RU2458367C2 - Optical device with pair of diffraction optical elements - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: when two diffraction optical elements are placed next and parallel to each other at a certain distance, the combination optically corresponds to a single diffraction optical element which acts as a lens, an axicon, a phase changer or a spiral retarder. If one of the diffraction optical elements rotates relative the other about a common central axis, the optical property of the device, for example, focal distance, refractive power, spiral index or phase shift, varies continuously.

EFFECT: high efficiency and compactness.

20 cl, 15 dwg

Description

The present invention relates to an optical device comprising a pair of diffractive optical elements, the optical device acting as a specific optical element having a continuously variable optical property.

Diffractive optical elements (hereinafter abbreviated as DOEs) are usually available in the form of optically transparent, flat plates made of glass, plastic, etc., which have an imprinted phase pattern of a special design with microscopically precise phase modulation. DOEs for specific tasks are commercially available, acting, for example, as line or pattern generators, holographic projectors, laser beam profile converters, lenses, lens arrays, axicons (ie ring-shaped prisms), etc. For example, diffractive lenses (Fresnel lenses) commercially used in conjunction with normal refractive lenses in high-quality photographic optics. One advantage of a DOE lens is that it is very thin and light in weight, consists of thin glass plates or even just a structured coating on top of a normal glass lens. In addition, with proper design, the dispersion properties of DOEs can compensate for the properties of normal refractive glass optics, which makes it possible to build dispersion-free optical systems that do not have chromatic aberrations.

In A.W. Lohmann, "A new class of varifocal lenses", Appl. Opt. 9, 1669-1671 (1970), a diffractive zoom lens system based on the transverse shift of two diffractive elements is considered. In an article by A. Kolodziejczyk and Z. Jaroszewicz, "Diffractive elements of variable optical power and high diffraction efficiency", Appl. Opt. 32, 4317-4322 (1993), describes the creation of diffractive elements of variable optical power based on the mutual displacement of diffraction structures with encoded pure-phase wave fronts. Mutual displacement can be carried out by parallel transfer, rotation, or zooming of one of the diffraction structures with respect to the other. The application of mutually rotated kinoforms of phase diffraction gratings is described, which creates conjugate wave fronts with linear phases, rotation of kinoforms at a certain angle in opposite directions, leading to a diffraction duplicate of the Risley-Herschel prism of variable strength.

The present invention is the provision of an optical device that acts as a special optical element having a continuously variable property, moreover, the device is compact and highly efficient. The optical device must additionally provide increased accuracy of the optical element over the entire range of variation of the corresponding optical property.

These tasks are solved using the features reflected in the claims.

The present invention is based on the general idea of sequentially placing a pair of special design diffractive optical elements (DOEs). Typically, DOEs are round plates with a diameter of about 50 mm or less, and their size can be up to 1 mm. When two DOEs are placed side by side and parallel to each other at a certain distance, preferably 10 μm or less, the combination optically corresponds to a single DOE and can perform similar tasks, acting, for example, as a lens with a certain focal length. If one of the DOEs rotates with respect to the other around a common central axis, a specific property of the optical device, for example, the focal length of the diffraction lens, continuously changes in a predetermined and advantageous manner.

The present invention specifically describes the construction of a DOE pair acting as a diffractive lens, the so-called Fresnel lens, with a focal length that continuously varies over a wide range due to the mutual rotation of the two DOEs. Such a lens can be used in the same way as a refracting glass lens, for example, for imaging (in cameras, telescopes, microscopes) or for beam projection (for example, in projectors, codoscopes, laser scanners), but with the additional advantage of a variable focal length . In imaging applications, this allows you to build systems that act like the human eye, i.e. can focus by changing the refractive power of the lens, instead of using bulky zoom lenses whose operation is based on the axial displacement of the lenses. The device of the present invention further provides a transverse compact structure.

Similarly, it is possible to construct a DOE pair acting as a diffraction axicon, i.e. spherical analysis of a prism that creates a ring of light due to refraction, in which the refractive power changes due to the mutual rotation of the DOE pair. Axicons are important optical elements in many scientific applications for beam forming, for example, for specialized microscope illumination systems (STED), in atomic traps, optical tweezers, in optical fiber compounds, etc.

In addition, the DOE pair of the present invention can act as a pure phase shifter, where the phase shift is very precisely controlled by the angle of rotation of the two DOEs. Such a device can also be used as a very accurate light beam frequency converter if one of the DOEs is continuously rotated with respect to the other. Such high-precision phase shifters and frequency converters have numerous applications, for example, in optical interferometers, which are used for quantitative measurements of object transformations with an accuracy of the order of the wavelength.

Another embodiment of the present invention provides for the construction of a DOE pair, which acts as a so-called spiral phase plate with an adjustable spiral index, transforming the incident light beam into a so-called bagel mode beam with a variable spiral charge. Such spiral phase plates are important elements in scientific applications, because they create light beams that carry angular momentum, which can be transferred to microscopic particles, for example, in atomic traps or in optical tweezers. Recently, such helical phase plates have also become important components of the new optical phase contrast microscopy method.

Due to the mutual rotation of the DOE, it is possible to form a section of the combined DOE having an optical property different from the optical property of the remaining region of the combined DOE, for example, a different focal length when the optical device acts as a lens. To avoid this effect, the optical device of the present invention may further comprise a sector-like absorber covering a sector having an undesirable optical property. The formation of such a sector having an undesirable optical property can also be avoided by eliminating the violation of continuity in the phase profiles of diffractive optical elements that arise along a radial line passing from the center to the edge. This can be done when calculating DOE by rounding the argument of the transfer function describing the phase profiles. In addition, an embodiment of the present invention provides for effective suppression of unwanted light emitted by an optical device if the efficiency of individual DOEs is below 100%, which may be, for example, a consequence of limitations in the manufacturing process.

In all applications of DOEs, it is preferable that they have a structure in which their combination provides the so-called reflective phase diffraction grating structures, for example sawtooth diffraction gratings, whose total diffraction efficiency is almost 100%. Preferably, the two DOEs of the optical device of the present invention are, at a certain angle of rotation, mirror images of each other, i.e. only two copies of the same main element are required, which are installed one after another facing each other, with the “face” being the side of the DOE on which the phase profile is imprinted. Thus, the transfer function of the optical phase of one of the DOEs is a mirror copy of the corresponding function of the other of them. Thus, two identical elements can be used, where the respective sides on which the phase profiles are imprinted are facing each other, which reduces the cost of production.

The present invention is described below in more detail with reference to the figures, where:

figure 1 - the basic structure of the optical device according to the present invention;

figure 2 - phase pattern of two DOEs, which after combining act as a Fresnel lens, the refractive power of which depends on their angle of mutual rotation;

figure 3 - the resulting phase pattern of the overlap of two DOE shown in figure 2, at different angles of mutual rotation;

figure 4 - phase pattern of two DOEs, which after combining form Fresnel lenses, the refractive power of which depends on the angle of mutual rotation by analogy with figure 2, modified in such a way that they do not create the effect of sectorization;

figure 5 - the resulting phase pattern of the overlap of two DOE shown in figure 4, at different angles of mutual rotation;

6 is a graph of diffraction efficiency and refractive power of a Fresnel lens in the absence of sectorization as a function of the angle of mutual rotation between two DOEs;

7 is a phase pattern of two DOEs that form a combined DOE, acting as a Fresnel lens with a variable focal length with a shift in its refractive power;

Fig. 8 is a phase pattern of two DOEs, which, after combining, act as an axicon, the refractive power of which depends on the angle of mutual rotation;

Fig.9 is a side view and a top view of a spiral phase element with a spiral index m = 1 and m = 5, respectively;

figure 10 - two identical spiral phase plates with a spiral index m = 1, which are located facing each other close to each other;

11 is a diagram explaining the principle of optimizing the signal-to-noise ratio of a DOE according to an embodiment of the present invention;

Fig - phase patterns of spiral phase elements with spiral indices m = 1 and m = -1 (upper row) and the same phase functions superimposed on the focusing and defocusing lens (lower row);

Fig - phase pattern of two DOEs, which, after combining, act as a spiral phase element, the variable spiral index of which linearly depends on the angle of mutual rotation between the two DOEs;

Fig - the resulting phase pattern of the overlap of the two DOEs shown in Fig, at different angles of mutual rotation;

Fig - optimized pair of DOEs, which forms the same spiral phase elements with variable spiral indices, which are shown in Fig.13, but with a significantly increased signal-to-noise ratio, obtained by superimposing the first and second DOEs on the scattering and collecting lens, respectively.

Principles of diffractive optical elements (DOE)

DOEs are commercially available optical elements consisting of microscopic phase structures in a transparent material. They can act as lenses, lens arrays or holograms (the so-called kinoforms), which can be designed to project certain paintings (lines, crosses, dot lattices, etc.) by illuminating with a plane light wave. Each pixel DOE shifts the phase of the incident light beam in the range from 0 to 2π. Therefore, if such a DOE is illuminated by an incident plane wave, the output wave behind the DOE plate carries a predetermined wavefront modulation. The corresponding DOEs are calculated according to well-known algorithms (for example, according to the kinoform algorithm, Gerhberg-Saxton algorithm, etc.), so that they carry out the necessary tasks. The output of the algorithms corresponds to the so-called transfer function T (x, y) of the desired DOE and, in this case, represents a “pure-phase landscape” in the form T (x, y) = exp (iF (x, y)), where Ф (x, y) is a matrix of pixels in the range from 0 to 2π, corresponding to the phase shift that the light beam acquires when passing through the corresponding spot.

In the next processing step, the calculated phase landscapes are imprinted in the material, for example, by photolithography, electron beam lithography or mechanical micromachining ("diamond turning"), so that each spot by the material delays the phase of the incoming light beam by the desired phase value. This can be achieved through a modulated surface profile (which, for example, is etched into a quartz plate) or by a spatially modulated refractive index in a material with a flat surface, such as, for example, in polymer films. The typical size of each pixel is in the range of 0.1 to 3 microns, and the typical diameter of the DOE is about 2-10 mm.

An important difference between DOEs generated on a computer and “normal” holograms (which are recorded by superimposing an object and a reference wave) is that the calculated phase structures of DOEs are usually not sinusoidal, i.e. structures Φ (x, y) locally look like sawtooth diffraction gratings. Due to this feature, the diffraction efficiency of DOEs can reach 100% of the incident light if they have the correct design, unlike phase holograms, which provide (due to their symmetrical diffraction grating structures) the maximum diffraction efficiency is only 40%.

Figure 1 shows the basic construction of an optical device according to the present invention. Two DOEs are located at a short distance one after another. They can mutually rotate around a central axis perpendicular to their surface. The DOE combination manipulates the wavefront of the incident light wave in a predetermined manner, depending on the angle of mutual rotation.

If two DOEs with transfer functions T ₁ (x, y) = exp (iF ₁ (x, y)) and T ₂ (x, y) = exp (iF ₂ (x, y)) are located directly one after the other on small distance, usually of the same order as the wavelength of light, the combination of two neighboring DOEs is optically equivalent to one single DOE with a transfer function

T _combi (x, y) = T ₁ (x, y) T ₂ (x, y), Lv. (one),

those. the phase landscape of the combined DOE is expressed as

Ф _combi (x, y) = Ф ₁ (x, y) + Ф ₂ (x, y), Lv. (2).

Thus, the combination of two DOEs can provide a transfer function for various specific purposes, which can, in addition, continuously change during the mutual rotation of the two DOEs. The basic principle applies to the same extent to the moire effect, which is considered, for example, in S. Bará, Z. Jaroszewicz, A. Kolodziejczyk, and V. Moreno, "Determination of basic grids for subtractive moire patterns," Appl. Opt. 30, 1258-1262 (1991), J.M. Burch and D.C. Williams, Varifocal moir zone plates for straightness measurement, Appl. Opt. 16, 2445-2450 (1977), or Z. Jaroszewicz, A. Kolodziejczyk, A. Mira, R. Henao, and S. Bar, "Equilateral hyperbolic moir zone plates with variable focus obtained by rotations," Opt. Express 13, 918-925 (2005). In the moire effect, a combination of two microscopically accurate diffraction grating structures with similar diffraction grating constants gives a macroscopically modulated diffraction grating structure. However, usually the moire effect is generated by absorbing rather than phase structures, and usually does not create the structure of a “reflective” phase diffraction grating in the combined transfer function, so that the full efficiency of the moire structures is limited to a few percent.

Zoom Fresnel Lens

A set of two sequential DOEs of a special design can act as a highly efficient (almost 100%) Fresnel zone plate (i.e., a lens), the focal length of which can continuously change over a wide (optional) range due to mutual rotation. Such a variable focal length lens element can replace heavy and bulky zoom optics built from glass lenses in an imaging system such as a camera, microscope eyepiece, telescope, and in beam control applications, such as fiber connectors, optical barcode scanners, or optical manipulators.

The corresponding transfer functions of the two DOEs are calculated according to the formulas:

T _lens1 (x, y) = exp [ia _lens r (x, y) ² θ (x, y)} and T _lens2 (x, y) = exp [-ia _lens r (x, y) ² θ (x , y)}, Lv. (3).

Let x and y be the Cartesian coordinates of the point (x, y) with the origin x = 0, y = 0, which corresponds to the center of the plate, and r (x, y) = (x ² + y ² ) ^{l / 2} and θ ( x, y) = angle (x + iy) are the corresponding polar coordinates of the same point. As will be shown below, the constant a _{lens is} proportional to the optical power of the combined DOE system.

To create two DOEs from these transfer functions at each point, only their phase should be calculated, and the result should be taken modulo (2π), so the result is an array of phase values ranging from 0 to 2π. The corresponding phase patterns of the two DOEs are built at 0.

In FIG. 2 and in all additional DOE schemes described below, gray to white values correspond to phase values ranging from 0 to 2π. Two transmission functions T _lens1 and T _lens2 are complex conjugate. From considerations of symmetry and images of DOE in figure 2 it follows that the two DOEs are mirror images of each other (if both of them are rotated 90 degrees). Therefore, two identical DOEs with the same transmission function T _1ens1 can be used. If they are facing each other, then the second of the two DOEs is turned upside down, and this actually corresponds to the desired mirror image function T _lens2 .

If the second DOE (with the transfer function T _lens2 ) is rotated by a certain angle φ, which can be adjusted from -2π to 2π, then the complete transfer function of the combined DOE can be expressed:

T _{combi, lens} = exp (ia _lens r ² θ) exp (-ia _lens r ² (θ-φ)), Lv. (four).

Note that such a transfer function T _{combi, lens} exactly corresponds to the transfer function of an ideal lens having refractive power

f ^-1 _lens = a _lens φλ / π Ur, (5)

(in diopters, corresponding to the reciprocal of the focal length f _lens ,), where λ is the wavelength of light. Therefore, the change in optical power linearly depends on the angle of mutual rotation of two DOEs, i.e. df ^-1 _lens / dφ = a _lens λ / π.

The corresponding phase transfer function of the combined DOE system corresponds to the kinoform transmission function of an ideal lens, which, due to its asymmetric, sawtooth structure of the phase diffraction grating (due to the operation modulo 2π), has unlimited diffraction efficiency, i.e. such a DOE element can be used for the indicated wavelength of light exactly as a “normal” glass lens without generating any undesirable light components.

The corresponding phase patterns created by the combined DOE system are shown in Fig. 3 for some positive and negative rotation angles φ, namely, -75, -30, -15 degrees in the upper row and +15, +30, +70 degrees in the lower row . Again, phase values are reflected as gray values, which actually correspond to phases ranging from 0 to 2π.

Figure 3 shows that the rotations of the second DOE provide Fresnel lenses with positive or negative refractive power, depending on the direction of rotation (positive and negative refractive powers can be distinguished in the diagrams in the direction of the radial phase change).

Interestingly, according to figure 3, in addition to the desired Fresnel lens, there is a sector of combined DOE, which demonstrates the pattern of the Fresnel lens with a different focal length. This sector is the region between the radial lines of two DOEs that exit from their centers in the direction of the polar angle π. The reason for the emergence of these sectors is the frequency of phase determination in Eq. (5), i.e. the fact that the rotation angle φ of one of the DOEs in relation to another is indistinguishable from another rotation by the angle φ ± 2π (the sign must be chosen so that | φ ± 2π | <2π). Two cases give according to Ur. (5) two Fresnel lenses with different refractive powers f ^-1 _{1, lens} = a _lens φλ / π and f ^-1 _{2, lens} = -a _lens (φ ± 2π) λ / π, respectively. Since the rotation angle φ is modulated only in the range from -2π to 2π, the corresponding two focal lengths f _{1, lens} and f _{2, lens} always have different signs, as well as different absolute values, except for the case φ = π, which gives a combined DOE, consisting of two symmetrical hemispheres, where the two halves act as convex and concave Fresnel lenses, respectively, with equal absolute values of their refractive power.

This effect can be disturbing in applications such as imaging, but it can be weakened by limiting the angle of mutual rotation to an interval that is significantly less than the full range of 2π. In addition, the “wrong” sector of the lens can be covered with a sector-like absorber before DOE. However, since the refractive capabilities of the two sectors are very different, and also have different signs, a couple of DOEs can also be used in many practical applications without masking.

Non-sectorial Fresnel Lens

The present invention further provides a method of counteracting sector formation. It can be shown that the occurrence of an undesirable sector is due to the fact that the formula for creating two DOEs is according to Ur. (3) (T _1,2 = exp [± ia _lens r ² θ]) is not continuous when passing from θ = π to θ = -π, i.e. on a radial line from the center to the left edge of each of the DOEs (see figure 2). However, this breaking line in the DOE can be avoided by slightly changing the formula for generating the DOE according to:

T _lens1 = exp [iround {a _lens r ² } θ] and T _lens2 = exp [-iround {a _lens r ² } θ], Lv. (6)

where round {...} is an operation meaning rounding an argument to the next highest integer.

An example of the corresponding phase patterns of two DOEs is presented in FIG. 4. The phase pattern looks similar to the phase patterns according to the first method shown in FIG. 2, however, the phase edges now look “rougher”. On the other hand, the discontinuity on the radial line θ = π disappears. Nevertheless, two DOEs have the property of mirror symmetry, i.e. they are identical to phase structures if one of them is turned upside down and placed above the other "facing it."

The corresponding phase values in the combined DOE system are shown in FIG. 5 for some positive and negative rotation angles φ, namely, -75, -30, -15 degrees in the upper row and +15, +30, +70 degrees in the lower row.

You can see that sector formation is no longer happening. The transition from negative refractive power to positive when the angle of mutual rotation φ changes from a positive value to a negative value is completely smooth, forming an almost "perfect" Fresnel reflective lens for small mutual rotation angles (as, for example, shown in the range from φ = -30 degrees to φ = + 30 degrees in figure 5). However, for larger angles of mutual rotation (for example, φ = 75 degrees), the combined Fresnel lens becomes somewhat less effective, i.e. the corresponding sawtooth diffraction grating becomes more and more binarized, losing smoothness.

The reason for this behavior is that, by analogy with the case described above, the latent ambiguity of the rotation angle between the rotation angle φ and one of φ ± 2π. These two cases actually form superimposed Fresnel lenses with refractive powers f ^-1 _{1, lens} = a _lens φλ / π and f ^-1 _{2, lens} = -a _lens (φ ± 2π) λ / π, respectively (the sign is chosen so that | φ ± 2π | <2π). The relative component of two lenses changes as a function of the angle of mutual rotation, i.e. for small angles φ, the main component has the refractive power f _{1, lens} , while for angles greater than 180 degrees (or less than -180 degrees), f _{2, lens} prevails. Thus, the diffraction properties of the combined DOE are similar to the diffraction properties of a binary Fresnel lens, which acts as a superposition of a convex and concave lenses, but with the difference that the diffraction efficiencies and the refractive powers of two superimposed lenses are not equal.

Figure 6 shows the calculation of diffraction efficiencies for two superimposed lenses as a function of the angle of mutual rotation in the range from -360 to 360 degrees. A curve having the formula cos ² connecting the left and right lower corners of the rectangle corresponds to the diffraction efficiency of the desired Fresnel lens. It has the highest efficiency at a zero angle of mutual rotation corresponding to a refractive power of 0, and the efficiency gradually decreases (less than 15%) in the range from -90 to +90 degrees. On the other hand, the efficiency of the second superimposed Fresnel lens in this interval is less than 15%. In addition, the refractive power of this unwanted lens is very different from the refractive power of the first, i.e. approaching the maximum value of f ^-1 _{2, lens} = ± 2a _lens λ. Interestingly, at a zero angle of mutual rotation, the refractive power of the second lens abruptly changes from f ^-1 _{2, lens} = + 2a _lens λ to f ^-1 _{2, lens} = -2a _lens λ. However, in the position of the jump, the efficiency of the corresponding lens drops to zero, and therefore this jump does not have a real impact on the experiment.

Thus, the graph in FIG. 6 demonstrates that the mutual rotation of two DOEs at an angle from -90 degrees to +90 degrees is possible, with a loss of efficiency of the combined Fresnel lens of only 15%.

If one of the two DOEs continuously rotates at a constant speed relative to the other, the corresponding Fresnel lens is periodically scanned in an adjustable range of focal lengths. This can be used in an imaging system (surveillance cameras) and beam scanning systems (barcode readers, etc.). For example, a lens based on a periodically rotating DOE, which is combined with an imaging system that uses a shutter phased with rotation of the DOE (i.e., the shutter always opens in a certain angular position of the DOE), clearly focuses on objects at a certain distance, which depends on the relative phase between the opening of the shutter and the rotation of the DOE. If the shutter is an electronic type (for example, a gated image intensifier), focusing can be controlled exclusively by electronic means, simply changing the phase between the rotation of the DOE and the shutter opening.

In practice, the maximum allowable focus limits in which the Fresnel lens obtained by combining two DOEs can be changed are given by Ur. (5), i.e. f ^-1 _lens = a _lens φλ / π, where φ can vary from -2π to 2π. However, there is a limitation for the maximum value of the constant a, due to the resolution of the physical DOE. To resolve the diffraction grating printed in the form of a pixel matrix, the maximum phase shift between two adjacent pixels must be less than π, i.e.:

where p is the minimum size of one pixel DOE, which is usually limited by the refraction process. The first and second conditions are satisfied for radial and tangential phase resolution, respectively. For DOEs forming Fresnel lenses, it turns out that the first condition is always stricter than the second, and therefore we will consider only it.

Since DOEs forming Fresnel lenses have the transfer functions T _1,2 = exp [± ia _lens r ² θ], we obtain Ф = a _lens r ² θ. Together with Ur. (7) this gives the condition:

2a _lens r _max θ _max <p- ¹ , Eq. (8)

where r _max is the maximum radius of the DOE, and θ _max is the maximum polar angle. Since θ is limited by the range from -π to π, θ _max corresponds to π, and the condition takes the form:

a _lens <(2pr _max ) ^-1 , i.e. a _{lens, max} = (2pr _max ) ^-1 , Ur (9).

Thus, the maximum value of a _lens , a _{lens, max} for a particular DOE depends on its pixel resolution and its maximum desired radius.

According to ur. (5) the refractive power of a combined DOE lens with such a _{lens, max} is expressed as: f ^-1 _lens = a _{lens, max} φλ / π.

In the last section, it was shown that to obtain an efficiency of about 85%

or higher, the rotation range φ should be limited to the interval from -π / 2 to + π / 2. Thus, substituting φ = π / 2, we obtain the restriction:

-f ^-1 _min <f ^-1 _lens <+ f ^-1 _min , Lv. (10)

Where

f _min = 4pr _max / λ, Lv. (eleven)

this is the smallest allowable focal length of the combined DOE system, which is achieved with a mutual rotation angle of ± 90 degrees, and this gives a diffraction efficiency> 85%.

As a practical example, a DOE with a typical pixel size p = 1 μm and a diameter of 2r _max = 5 mm will have a refractive power adjustable from -50 to +50 diopters (corresponding to a range of focal lengths from ± 2 cm to ± ∞), by wavelength of 500 nm. It can be shown that this limit of the useful range is only 2 times more limited than the limit for a Fresnel lens based on a single DOE under the same conditions, i.e. in the above example, a single DOE should have a minimum focal length of 1 cm.

In many cases, it may be desirable to change the refractive power of the lens symmetrically not relative to the zero force, but relative to a certain offset value. This can be achieved by placing the combined DOE element directly behind the “normal” glass lens, which acts as a “shift” of refractive power. However, this can also be achieved by creating such an offset directly on the combined DOE element.

This can be done by multiplying both transfer functions of two DOEs by a member of the lens shift, each of which is equipped with half the necessary offset refractive power f ^-1 _offs , i.e.:

By setting a _offs = π / λf _offs , one can obtain the following equations (12b):

It can be shown that in this case the focal length of the combined DOE changes as a function of the angle of mutual rotation, as before, in particular, with the same efficiency and the same change in refractive power as a function of the angle of mutual rotation, but now it has a shifted focal length, corresponding f _offs . Advantageously, the two elements are still mirror images of each other (at a certain relative rotation), and thus it is enough to create two identical elements with the transfer function T _lens1 , set facing each other. Figure 7 shows an example of two DOEs that, after combining, form a Fresnel lens with a variable focal length with offset refractive power.

Axicons with variable refractive power

By analogy with an adjustable Fresnel lens, another set of two DOEs of a special design can act as an axicon (or axicon lens), the refractive power of which is regulated by the angle of mutual rotation. Such axicons are mainly needed in beam control applications for fiber connectors, optical tweezers and laser cutting systems to obtain an axially extended region of small focus (the so-called "Bessel beams").

The formula for calculating the transfer functions of two DOEs is as follows:

An example of a DOE corresponding to such a transfer function is shown in FIG. 8 (A). By analogy with the last section, the refractive power of an axicon formed by combining these two DOEs is proportional to the constant a _axic and the angle of mutual rotation φ. All DOEs shown in FIGS. 8 (A), (B), and (C) should be combined with a second DOE, which is identical to the first but upside down.

An example of a combined DOE after the mutual rotation of T _axi1 and T _axi2 by an angle of 25 degrees is shown in Fig. 8 (D). Again, there is the effect of forming an undesirable sector, the angular extent of which corresponds to the angle of mutual rotation and which contains an axicon lens with a different refractive power. By analogy with the last section, the formation of such a sector can be avoided, including the rounding operation in its formula, i.e.:

Fig. 8 (E) shows an example, again for a rotation angle of 25 degrees. By analogy with the last section, a second axicon structure appears, superimposed on the first and having a different refractive power. However, the relative efficiency of the second axicon structure, again, is below 15%, in the case when the angle of mutual rotation is limited by the interval from -90 to +90 degrees.

Finally, you may need to create an “axicon lens," i.e. axicon with variable refractive power, superimposed on the "normal" collecting or scattering Fresnel lens with a fixed focal length f ^-1 _offs . The corresponding formula for obtaining such an element is:

By setting a _offs = π / λf _offs , one can obtain the following equations (15b):

and an example of the corresponding DOE is presented in 0 (C).

Such a structure can, for example, create an annular distribution of light intensity from an incoming plane light wave, which focuses at a certain distance behind the element. In addition, the advantage of such a structure is that the desired light field generated by the axicon has a different discrepancy than the residual light that has not experienced diffraction. Such residual light corresponds to the zero diffraction order of the element of the combined DOE and can occur if the DOE is improperly manufactured, for example, if the calculated phase values are incorrectly reconstructed in the physical structure of the DOE. Due to different discrepancies between the desired axicon-like diffracted light and unwanted zero-order non-diffracted (i.e. transmitted) light, the two components can be easily separated, for example, by applying an aperture diaphragm in the focal plane of the undiffracted light.

Continuously adjustable phase shifter and frequency converter

Two consecutive DOEs of a special design can act as an accurate, continuously adjustable phase shifter when they mutually rotate. This ability to phase rotation is necessary, for example, in optical interferometry, interference microscopy, holography, as well as in many scientific applications (for example, continuous phase rotation of standing light waves in atomic traps, etc.). A DOE system can also act as a continuous frequency converter for a transmitted light beam if one of the DOEs rotates continuously with respect to the other. Such a frequency converter is necessary in interferometric ("heterodyne interferometry") and scientific applications.

To this end, the corresponding DOEs consist of the so-called spiral phase plates assigned to the transfer functions:

where the so-called spiral index (sometimes referred to as the "spiral charge") m _phas is an integer. An example of such spiral phase plates is shown in FIG. 9 for spiral indices m _phas = 1 and m _phas = 5.

If two identical spiral phase plates are located close to each other close to each other, their effective spiral indices have opposite signs, and the combined DOE acts as a variable phase shifter. An example of such a combined DOE for the case m _phas = 1 is shown in FIG. 10. According to figure 10, two identical spiral phase plates with a spiral index m _phas = 1 are located close to each other close to each other. Due to the inversion of one of the DOEs relative to the other, the effective helicities of the two DOEs in this configuration have the opposite sign. If one of the DOE rotates with respect to the other, the phase of the transmitted light wave is continuously shifted in the range from 0 to 2π.

If one of the DOE rotates with respect to the other by an angle φ, the combined transfer function takes the form:

Thus, the combined transfer function corresponds to a static, flat phase shift m _phas φ, which is acquired by the incoming wave passing through the element of the combined DOE.

If one of the two elements rotates continuously with a constant angular velocity ω _rot , then the static angle of rotation φ in Eq. (17) is expressed as ω _rot t, and the system acts as a frequency converter that changes the light frequency of the incoming optical beam, ω _in , in:

The sign of the frequency shift (or phase) can be selected according to the direction of rotation. Ur (17) and Lv. (18) also show that the helical index m _phas acts as an internal gear ratio, i.e. the phase or frequency shift of the transmitted beam corresponds to the coefficient m _phas , which is multiplied by the rotation angle or the rotation frequency of the rotating DOE, respectively.

Combined DOE with increased relative efficiency

If a DOE pair, for example, the phase shifter described in the last chapter, is correctly manufactured, both DOE pairs and the combined DOE element have an efficiency q = 100%, i.e. there is no front section of the incoming wave that has not been given the desired shape. However, if, due to the practical limitations of the physical element of DOE, the actual phase shift of each pixel does not correspond directly to the planned values, then the diffraction efficiency of two separate DOEs and combined DOE will increase. The main undesirable component of improperly manipulated light on each of the two DOEs will be the so-called zero diffraction order, i.e. part of the light that passed through DOE without change. Even if this component of the zero order is small, it can have a significant effect on the total transmitted wave, since it coherently interferes with the rest of the light, which usually leads to spatial modulation of the intensity of the transmitted light wave, which changes with the mutual rotation of the DOE.

For the case of a “normal” pair of DOEs, where each DOE has an efficiency q <1 (the efficiency is defined as the fraction I _{mod of} incident light I _in , which is modulated accordingly, i.e. q = I _mod / I _in ), the total efficiency of the combined DOE is equal to q ² , and, accordingly, the unwanted light wave I _noise consists of the rest of the light, i.e. has an efficiency of 1-q ² . Thus, the signal-to-noise ratio of a normal DOE pair is specified as:

However, to the detriment of the properties of the DOE pair for composing two mirror-symmetric elements, there is a way to significantly increase the performance of the DOE pair. The idea is to separate the unwanted zero-order component from the desired manipulated light wave by introducing different beam divergences into the two wave components. For example, in this case, the correctly manipulated beam will pass through the element of the combined DOE without changing its divergence, while the undesirable component diverges greatly and thus “dissolves” at a certain distance behind the DOE.

This can be achieved by applying one of the two DOEs to a strongly scattering lens, while simultaneously applying the other DOE to a strongly collecting lens, which exactly compensates for the effect of the first. Thus, the planned light wave does not change its total discrepancy after passing through two DOEs, while the components of the zero-order beam, which only diffract on one of the two DOEs, will diverge. The only perturbing component of the beam that maintains the same discrepancy as the desired wave is the component that passes through both DOEs as a zero-order wave (i.e., an undiffracted wave). However, the likelihood of this effect gradually decreases as a function of DOE efficiency, compared with a purely linear decrease if DOEs are created without lens parts. The principle of the method is shown schematically in FIG. 11.

11 is not to scale, since the actual distance between the two DOEs should be as small as possible in order to exclude the actual beam expansion. This figure shows a set of two DOEs, where the first has a superimposed scattering lens (in addition to its adjusted phase profile), while the second has a superimposed collecting lens of the same absolute refractive power. Part (A) of the figure shows the ideal case where a light wave diffracts into the desired first diffraction order by both DOEs. As a result, the outgoing wave has the same divergence as the incoming wave, and additionally acquires the desired modulation of the phase front. Assuming that the diffraction efficiency of each individual DOE is q, we obtain that the total diffraction efficiency is q ² .

The following two schemes (B) and (C) in FIG. 11 demonstrate the situation with the second after the highest probabilities, where diffraction occurs only on one of the two DOEs, while the corresponding other DOE simply transmits a wave. In both situations, the output wave diverges, and thus, it can be easily separated from the wave of the useful signal in (A).

The last circuit (D) in FIG. 11 shows a single component of unwanted light that has the same divergence as the signal wave, and thus acts as “noise”. It is formed by a part of the incident wave, which passes through both DOEs without diffraction. The corresponding efficiency in this situation is (1-q) ² . Thus, we obtain the total "signal-to-noise ratio" S _{optimal of the} optimized DOE pair in the form:

Thus, an increase in the signal-to-noise ratio between the optimized and normal (Eq. (19)) DOE pairs is expressed as:

As an example of the increased performance of each collecting / scattering pair of DOEs, we consider the case where each unit DOE has a diffraction efficiency of 90% (q = 0.9), while 10% of the incident light passes without diffraction in zero order. Diffraction into other diffraction orders can be neglected for the reflective structures of DOEs.

In this case, the signal-to-noise ratio of the normal DOE pair (according to Eq. (19)) should be S _normal = 4.3. On the other hand, the corresponding signal-to-noise ratio of the optimized DOE pair will be equal to (Eq. (20)) S _optimal = 81, which corresponds to an almost 20-fold increase. If this method is applied to a continuous phase shifter and a frequency converter described in the previous section, the corresponding transfer functions of the two DOEs are expressed as:

Setting a _div = π / λf _div , we can obtain the following equations (22b)

Here, f _div is the focal length of the superimposed collecting and scattering Fresnel lenses, and is preferably chosen as small as practicable (with a limitation according to Eq. (11)) to achieve rapid “dissolution” of the undesired beam components due to divergence behind the DOE pair.

The element of the combined DOE has the same transfer function T _combi = exp (-im _phas θ) as the function of the mutual rotation angle φ for the “normal” pair of DOE, expressed by Eq. (16). The corresponding DOE elements of the “normal” DOE pair and the optimized pair are compared in FIG. 12, which shows the phase patterns of spiral phase elements with spiral indices m _phas = 1 and m _phas = -1 (upper row) and the same phase functions superimposed on the focusing and a defocusing lens (bottom row). The phase shifter / frequency converter formed by the DOE pair in the bottom row has a significantly increased relative efficiency between the desired phase-shifted wave components and the residual undiffracted components.

DOE pairs for spiral phase elements with variable spiral index

Spiral phase elements have important applications in beam formation for generating so-called bagel beams, which are used in optical traps (laser tweezers and atomic traps), for transferring the angular momentum to microscopic particles (optical pumping), and, more recently, for spiral phase-contrast formation images in microscopy and interferometry.

A set of two consecutive DOEs has been constructed, which can act as a spiral phase element, the spiral charge of which can be continuously adjusted (and even inverted) in the selected range by adjusting the angle of mutual rotation of the two DOEs.

The main transmission functions of the corresponding DOEs are expressed as:

where a _spir is a constant that determines the change in the spiral index of the combined DOE as a function of the angle of rotation. Thus, the transmission function of the combined DOE at the angle of mutual rotation φ has the form:

This corresponds to the transfer function of the spiral phase plate with the spiral index m = 2a _spir φ (first coefficient), combined with an additional pure phase shifter by the value -a _spir φ ² (second coefficient). If such a combined DOE is not used for interferometry, but only as a mode converter, the phase-shifting term can be neglected, and the spiral index of the combined spiral phase element linearly depends on the angle of mutual rotation φ.

Example of a set of DOEs with transmission functions according to Eq. (24) is shown in FIG. 13.

Permission requirements are most stringent in the area around the DOE center. The resulting combined transmission function after the overlapping of two DOEs is shown in Fig. 14 for several angles of mutual rotation, namely -25, -5, -1.5, +1.5, +5 and +25 degrees. The corresponding transfer functions correspond to spiral phase elements with spiral indices of approximately −13, −3, −1, +1, +3, and +13, respectively.

It turns out that again there is a sector formation similar to sectorization in figure 3. The reason for this is the same, i.e. sectorization due to the ambiguity of the angle of rotation φ, which takes place in Eq. (24) for the transmission function of the combined DOE, i.e. the rotation φ cannot be distinguished from φ ± 2π, and therefore the combined spiral phase element contains two spiral indexes at the same time, however, an undesirable second index is contained in the sector included in the angle of mutual rotation of the two DOEs. Thus, the use of a large coefficient a _spir in the design of DOEs still allows us to create DOE elements that generate a significant change in spiral indices at a sufficiently small angle of mutual rotation, so that the area of the undesired sector will be negligible in many practical applications.

By analogy with the DOE optimization method described in the last section, it is also possible to increase the contrast in the signal-to-noise ratio by applying a scattering and collecting lens (with focal length ± f _div ) by two DOEs. In this case, two optimized transfer functions are:

An example of two DOEs calculated according to these transmission functions is shown in FIG.

The transfer function of the combined DOE corresponds exactly to the transfer function of the previous DOE pair (see Fig. 13), but with an increased signal-to-noise ratio if the diffraction efficiency of individual DOEs is less than 1, which is specified in Eq. (21).

The above detailed description describes the design of optical elements such as continuously adjustable phase shifters, adjustable Fresnel zone plates and axicons and adjustable spiral phase elements, which are controlled by the mutual rotation of two successive DOEs, which provides a technical solution for many problems in optical systems. The following are technical solutions for the various applications mentioned in the previous sections.

Fresnel zone plate as a continuously adjustable focal length lens

Such lenses are in high demand in technical applications. By way of example, such a lens makes it possible to realize an imaging system that focuses, similarly to the human eye, not by adjusting the distances between the optical elements (by axial movement of the lenses), but by changing the focal length of the lens to form images. The creation of zoom lenses is still at the research stage, i.e. There are approaches to using liquid droplets as lenses, where the curvature of the droplet can be adjusted using an electric field. Similar approaches use the interface between two liquids, which can be curved by an electric field. Finally, there are specially designed liquid crystal systems that can act as zoom lenses. Compared with the present invention, all these approaches are more complex and sophisticated, and also do not allow to periodically scan the focal plane (by continuously rotating one DOE relative to another). In approaches using a pair of two moire patterns that can mutually rotate to realize a zone plate with a variable focal length, the efficiency is limited to 16%, while the present invention allows to achieve almost 100% efficiency.

Axicons with adjustable refractive power

Technical approaches to the implementation of axicons with adjustable refractive power (mainly necessary for beam control applications) are similar to those mentioned in the last section. Again, it is proposed to implement such elements with moire patterns, which, however, leads to a limited efficiency of 16%.

Spiral phase element with variable spiral charge

Spiral phase elements for generating bagular beams or for spatial filtering are usually made in the form of holograms or DOEs. However, in this case, the corresponding spiral charge is not variable. Spiral phase cells with a variable spiral charge can be programmed on standard high-resolution liquid crystal displays or on specialized liquid crystal displays. Both methods are very expensive. In addition, it was reported about the creation of an alternating spiral phase element by controlled deformation of a special Plexiglas disk by mechanical means. However, this method requires very complex mechanisms and manufacturing methods.

Phase shifters and frequency converters for interferometers, etc.

For standard phase-rotation applications, solutions are known for changing the optical path length with mirrors mounted on piezoelectric crystals, inserting a pair of complementary glass wedges into the beam path that are shifted in the transverse direction, or inserting a glass plate that can be inclined relative to the horizontal or vertical axis.

All of the above methods have the disadvantage that phase rotation is not continuous, i.e. the optical elements used have a “stop position”, from where they must be returned to their respective “initial positions” before performing phase rotation. This also prevents the use of these methods for adjustable frequency converters, which can be done by continuous mutual rotation of the DOE according to the present invention.

A known method of achieving a continuous phase shift is to use the so-called Pancaratnam phase due to the mutual rotation of the combination of two quarter-wave and one half-wave plate in the beam path. However, unlike the present invention, the Pancaratnam method is implemented only for light in a certain state of polarization. A particular advantage of the present invention is its high phase rotation accuracy and the possibility of realizing a “gear”, i.e. create pairs of DOEs that provide a frequency shift of +/- nf (where n is an integer that depends on the design of the DOE and can reach 1000), with spatial rotation with a frequency f Hz.

Claims (20)

1. An optical device containing a pair of plate-like diffractive optical elements (DOE) with transmission functions T ₁ (r, θ) = exp [iF ₁ (r, θ)] and T ₂ (r, θ) = exp [iF ₂ (r , θ)], where r and θ are polar coordinates, r is the radius and θ is the polar angle, and Φ _1,2 (r, θ) are phase profiles in the range from 0 to 2π, imprinted in the DOE, and two DOEs sequentially mounted parallel to each other with a sufficiently small spacing between them, so that the combination of two neighboring DOEs optically corresponds to one DOE with the transfer function T _combi = T ₁ · T ₂ , and at least at least one of the DOEs is made to rotate relative to the other around an axis perpendicular to its surface, so that the pair of DOEs acts as DOEs with a transfer function
T _combi = T ₁ (r, θ) · T ₂ (r, θ-φ) = exp [iФ ₁ (r, θ)] · exp [iФ ₂ (r, θ-φ)] = exp [i {Ф ₁ (r, θ) + Ф ₂ (r, θ-φ)}],
where φ is the relative angle of rotation, and the phase profiles Ф _{1,2 of} two DOEs are determined by the following expressions:
Ф ₁ (r, θ) = mod _2π {[a _lens r ² θ + a _axic rθ + m _phas θ + a _spir θ ² ] + (a _offs / 2) r ² + а _div r ² } and
Ф ₂ (r, θ) = mod _2π {- [a _lens r ² θ + a _axic rθ + m _phas θ + a _spir θ ² ] + (a _offs / 2) r ² -a _div r ² },
where mod _2π {...} means the operation modulo 2πi, the coefficients a _lens , a _axic , a _spir , a _offs and a _div are real numbers, which can also be zero, and the coefficient m _phas is an integer that can also be zero, with at least one of the coefficients a _lens , and _ahic , m _phas and a _{spir is} not zero, while a _{lens is} proportional to the refractive power of the lens, a _{axic is} proportional to the refractive power of the axicon, a _{spir is} proportional to the spiral index of the spiral phase element , a _{offs is} proportional to the refractive index features of the lens superimposed with displacement, and _{div is} proportional to the refractive power of the scattering lens and m _phas is the spiral index of the spiral phase plate. 2. The optical device according to claim 1, in which the phase profiles f _{1,2 of} two DOEs are determined by the following expressions:
Ф ₁ (r, θ) = mod _2π {[a _lens r ² ] θ} and
Ф ₂ (r, θ) = mod _2π {- [and _lens r ² ] θ}. 3. The optical device according to claim 1, in which the phase profiles f _{1,2 of} two DOEs are determined by the following expressions:
Φ ₁ (r, θ) = mod _2π {[a _axic r] θ} and
Φ ₂ (r, θ) = mod _2π {- [a _axic r] θ}. 4. The optical device according to claim 1, in which the phase profiles f _{1,2 of} two DOEs are determined by the following expressions:
Φ ₁ (r, θ) = mod _2π {a _spir θ ² } and
Φ ₂ (r, θ) = mod _2π {-a _spir θ ² }. 5. The optical device according to claim 1, in which the phase profiles f _{1,2 of} two DOEs are determined by the following expressions:
Φ ₁ (r, θ) = mod _2π {m _phas θ} and
Φ ₂ (r, θ) = mod _2π {-m _phas θ}. 6. The optical device according to one of claims 2-5, in which an offset term (a _offs / 2) is added to the argument of the mod _2π function of the two phase profiles Φ ₁ (r, θ) and Φ ₂ (r, θ) of the two DOEs r ² . 7. The optical device according to one of claims 2 to 5, in which the term a _div r ^{2 is} added to the argument of the mod _2π -function of the phase profile Φ ₁ (r, θ) of the first DOE and in which the same term a _div r ² is subtracted from argument mod _2π -function of the phase profile Ф ₂ (r, θ) of the second DOE. 8. The optical device according to one of claims 1 to 5, in which the maximum relative rotation of the two DOEs is limited by extreme rotation angles φ _min and φ _max that satisfy the condition [-90 ° ≤φ _min <φ <φ _max ≤90 °], where φ is the relative rotation angle. 9. The optical device according to one of claims 1 to 5, further comprising a sector-like absorber that covers the sector of the DOE pair. 10. The optical device according to claim 9, in which said sector, covered with a sector-like absorber, has an angle corresponding to the angle of rotation of one DOE relative to another. 11. The optical device according to one of claims 1 to 5, in which one of the DOE is made with the possibility of continuous rotation with an angular frequency ω _rot to continuously change the optical properties of the optical device that depend on the angle of rotation φ. 12. The optical device according to one of claims 1 to 5, in which the phase profile imprinted in the DOE has pixels of 10 μm or less, preferably of the order of the wavelength of light or more. 13. An optical device containing a pair of plate-like diffractive optical elements (DOEs) with transmission functions T ₁ (r, θ) = exp [iF ₁ (r, θ)] and T ₂ (r, θ) = exp [iF ₂ (r , θ)], where r and θ are polar coordinates, r is the radius and θ is the polar angle, and Φ _1,2 (r, θ) are phase profiles in the range from 0 to 2π, imprinted in the DOE, and two DOEs sequentially arranged in parallel to one another with a sufficiently small spacing between them, so that the combination of the two neighboring DOEs optically corresponds to one DOE with transfer function T _combi = T ₁ · T _2, wherein at Men it least one of the DOE is rotatable relative to the other about an axis perpendicular to its surface so that the DOE functions as a pair of DOEs with transfer function
T _combi = T ₁ (r, θ) · T ₂ (r, θ-φ) = exp [iФ ₁ (r, θ)] · exp [iФ ₂ (r, θ-φ)] = exp [i {Ф ₁ (r, θ) + Ф ₂ (r, θ-φ)}], where φ is the relative rotation angle, and the phase profiles Ф _{1,2 of} two DOEs are determined by the following expressions:
Ф ₁ (r, θ) = mod _2π {[round {a _lens r ² + a _axic r} θ + m _phas θ] + (a _offs / 2) r ² + а _div r ² } and
Ф ₂ (r, θ) = mod _2π {- [round {a _lens r ² + a _axic r} θ + m _phas θ] + (a _offs / 2) r ² _div r ² },
where mod _2π {...} means the operation modulo 2π, round {...} means rounding the argument to the next larger number, a _lens , a _axic , a _offs and a _div are freely selectable coefficients, which can also be zero, and the coefficient m _phas is an integer, which can also be zero, while at least one of the coefficients a _lens , a _axic or m _{phas is} not zero, while a _{lens is} proportional to the refractive power of the lens, a _{axic is} proportional to the refractive power of the axicon, a _offs proportional to the refractive power of lenses s, superimposed with bias, a _{div is} proportional to the refractive power of the scattering lens and m _phas is the spiral index of the spiral phase plate. 14. The optical device according to item 13, in which the phase profiles f _{1,2 of} two DOEs are determined by the following expressions:
Ф ₁ (r, θ) = mod _2π {[round {a _lens r ² } θ} and
Ф ₂ (r, θ) = mod _2π {- [round {a _lens r ² }] θ}. 15. The optical device according to item 13, in which the phase profiles f _{1,2 of} two DOEs are determined by the following expressions:
Φ ₁ (r, θ) = mod _2π {round {[a _axic r]} θ} and
Φ ₂ (r, θ) = mod _2π {-round {[a _axic r]} θ}. 16. The optical device according to item 13, in which the phase profiles f _{1,2 of} two DOEs are determined by the following expressions:
Φ ₁ (r, θ) = mod _2π {m _phas θ} and
Φ ₂ (r, θ) = mod _2π {-m _phas θ}. 17. The optical device according to one of claims 14-16, in which an offset term (a _offs / 2) is added to the argument of the mod _2π function of the two phase profiles Φ ₁ (r, θ) and Φ ₂ (r, θ) of the two DOEs r ² . 18. The optical device according to one of claims 14 to 16, in which the member a _div r ^{2 is} added to the argument of the mod _2π -function of the phase profile Φ ₁ (r, θ) of the first DOE and in which the same member a _div r ² is subtracted from argument mod _2π -function of the phase profile Ф ₂ (r, θ) of the second DOE. 19. The optical device according to one of paragraphs.13-16, in which one of the DOE is made with the possibility of continuous rotation with an angular frequency ω _rot to continuously change those optical properties of the optical device that depend on the angle of rotation φ. 20. The optical device according to one of claims 13-16, wherein the phase profile imprinted in the DOE has pixels of 10 μm or less, preferably of the order of the wavelength of light or more.

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