiplication with the exponential function shown on the right side of the equation. Alternatively, the complex interference term IF can also be determined with a measurement.

[0146] In the exemplary embodiment, a square with an edge length D and MN pixels is assumed as detection surface 124. For example, the edge length is 6 mm. In the x direction, the indexing is x.sub.m=D/2+mD/M, where m=0, 1 . . . M1. In the y direction, the indexing is y.sub.n=D/2+nD/N, where n=0, 1 . . . N1. It is in particular possible for the total number of pixels in the x direction to be the same as the total number of pixels in the y direction (M=N). For even M and N, the center of the detection surface has the indices (N/2, m/2) with (x, y)=(0,0). The polarization of the reference beam remains unchanged in this case.

[0147] The amplitude of the propagated main ray field corresponds to the absolute value of the plane wave and can be determined, for example, by blocking the reference beam within the second interferometer arm and merely measuring the propagated main beam on the detection surface. For example, the main beam is a vector field polarized in the x direction, wherein |E1.sub.x|=1, |E1.sub.y|=0 and |E1.sub.z|=0 at z=0.

[0148] In a further method step of the exemplary embodiment of the method, the first polarization matrix, the second polarization matrix and thus the polarization matrix are determined by means of a straight line direction, for example a linearization, of the interferometer system by means of a so-called tunnel diagram. In particular, all reflections on mirrors and other optical elements are unfolded. The beam propagation through the two interferometer arms is further determined from a propagation in free space with refractive index n=1 (vacuum), once over the first length L1 and once over the second length L2. In the second interferometer arm, the displacement by the path length dx must be taken into account.

[0149] The propagation matrix of the exemplary embodiment then corresponds to a solution of the Helmholtz equations in Fourier space. The propagation matrix is equivalent to the Rayleigh-Sommerfeld propagator. For propagation between different planes in which z is kept constant, the Rayleigh-Sommerfeld propagator assumes a particularly simple form in Fourier space (see, for example, Diffraction, Fourier Optics and Imaging, Okan K. Ersoy, Wiley Interscience 2007, Angular Spectrum of Plane Waves, chapter 4.3, and Fast Fourier Transform (FFT) Implementation of the Angular Spectrum of Plane Waves, chapter 4.4.).

[0150] The first propagation matrix and the second propagation matrix, in which the rotation was not yet taken into account, read as follows in Fourier space:

[00006]$\hat{U}.Math..Math.1_{t,{\mathrm{pm}}^{},{\mathrm{qn}}^{},s,p.Math..Math.m,q.Math..Math.n,}={}_{s,t}.Math.{}_{\mathrm{pm},{\mathrm{pm}}^{}}.Math.{}_{q.Math..Math.n,q.Math..Math.n^{}}.Math.e^{i.Math..Math.L.Math..Math.1.Math.\sqrt{{\left(/c\right)}^2-{\mathrm{pm}}^2-q.Math..Math.n^2}},.Math.\hat{U}.Math..Math.{2^{}}_{t,{\mathrm{pm}}^{},{\mathrm{qn}}^{},s,p.Math..Math.m,q.Math..Math.n,}={}_{s,t}.Math.{}_{\mathrm{pm},{\mathrm{pm}}^{}}.Math.{}_{q.Math..Math.n,q.Math..Math.n^{}}.Math.e^{i.Math..Math.L.Math..Math.2.Math.\sqrt{{\left(/c\right)}^2-{\mathrm{pm}}^2-q.Math..Math.n^2}}.$

[0151] Here, pm, pm, qn, qn are the k vectors for the indices in Fourier space, matching the selected indexing in the position space, and .sub.pmm pm, .sub.qn,qn are Kronecker functions. The following applies:

[00007]$p.Math..Math.m=\frac{2.Math..Math.}{D}.Math.m.Math..Math.\mathrm{for}.Math..Math.m=0,\mathrm{.Math.}.Math.,M-1$$\mathrm{and}$$\mathrm{qn}=\frac{2.Math..Math.}{D}.Math.n.Math..Math.\mathrm{for}.Math..Math.n=0,\mathrm{.Math.}.Math.,N-1..Math.$

[0152] In the exemplary embodiment, U1 and U2 in Fourier space are thus unitary diagonal matrices.

[0153] The Fourier transformation FT of the field E.sub.s,m,n in the position space into the field .sub.s,pm,qn in Fourier space and the corresponding Fourier backtransformation FT.sup.1 read as follows:

[00008]${\hat{E}}_{s,p.Math..Math.m,q.Math..Math.n}=\underset{m^{}=0}{\overset{M-1}{.Math.}}.Math..Math.\underset{n^{}=0}{\overset{N-1}{.Math.}}.Math.E_{s,m^{},n^{}}.Math.e^{-2.Math..Math..Math.i.Math..Math.m^{}.Math.mM}.Math.e^{-2.Math..Math..Math.i.Math..Math.n^{}.Math.nN}.Math.\text{=}.Math.\text{:}.Math..Math.\mathrm{FT}.Math.\left(E\right),.Math.E_{s,m,n}=\frac{1}{M.Math..Math.N}.Math.\underset{m^{}=0}{\overset{M-1}{.Math.}}.Math..Math.\underset{n^{}=0}{\overset{N-1}{.Math.}}.Math.{\hat{E}}_{s,p.Math..Math.m^{},q.Math..Math.n^{}}.Math.e^{2.Math..Math..Math.i.Math..Math.m^{}.Math.mM}.Math.e^{2.Math..Math..Math.i.Math..Math.n^{}.Math.nN.Math.}.Math..Math.\text{=}.Math.\text{:}.Math..Math.{\mathrm{FT}}^{-1}\left(\hat{E}\right).$

[0154] Here, the constant index was suppressed.

[0155] The product of the second propagation matrix and the inverse of the first propagation matrix must be formed in order to determine the propagation matrix. In addition, the displacement by the path length dx and the rotation by 180 must be taken into account.

[0156] According to the Fourier theory, the displacement by dx in Fourier space corresponds to a multiplication with a phase factor exp(i.Math.dx.Math.pm). The rotation in position space corresponds to the transformation (x,y).fwdarw.(x,y) (see above); in the indices of the detection surface, this corresponds to the transformation (m,n).fwdarw.(Mm,Nn). For even M and N, the doublet (m,n)=(M/2,N/2) is a fixed point of rotation. Rotation by 180 in position space also results in a rotation by 180 in Fourier space, i.e. (p.sub.m,q.sub.n).fwdarw.(p.sub.m,q.sub.n).

[0157] Below, index points outside the detection surface, that is to say outside the index range, are brought into the index range by addition/subtraction. For example, the pixel having the index (M,N) corresponds to the pixel having the index (0,0). The pixel with the index (0,0) can be a further fixed point in the modeling of the rotation. This can be caused by the periodic Fourier boundary conditions. This addition/subtraction does have an influence on the choice of propagation matrix but not on the determination of the input beam. In particular, in the exemplary embodiment, the main beam and the reference beam and thus also the input beam are real fields in which no boundary conditions are assumed at the outer regions of the detection surface.

[0158] Overall, the propagation matrix in Fourier space and, by corresponding Fourier backtransformation, the propagation matrix in position space thus result:

[00009]$.Math.{\hat{U}}_{t,{\mathrm{pm}}^{},{\mathrm{qn}}^{},p.Math..Math.m,q.Math..Math.n,}=e^{-i.Math..Math.\mathrm{dx}.Math..Math.\mathrm{pm}}.Math.{}_{-\mathrm{pm},{\mathrm{pm}}^{}}.Math.{}_{-q.Math..Math.n,q.Math..Math.n^{}}.Math.e^{i.Math..Math.\left(L.Math..Math.2-L.Math..Math.1\right).Math.\sqrt{{\left(/c\right)}^2-{\mathrm{pm}}^2-q.Math..Math.n^2}}$$U_{t,m^{},n^{},s,m,n,}=\frac{{}_{s,t}}{M.Math..Math.N}.Math.\underset{m^{}=0}{\overset{M-1}{.Math.}}.Math..Math.\underset{n^{}=0}{\overset{N-1}{.Math.}}.Math.e^{-2.Math..Math..Math.i.Math..Math.m^{}.Math.m^{}.Math.mM}.Math.e^{-2.Math..Math..Math.i.Math..Math.n^{}.Math.n^{}N}.Math.e^{-i.Math..Math.\mathrm{dx}.Math..Math.\mathrm{pm}}.Math.e^{i.Math..Math.\left(L.Math..Math.2-L.Math..Math.1\right).Math.\sqrt{{\left(/c\right)}^2-{{\mathrm{pm}}^{}}^2-q.Math..Math.{n^{}}^2}}.Math.e^{-2.Math..Math..Math.i.Math..Math.m^{}.Math.mM}.Math.e^{-2.Math..Math..Math.i.Math..Math.n^{}.Math.nN}$

[0159] For example, these formulae can be calculated and/or applied by means of so-called Fast Fourier Transform (FFT) routines. For example, Python Scipy is suitable for this purpose.

[0160] In a further method step of the exemplary embodiment, the complex phase-matching factor is determined by phase matching. For this purpose, the complex interference term IF and the propagation matrix are calculated approximately for exactly the same difference between the first and the second length. This results in a phase-matching factor f(w)=1.

[0161] In a further method step, the phase of the input beam is determined by means of the propagation matrix of the optical system and the measured interferogram. For this purpose, the conditional equation (6) is solved, wherein the index is suppressed. The equation to be solved thus reads as follows:

[00010]$\begin{array}{cc}\stackrel{\_}{{\mathrm{IF}}_{t,m,n,}}.Math.E.Math..Math.1_{t,m,n,}=\underset{s,u,v}{.Math.}.Math.U_{t,m,n,s,u,v,}.Math.E.Math..Math.1_{s,u,v},& \left(6.Math..Math.b\right)\end{array}$

[0162] wherein the conjugate of the complex interference term IF in Fourier space generates the following transformation of the main beam E1:

1.sub.s,pm,qn.fwdarw.e.sup.i(k.sup.x.sup.dxk.sup.z.sup.[L2L1])1.sub.s,pm2kx,qn2ky.

[0163] Thus, the reformulated equation (6b) in Fourier space reads:

[00011]$e^{i\left(-k_x.Math.\mathrm{dx}+k_z\left[L.Math..Math.2-L.Math..Math.1\right]\right)}.Math.\hat{E}.Math.1_{s,\mathrm{pm}-2.Math.\mathrm{kx},\mathrm{qn}-2.Math.\mathrm{ky}}-e^{-i.Math..Math.\mathrm{dx}.Math..Math.\mathrm{pm}}.Math.e^{i\left(L.Math..Math.2-L.Math..Math.1\right).Math.\sqrt{{\left(/c\right)}^2-{\mathrm{pm}}^2-{\mathrm{qn}}^2}}$$\hat{E}.Math.1_{s,-\mathrm{pm}-\mathrm{qn}}=0.$

[0164] This equation is an eigenvalue equation for E1.sub.s,pm,qn for eigenvalue 0. Inserting shows that the eigenvalue equation is satisfied for p.sub.m2k.sub.x=p.sub.m and q.sub.n2k.sub.y=q.sub.n.

[0165] This means that for k.sub.x=2.Math.m.sub.in/D and k.sub.y=2.Math.n.sub.in/D, where m.sub.in and n.sub.in are integers and m.sub.in=0, . . . M1; n.sub.in=0 . . . N1, the system of equations has the following solution for the reconstructed field with the exception of one phase (and the normalization):

[00012]$r.Math..Math.e.Math..Math.c.Math..Math.\hat{E}.Math..Math.1_{s,p.Math..Math.m,q.Math..Math.n}=\{\begin{array}{cc}1& \mathrm{for}.Math..Math.m=m_{\mathrm{in}},n=n_{\mathrm{in}},s=1\\ 0& \mathrm{otherwise}\end{array}$

[0166] This corresponds to the Fourier transform of the main beam. A numerical calculation can also show that the solution is also stable. There is thus no further solution to the equation, for example for L2L1=1 cm. Moreover, it can be shown numerically that for non-integer m.sub.in and/or n.sub.in, equation (6b) has a unique approximate solution. The solution for the lowest eigenvalue is sought in this case. The numerical results for m.sub.in=4.5 and n.sub.in=0 are shown in FIG. 3.

[0167] FIG. 3 shows the amplitude of the real part of the field for recE1.sub.1,m,N/2 and for a wave in which the phase of recE1.sub.1,m,N/2 but the amplitude of the main beam 112 is used, along the x axis, i.e. for the indices m=0 . . . M for y=0. In the latter solution, in which only the phase of the reconstructed solution is used, the system of equations (6b) is thus used only for phase determination. m.sub.in=4.5 was used for this numerical example. This can be seen in FIG. 3 by counting the nodal lines. The input field E1 is thus not periodic in the viewing window of the discretization. However, the numerical method provides the correct result for the phase. In particular, the dashed line corresponds to the input field E1. This shows the high stability of the method, even if the data or the propagator mapping are in particular slightly erroneous. As a result, this provides a solution that corresponds in the phase to the main beam E1 up to 2.Math.10.sup.4 rad. The reason for the deviation can be the different treatment of the fields at the edge of the detection surface.

[0168] FIG. 4 shows a measurement of a first and a second data set for a field along the x axis, i.e. for the indices m=0 . . . M for y=0. Two data sets in the conditional equation (6b) were used in the measurement. In the first data set, the path length dx=0 pixels, and in the second data set, the path length dx=4.9 pixels. The reference beam was rotated by a rotational angle of 176.1. The path difference L1-L2 is 1.65 mm. The phase of the complex interference term IF was determined according to the phase shifting method using an 8-bit detector.

[0169] The input beam of the exemplary embodiment is a Gaussian beam diffracted on an iris diaphragm. The diffraction on the iris diaphragm corresponds to a so-called Fresnel diffraction. By using a method described herein, the measured Fresnel diffraction can be compared to the theory. Conclusions about the effectiveness of the method can be drawn thereby.

[0170] The beam incidence is slightly obliquely to the optical axis, comparable to the incidence of the plane wave in the example of FIG. 3. The structuring of the plane wave interference visible in FIG. 4 is reproducible and represents the influence of the Fresnel diffraction. The diffraction effects are shown in more detail in FIG. 5. For this purpose, the phase influence coming from the oblique incidence is compensated and the results are shown in relation to a straight incidence.

[0171] FIG. 5 shows the phase and amplitude of the main beam on the detection surface, which is arranged 200 mm after the iris diaphragm for Fresnel diffraction. The variations and oscillations shown, both in the amplitude of the main beam |E1| and in the phase .sub.E1, can be reproduced up to 1.Math.10.sup.2 rad by means of independent measurements. The measured data are compatible with a so-called boundary wave simulation of the diffraction on the iris diaphragm. For this verification, the measured field was numerically propagated in the z direction up to the position of the diaphragm. A good match to the theoretical expectation, which is a cut-off field on the diaphragm, results.

[0172] The explicit calculation and the results of the method for the exemplary embodiment shown in the figures show that, even under unfavorable conditions, the reconstructed field recE1.sub.1,m,n, contains the solution for the correct phase. However, the reconstructed field does not necessarily contain the correct amplitude.

[0173] The phase determination described herein without a reference ray, in particular the scalar and vectorial phase determination without a reference ray, results in several advantages. In particular, the elimination of an in particular external can lead to a maximum flexibility in the design of the interferometer systems to be evaluated, since even complicated situations can be detected without approximation in a phase determination without a reference ray. The requirements for the coherence length of the field used are also reduced; as a result, the Fourier transform property of interferometers in the time/frequency domain can, for example, be used. Field measurement at different frequencies is also conceivable. The exact systematic method in the formalism is also compatible with Fourier optics and pulsed lasers. The method described here allows the digital method enables the numerical determination of the phase of any input field.

[0174] In addition, the method can be used in metrology, in the quality control of workpieces in production, in three-dimensional surface inspection, in the testing of the authenticity of banknotes, in holographic microscopy with three-dimensional, in particular high resolution (so-called super resolution), in three-dimensional diffraction tomography, x-ray tomography and/or neutron beam tomography. Relative distance determinations in semitransparent objects, for example in terahertz separation of an object behind a semitransparent wall, are also possible.

[0175] The method for the determination of the phase without a reference ray has been tested successfully both numerically and experimentally (see FIGS. 4 and 5 in this respect). The phase reconstruction takes place with an accuracy determined by the detector used, i.e. an 8-bit detector allows a phase determination with a highest accuracy of approximately 2E-8 bits, i.e. 0.004 rad. An accuracy of about 0.01 rad was achieved experimentally under these conditions. This constitutes a favorable agreement with the experimental results.

[0176] The description on the basis of the exemplary embodiments and/or the description on the basis of the theoretical models does not limit the invention thereto. Rather, the invention encompasses any new feature as well as any combination of features, including in particular any combination of features in the claims, even if this feature or combination itself is not explicitly specified in the claims, the exemplary embodiments and/or the theoretical models, for example, the construction of ultra-compact interferometer systems for phase measurement, which can optionally also be designed as internal reflection devices, and can digitally measure the phase of a light field at one or more wavelengths.

[0177] Various applications of the method are conceivable as a result of the described advantages. For example, the physical method using a physically stored hologram enables the determination or detection of a complex input field having a phase by checking the input field against the hologram. Only when reading the hologram indicates a match with the internally generated test ray, is the input field the field used for the generation of the hologram. The input field cannot be derived from the hologram itself. Such a method is suitable for authenticity verification and cryptography, for example in the context of asymmetric encryption methods.

LIST OF REFERENCE SIGNS

[0178] 108 Input ray [0179] 110 Input beam [0180] 112 (Propagated) main beam [0181] 114 (Propagated) reference beam [0182] 116 First main ray [0183] 118 First comparative ray [0184] 120 Diaphragm [0185] 124 Detection surface [0186] 126 Hologram [0187] 130 Optical element [0188] 132 Lens [0189] 134 Second lens [0190] 136 First lens [0191] 138 Diffractive optical element [0192] 140 Beam splitter [0193] 142 First deflection element [0194] 144 Beam combiner [0195] 146 Second deflection element [0196] 150 Central interference ray [0197] 152 First interference ray [0198] 154 Second interference ray [0199] 170 Test beam [0200] 172 First beam [0201] 174 Second beam [0202] 190 First path length [0203] 192 Second path length [0204] 194 Third path length [0205] 200 Further beam splitter [0206] 202 First further deflection element [0207] 204 Second further deflection element [0208] 206 Further beam combiner [0209] 210 Reference unit