The present invention is the measurement setup for determining position of focal plane and effective focal length of an optical system and the method of determining position of focal plane and effective focal length of an optical system, in particular, for multiple-lenses and multiple-elements optical setups applied in precise optical and optoelectronics devices. The solution according to the invention concerns imaging optical setups of a finite and positive effective focal length.

The effective focal length is a primary feature characterizing optical setups. It determines i.a. a distance where a collimated beam is focused, and also what are magnifying properties of the optical setup. The simplest, yet idealized, optical setup consist of a single thin lens. A collimated beam of light or a bundle of optical rays passing through the lens focuses in a point situated on the focal plane. In this case the effective focal length is the distance between the focal plane to the lens. Real optical setups starting from thick lenses to complex objective lenses constructed with multiple lenses, are spread in space. Each optical setup has two, so called, principal planes - the frontal and the back one. The optical setup illuminated by a collimated light focus the light in a point, and the effective focal length is a distance from the front (or back) principal plane to the focal plane, depending on direction of illumination, respectively. In case of real optical setups the position of principal planes is unobvious and in general it is neither front, nor back plane of the optical setup itself.

The effective focal length ( ) for each optical setup is determined by a ratio of a distance (x) between the optical axis to a parallel incoming/outgoing ray (fig. 2a or fig 2b, respectively) to the angle of ray (Θ) leaving/incoming to the setup (fig. 2a or fig 2b, respectively), or a ratio incoming to the optical (fig. 2a and fig. 2b, respectively). In the commonly used approximation of paraxial rays, the relation between the distance (x) from axis and the angle (Θ) is as follows: χ [θ) = f · Θ.

In the state of the art, the following academic or industrial methods for determining of the focal plane and effective focal length are known. For optical setups consisting of only a single lens, the determination of a focal length is pursued following, so called, "lens formula" [T. Drynski, Cwiczenia laboratoryjne z fizyki, PWN, Warszawa 1980; H. Szydlowski, Pracownia fizyczna, PWN, Warszawa 1999; Sz. Szczeniowski, Fizyka doswiadczalna, torn IV - Optyka, PWN, Warszawa 1983] which is the most fundamental formula in the geometrical optics. For determination of the focal length ( ) one measures in experiment a distance between a lens and object ( ) and a distance to a sharp image (y):

1 1 1

J ^~ x y

In case, where an optical system consists of two lenses one can apply the following formula to determine the effective focal length:

1 _ 1 1 d

where: / - effective focal lengths, f _lt f ₂ - focal length of respective lenses and d - distance between component lenses.

The second experimental and widespread method for determining a focal length of a single lens is measurement of magnification. The methods allows to determine a focal length of an investigated lens, for a known size of the object (K) and its image size (P) applying the following lens formula: The third method to determine the focal length of a single lens is the Bessel method. This method takes an assumption that for the given distance "object - screen" (I) one can find two position of the lens where the sharp image is formed, either magnified or demagnified. From these two measurements one can find the focal length applying the formula:

I ² + a ²

f — where: a stands for a position change of lens to achieve a sharp image; - distance „object - screen".

The Bessel method can be applied for setup with two lenses as well. Then the distance between the object and a lens reads:

I— x + k + y

where x and y are distances as defined in the„lens formula" and k is a distance between the lenses. If the distance of translation of the optical setup a = y -x, then we obtain:

V - k) ² - a ²

† ^~ 4(1 - k)

From the patent description US6486457B1, a device and a method for automated focusing is known, where the device consists of a laser, an optical setup and a detector.

The subject of the patent application CN 104122609 A is an automated method for determining an actual focal length of the varifocal lens based on the liquid crystal spatial light modulator.

In the prior art there are also known methods for determining focal length using collimated light, in micro-optics setups following the Patent CN102494873B, as well as systems with long focal lengths described in CN203216702U, CN102331336B.

There are also known methods for determining focal lengths of the optical systems utilizing collimated light, the movable detector and a single diffraction grating, for example in the documents CN102288392A, CN102313642B, CN103063414A and US7106428B2. Methods of determining the position of the focal plane and the effective focal length known in the prior art involve laborious experimental measurements of distances of objects and images (magnified or demagnified) in front and behind of the optical system. Finding these two parameters, especially for complex multi-element optical systems, is time consuming and subject to large measurement errors.

The methods of measurement of position of a focal plane and an effective focal length known in the art are subject to inaccuracy, or do not allow the simultaneous, precise and rapid measurement of the focal plane and a focal length. There exist also measurement methods based on the source of collimated light. Speaking strictly, such collimated light do not exist due to diffraction phenomenon. Gaussian laser beams, which have the least deviation in relation to the diameter of the beam also undergo a diffraction phenomenon. Consequently, the Gaussian beam transmitted through the optical system does not focus to a mathematical point but to a finite spot size. Moreover, the plane where the spot reaches the smallest dimension is not exactly the focal plane of the system. It would be so only in the case of beams of infinitely large diameter which contradicts the finite diameter of the optical components. At the same time it is very troublesome to prepare and use a bundle of parallel rays, and direct measurement of angles at which they leave the optical system. Another problem is the measurement of the focal length of the optics of short focal length. Such an optical system converges a beam of light on the surface of several pixels. The assessment of where spots are the smallest is difficult, which prevents accurate determination of the focal length.

The aforementioned problems have been solved by using the properties of optical systems in their focal planes and the wave nature of light diffraction diffracted on gratings. Instead the prior-art, time-consuming methods for determining the focal plane, the following observation has been applied: the position of the light beam exactly in the focal plane depends only on the angle of the ray (Θ) with respect of the axis at which the beam enters the system and is independent of the position and the distance the system axis (x). The measurement setup for detemnning the position of a focal plane and an effective focal length of an optical system, is characterized that at least two identical diffraction gratings are placed between the light source and the optical system.

Preferably, at least two identical diffraction gratings are set to their planes parallel to each other and perpendicular to the optical axis, wherein the diffraction gratings are rotatable about the optical axis of the optical system.

Preferably, the rulings of diffraction gratings are rotated relative to each other along the optical axis and forming angles ranging from 0° to 20°, favorably 15 °.

Preferably, the first diffraction grating is located at such a distance from the second diffraction grating that at least 1 and -1 row of diffraction rays on the first diffraction grating enters the aperture of the second diffraction grating.

Preferably, the second diffraction grating is located at a distance from the optical system that at least 1 and -1 row of diffraction rays in the second diffraction grating enters the aperture of the optical system, favorably the second diffraction grating is located by the optical system.

Preferably, the diffraction gratings are of the Ronchi type characterized by a sparse grating with a number of rules between 1 / mm to 10 / mm, favorably 3 / mm.

Preferably, the light source is monochromatic laser light, favorably with a Raighley range greater than the focal length of the optical system under study.

Preferably, the detector is placed on the translation stage and is movable along the optical axis.

The method for determining the position of a focal plane and an effective focal length of an optical system is characterized in that:

a) an optical system placed in the measurement setup between the diffraction gratings and the detector;

b) at least one diffraction grating is rotated in the axis of the system to obtain an angle different from zero between the rules of diffraction gratings; c) for changing the position of the detector along the optical axis of the optical system to the position of the focal plane, wherein the recorded spots on the detector are provided above each other;

d) the two diffraction gratings are again set with rulings parallel to each other; e) then one measures the distance between the recorded spots on the detector.

Preferably, at least two identical diffraction gratings are placed between the light source and the optical system.

Preferably, at least two identical diffraction gratings are set their planes parallel to each other and perpendicular to the optical axis, wherein the diffraction gratings are rotatable around the optical axis of the optical system.

Preferably, the rulings of diffraction gratings are rotated relative to each other along the optical axis and forming angles ranging from 0° to 20°, favorably 15 °.

Preferably, the first diffraction grating is located at such a distance from the second diffraction grating that at least 1 and -1 row of diffraction rays on the first diffraction grating enters the aperture of the second diffraction grating.

Preferably, the second diffraction grating is located at a distance from the optical system that at least 1 and -1 row of diffraction rays in the second diffraction grating enters the aperture of the optical system, favorably the second diffraction grating is located by the optical system.

Preferably, the diffraction gratings are of the Ronchi type characterized by a sparse grating with a number of rules between 1 / mm to 10 / mm, favorably 3 / mm.

Preferably, the light source is monochromatic laser light, favorably with a Raighley range greater than the focal length of the optical system under study.

Preferably, the detector is placed on the translation stage and is movable along the optical axis.

The measurement setup and the method for determining the position of a focal plane and an effective focal length of an optical system have advantages, which include: - high precision and easy criterion for the determination of both focal plane and the effective focal length;

- versatility of the method according to the invention - for any optical system may be used the same laser, the same diffraction grating and the detector. In the case of systems with very small or very long focal length, one can adjust the grating constant (λ _δ ). In the case of a long focal length systems - the grating constant (λ ₅ ) can be reduced and, in the case of systems with a very short focal length the grating constant (λ _δ ) can be increased;

- the movable elements in the process of the invention is one of the "sparse" transmission diffraction gratings, which should be rotated for determination of the focal plane and the detector which should be moved to determine and calculate the correct effective focal length. Other elements of the optical system remain stationary during the determination of the plane and focal length;

- Simple measurements - one of the diffraction grating is rotated and the detector is moved, so spots corresponding to that successive orders of diffraction on two diffraction gratings overlap, and then one measures their mutual position.

The additional advantage of the proposed solution according to the invention are following conveniences - all components used in the invention are very cheap and widely available: a laser (this may be any laser pointer), two "sparse" transmission diffraction grating (may be printed on the transparent film or glass as black uniformly separated stripes); camera without a lens attached to a computer (can be any webcam) of known size and separation of pixels, such that one can determine beams separation in space in micrometers [μηι].

The invention is shown in the drawing, in which:

- Fig. 1 presents a simplified measurement system according to the invention in a preferable embodiment;

- Fig. 2a and 2b - a simple optical system showing the relationship x [Θ) = f · Θ in two cases, on example of a single thin optical lens into which the light beam enters a) beam of light entering parallel to the axis of the system; b) beam of light entering the system at an angle Θ;

Fig. 3 - detailed diagram of the measurement setup according to the invention in a preferable embodiment;

Fig. 4a - presents an image from the detector of the correct measurement of focal length and the focal plane of the 300 mm lens, the two diffraction gratings are set with planes and the rulings parallel to each other;

Fig. 4b - graph of the intensity with the localized points of local maxima of the position of the spots of consecutive orders on the detector; properly conducted measurement for 300mm lens; rows of diffractive fringes from -3 to 3 are seen; Fig. 5 - presents the position of successive rows of deflection as a function of their angles, for the focal length of the tested lenses, with a simple adjustment, depending on the corresponding x Θ) = f · Θ; Measurements were made for a lens with the effective focal length of 300 mm and for commercial Nikkor 55-

200mm (1 : 4-5.6G ED) for setting 200mm, 135mm, 55mm;

Fig. 6 - image from the detector placed in the focal plane, the correct measurement of a lens with the effective focal length of 300 mm, where the distance of the detector corresponds to the focal length, and one of the diffraction grating is rotated by an angle of 1 1°;

Fig. 7 - image from the detector, which is not in the focal plane, the measurement for a lens with the effective focal length of 300 mm, where one of the diffraction grating is rotated by an angle of 12°;

Fig. 8 - image from the detector, which is not in the focal plane of the measurement for the lens having the effective focal length of 300 mm, where the diffraction gratings are arranged in parallel by planes and rulings to each other;

Fig. 9 - image from the detector, in another preferable embodiment, where the plane of the two diffraction gratings are parallel, while the rulings are not parallel

(tilted by angle of 18°) and they are not arranged vertically. The measurement setup for determining the position of a focal plane and an effective focal length of an optical system 4, in the embodiment of Fig. 1 and 3, includes a light source 1, two identical transmission diffraction gratings 2 and 3 and a movable detector 5 placed on the optical axis of the optical system 4. Diffraction gratings 2, 3, placed between the light source 1 and the optical system 4, they are set with their surfaces parallel to each other and perpendicular to the optical axis, and additionally are rotatable around the optical axis of the optical system 4. The first diffraction grating 2 is located at such a distance from the second diffraction grating 3 that at least 1 and -1 diffraction orders rays from the first diffraction grating 2 enter the second aperture of the diffraction grating 3. In contrast, the second diffraction grating 3 is arranged at a distance from the optical system 4, that at least 1 and -1 diffraction orders rays from the second diffraction grating 3 enter the aperture of the optical system 4. In the preferred embodiment, the second diffraction grating 3 is located by the optical system 4.

To solve the problems known in the prior art, at first the simplest optical system was analyzed - Fig. 2b - the thin lens followed by the screen or detector 5 in the distance d. The distance x between the ray to the optical axis after leaving the optical setup and seen on the screen, is related with the distance x ' between the optical axis and the ray entering the setup with the angle Θ:

If in the above equation is relation d ~ f, means the distance from the plane of the beam is equal to the lens focal length / of the lens, then the above equation simplifies to:

x = f - θ

The above equation implies that the x position of the ray on the screen depends only on the angle Θ to the axis of the incoming beam, and does not depend on the distance x' from the axis. In contrast, the proportional coefficient between the distance x and the angle #,/is the effective focal length to be determined, as shown in Fig. 2b. Theoretically, it is possible to use the equation above in a straightforward manner, using multiple beams of light entering the system at different angles, provided that for the specific plane the above relation is fulfilled. At the same time, formation of rays or beams at well-defined angles relative to the optical axis is a time-consuming and cumbersome task requiring a good calibration of the system. Procedure would require the repetition of measurements for each plane and for many different angles of entering rays.

Application of "sparse" transmission diffraction grating illuminated with laser light allows to obtain equally-spaced apart spots on the screen behind the gratings due to the diffraction of the incident beam, which is the ideal scale to determine the effective focal length of the optical system under study. The transmission diffraction grating 2 and 3 used in this embodiment are a sparse type of Ronchi gratings having from 1 to 10 rulings per mm, as shown in Fig. 3. Monochromatic laser light from any laser light source 1, for example, the laser diode as TOPTICA DL100 with a beam power 20mW and the wavelength of 795nm, favorably within Raighley range greater than the effective focal length of the test system 4, shining perpendicularly to the plane of the sparse transmission diffraction grating 2 and 3, after passing through the diffraction grating 2 and 3 is diffracted, as shown in Fig. 3. For transmission gratings whose rulings are sparsely separated, the beam is diffracted approximately by following angles a, 2a, 3a, 4a etc. and -a, -2a,-3a, 4a etc. For a grating with a well- defined grating constant, wherein the grating constant λ ₈ can be readily determined e.g. by calipers, the angles at which the grating deflect the light rays are well known and are regularly distributed, as shown in FIG. 3. The sparse transmission diffraction grating can be easily made from commercially available materials such as etched metal on a transparent substrate e.g. glass or a film made of transparent polymer coating, or, for instance by printing black stripes on a transparent film. Modern technology allows precise preparation of such a sparse grating of at least 300 rulings per inch. Commonly available laser printers are able to print up to 200 lines per inch, this is the so-called parameter LPI (lines per inch), used for many computer printers. After passing the beam of a monochromatic source of light such a laser light 1, through two transmission diffraction gratings 2 and 3 shown in Fig. 3, followed by the tested optical system 4 on the screen or detector 5, for example, on the camera, in the vicinity of the plane the focal length, one can observe several equally- spaced spots, as shown in Fig. 4a and 4b. When the detector 5 is arranged ideally in the focal plane, by measuring the distance between two dots, and based on the mathematical methods e.g. the least squares method, as shown in Fig. 5, the focal length can be determined. It should be noticed that the measurement of light in a plane other than the focal length gives a very similar result, several equidistant spots, but otherwise arranged on the screen / detector 5, as illustrated in Fig. 7 and 8. Therefore, an additional step which determines whether the detector 5 is in the focal plane, is required. In determining the focal plane used for the fact that the distance in the focal plane of rays emerging from the -axis depends only on the angle Θ ray entering the system. Therefore, when applied as a second identical transmission diffraction grating 3 arranged in the largest possible distances from the first grid 2 and the optical system 4 then the tested optical system 4 was illuminated by beams at the same angles i.e. the angles a of equal separation, but at other positions on the test plane of the optical system, making a series of equally spaced spots on the screen / detector 5. The greater the distance between the diffraction gratings 2 and 3, the grater difference between positions of different entering rays, the set of angles a remains the same. Thus, if the detector 5 is not in the focal plane, one can record the two sets of spots equally spaced from each other, but with different separations. Fig. 8 shows an image of the detector 5 located in the wrong position, that is, when the detector 5 is not in the focal plane 4 of the system under study.

In case of sparse gratings used in the present invention, the diffraction angles are small, therefore, there are visible following orders of diffraction - always appearing in pairs 1 and -1, 2 and -2, etc., plus the zero diffraction order. Thus, the determination of the focal length can be made from 3, 5, etc. spots - an odd number. In a natural way also the more spots were recorded, the higher is fit precision x (Θ) = f - Θ and thus the accuracy of the determination of the effective focal length.

In the method of determining the position of a focal plane and an effective focal length of an optical system 4, in the embodiment, first two sparse transmission diffraction gratings 2 and 3 are set with surfaces rulings parallel to each other. Then one moves the detector 5, here IDS Imaging camera brand, model UI-1240LE used without the lens, of a known size of the image sensor and the separation of pixels, so that one can determine the beams separation in space in micrometers. The detector 5 is set on the movable board, driven manually or by a mechanical or electronic mechanism. The further is the detector 5 distanced from the plane, the more spots - more orders of diffraction - pairs distanced and turn the detector 5 closer to the focal plane, the dots - more rows of deflection - and eventually converge at the focal plane coincide perfectly. Then rotates a transmission diffraction grating 2, 3 with a small angle from 0° to 20°, 15° here. Due to the finite spot diameter, it is favorable to rotate one of the diffraction grating 2, 3 by a small angle which facilitates identifying when spots overlap e.g. in the horizontal direction, since upon rotation of one of the diffraction gratings the spots move apart vertically, which can be seen by comparing the Fig. 6 and Fig. 9. The proper setting of gratings and camera position in the method according to the present invention is presented in Fig. 4a whereas Fig. 8 shows an improper setting in the method according to the invention. When the spot exactly coincide, the detector 5 is precisely positioned in the focal plane, and according to the positioning of the dots can be accurately determine the value of the focal length / using, for example, by fitting with a simple method of least squares as illustrated in Fig. 5.

The upper part of Fig. 3 presents example of "sparse" transmission diffraction grating 2, 3, used to calibrate the optical system 4, with n I mm rulings. During the determination of the focal length and the focal plane, the diffraction gratings 2, 3 were used that contained a number of rulings 1 / mm to 10 / mm. The optimal number of rulings is a 3 / mm. Note that, a typical reflective diffraction grating has 600 rulings per mm, so the grating constant equals λ _δ = 1,667 μm (d = 1000 μιη/600). A plane wave illumination on a typical diffraction grating 2, 3 is decomposed into constituent waves that are visible on the screen in the form of a diffraction pattern. In the case of the applied of "sparse" transmission diffraction grating according to the invention - of the 1-10 rulings per mm - the grating enables aforementioned diffraction/deflection of the incident light wave.

For the method of the invention, where λ ₈ is the constant of the diffraction grating, for example a grating comprising 3 rulings per mm, λ _δ = 333.33 μιη; from the formula λ ₅ = l/n [mm] _L - laser wavelength; the diffraction grating K = 2π / λ ₅ for grating = 2πη for light wave = 2π / X _L , the following relationship is true for small angles of incidence of the light beam:

K 2π _L λι

~ k λ _$ 2π λ ₅

Fig. 3 presents the effect of the diffraction grating 2, 3 with a small grating constant, which diffracts/deflects a beam of visible light at the characteristic angles a of the grid, thereby giving, following orders of diffraction. For instance, in a preferred embodiment, for 795nm laser wavelength used for diffraction grating n = 1/3 mm:

795 nm 795 nm

a ¾— =——— = 0,265 mrad = 0,015°

3 mm 3000000 nm

Fig. 5 presents a solution according to the linear method of least squares using the following formula:

*(0) = f - e + x ₀

where xo means the position of the deflection zero-order on camera

when x _Q = 0 ^{men χ} {β} - f ' Θ

θ _η = η · a

for which there is a relationship: x _x = fa ; x ₂ = f a ; x ₃ = f3a therefore, (/) it is the coefficient of the linear fit. In the method of the invention, an approximation was used that ignores the double deflection beam, namely the first grid 2 and the second grid 3 of diffraction because it is very weak and difficult to record onto a detector, which show Fig. 4a, Fig. 6, Fig. 7 and Fig. 9. Double deflection are of less intensity and it is difficult to confuse them with the beam deflections to only one grid. Any possible angles a of successive orders of diffraction on the second grating 3, correspond to the angles a deflection on the first diffraction grating 2.

In summary, the method of determining the position of a focal plane and a focal length of an optical system in the embodiment comprises the following steps:

• positioning a sparse transmission diffraction grating 2, 3 in parallel planes and rulings to each other and such that the first diffraction grating 2 is located at such a distance from the second diffraction grating 3 that at least 1 and -1 diffraction orders on the first diffraction grating 2 enters the aperture of the second diffraction grating 3;

• positioning of the test optical system 4 at a predetermined distance from the second diffraction grating 3, so that first order diffraction light beams do not go beyond the aperture of the optical system 4;

• positioning the detector 5 at a distance which allows to distinguish between successive orders of diffraction of light beams, an initial image shown in Fig. 8;

• rotate one of the diffraction gratings 2, 3 by a small angle range from 0° to 20°, here about 15° so that the first orders of diffraction were right above them;

• moving the detector in a position in which the spot originating from the diffraction of consecutive orders will be located above each other;

• rotating the diffraction grating to the initial setting, i.e., to the setting in which the surfaces and rulings of the diffraction grating 2 and 3 are parallel to each other, see Fig. 4a;

• collecting and analyzing the data from the detector 4, the intensity cross-section and finding the local maxima, as illustrated in Fig. 4b; • to increase the precision of determining an effective focal length one performs fitting the position of subsequent orders of diffraction as a function of their angles for the effective focal length of the optical system, by applying a linear shown in fig. 5.

The invention may find numerous applications in precision optical devices comprising of multiple lenses and / or multiple optical elements, in particular microscopes and telescopes, objective lenses in photography, sighting systems, ocular optics, and medical applications of precision optics. In addition, the solution can be used in all other devices that use optical setups and devices.