Laser interferometers for control of optical surface shape
you cannot improve it".
W. Thompson (Lord Kelvin)
Interferometers are one of the most accurate measuring devices, which use the principle of light wave interference . Microelectronics, optics, laser engineering, telecommunication, astronomy, precision mechanics, measuring equipment could not exist without use of various types of interferometers. Quality of optical surfaces is determined by the methods of their control. When making the modern optical systems, it is required to control the shape of surfaces (plates, mirrors and lenses) with the accuracy of units and even fractions of nanometer. At the same time, the area of surface under test can be equal to several tens of square meters.
There are many types of wavefront sensors  and interferometers, of which the most common are configurations of Fizeau and Twyman-Green. Perfect interferometer must form the map of three-dimensional optical surface without distortion (error) irrespective if the surface is flat, spherical or asphercial. The perfect system must be resistant to vibrations, temperature differences, it must be easy to use and it should operate without failures. For the practical application of interferometers it is necessary to know and understand their peculiarities, application ranges and areas of their improvement.
MODERN FIZEAU INTERFEROMETERS FOR OPTICAL SURFACES TESTING
Interferometers with large field are usually based on Fizeau principle, which requires minimum optical components: source and detector of radiation, collimating lens, reference plate and object under test. Fizeau interferometer uses the common path for test and reference beams, and therefore there are no special requirements to the quality of optical components.
On one hand, light field diameter determines the dimensions of tested components, and on the other hand, it determines the dimensions and cost of device. At the present time, interferometers with the operational field from 5 to 150 mm are produced. Larger diameter of the field is provided by beam expanders. Devices with the field of 102 mm are the most common devices.
Single-frequency He-Nе – or semiconductor lasers are usually used as the light source. One of the main characteristics includes coherence length, which can reach 100 m. Sometimes single-frequency 532/1064-nm lasers are used in interferometers. However, for high-precision testing of aspherical surfaces with the use of CGH, He-Ne lasers are selected, as its wavelength is determined with high accuracy.
Currently, Fizeau interferometers are commercially available from many companies. Some models of commercially available devices with the 102 mm field, which are compatible with reference flats and spheres of Zygo standard, are specified in the table. Modern devices are controlled by computer and allow generating 2- and 3-dimensional maps of tested optical surface. The algorithms of phase shift (PS) or spectral analysis (SA) are used for processing of interferograms . PS method is the most precise method; however, it is not applicable under the conditions of vibrations because the measurement time is up to 0.3 sec. SA method uses one interferogram, and therefore the measurement time is reduced to tens of microseconds. However, the key feature of this method is registration of interferogram with many fringes, and this fact brings measuring errors caused by the disturbance of common path principle and requires high-resolution CCD cameras without protective glass . When testing the asphercial surfaces, especially with large curvature, in case of inclination the aberration of coma type occurs. This aberration must be excluded from the measurement results via software, but it is not always convenient.
Over the recent years, the devices implementing instantaneous phase shift algorithms (IPS) have been developed in relation to Twyman-Green type interferometers  as well as Fizeau-type interferometers . These devices allow performing the measurements under the conditions of high level of vibrations; however, they require careful calibration because the common path principle is disturbed.
Often, interferometers are built into the measuring complexes with vertical arrangement of device and component . Complexes of Zygo  and QED  companies refer to one of the highest-technology devices, which allow controlling the shape of spherical and aspherical surfaces. The cost of such systems reaches millions of dollars.
Earlier, in Russia the interferometer IKD-110 was commercially available . At the present moment the devices of following models are produced: OPTOTL-ICO-60 (field – 60 mm, λ = 0.532 µm, measurement error δ ~ 1/20 λ) , RIF (field – 95 mm, λ = 0.65 µm, δ ~ 1/20 λ) , IFA-300 (field – up to 300 mm, λ = 0.65 µm, δ ~ 1/100 λ)  and FTI-100PS  (see Table 1).
In order to test flat and spherical surfaces the standard commercial interferometers are used, and in order to test aspherical surfaces (AS) they are complemented  CGH. However, use of CGH has a number of peculiarities, such as presence of parasitic diffraction orders (DO), low diffraction efficiency, required high accuracy of CGH adjustment, etc. . Use of CGHs in combination with commercial interferometers  do not always allow taking into account these factors, which causes the decrease of measurement accuracy and sometimes gives wrong measurement result.
PECULIARITIES OF FIZEAU INTERFEROMETERS FOR ASPHERE TEST
Simplified optical layout of IF is shown in Fig. 1a. Optical radiation from the light source (S) using beam splitter (BS) and collimating lens (L1) is directed to the transmission flat (TF) and then to the surface under test (plane P1). Radiation reflected from this surface and from the side A of transmission flat (side B has wedge α) is focused by the lens L1 in the plane of diaphragm D (point S), and then with the help of lens L2 it is directed to CCD camera VC1 located in the plane P2. Diaphragm D, which serves for blocking of parasitic radiation reflected from optical elements, in particular, from the side B of transmission flat (dashed line and point S''), is one of the key elements of IF. The diameter d of diaphragm determines the spatial resolution of device. If the transmission flat has wedge α = 15 angular minutes, then at the focal length of lens L1 f1 = 600 mm, the diaphragm diameter must be d < 4 α f1 ~ 10 mm. However, if IF is used together with CGH, the diaphragm size must be determined on the basis of diffraction angles of DO. In Fig. 1b the example of DO position on diaphragm surface is shown during the AS testing using CGH . In order to block unwanted DO, the diaphragm diameter must be d < 2f1λ/Tmax, where Tmax is maximum period of the CGH pattern. At λ = 633 nm and Tmax ~ 0.5 mm, the diaphragm diameter must be d < 1.5 mm. Thus, the reasonable selection of diaphragm diameter d allows minimizing the influence of parasitic DO. Diaphragm can also be used for interferometer adjustment. If diaphragm surface has sufficient area, then the position and movement of autocollimating spots can be controlled using the second CCD camera VC2, installed in front of it . This adjustment method is implemented in IF of FTI-100 type.
Interferometer with TF (see Fig. 1a) allows testing the surfaces only with low deviation from plane. In order to test the spherical surfaces, instead of TF the transmission sphere (TS) consisting of several lenses is installed; the output lens has high-quality aplanatic surface, which reflects the reference wavefront . Currently, such TS with the field of 102 mm and 153 mm and aperture of f/0.65 are produced by many companies (Zygo, MPF, Marh etc.).
In order to test AS, the transmission spheres are complemented with wavefront correctors based on CGH. Hologram-corrector transforms the wavefront W1 at TF output into the wavefront W2 conjugated with the shape of AS as shown in Fig. 2a. This layout is widely used for AS testing ; however, it has disadvantage: the accuracy is always lower than the accuracy of flat or spherical surfaces testing. In particular, it is caused by the fact that CGH substrate is not included into the common path of IF, and therefore its error is added to the measurement result. Also, the accuracy of CGH alignment relative to IF and surface has essential role.
One of the ways for elimination of specified disadvantages consists in the application of so-called "combined computer-generated holograms" [14, 21]. Such holograms allow generating two and more independent wavefronts using one element. In this case the need in TS or TF is eliminated. In Fig. 2b the layout of spherical or aspherical surface testing using the combined CGH operating in parallel beam is shown . The reference flat wavefront is formed during the reflection from side A of substrate. Side B has low wedge and does not influence the measurements. Use of parallel beam significantly simplify the hologram alignment. In suggested layout, the substrate of combined CGH is included into the common path and its inhomogeneities have no effect on measurement accuracy. Combined CGHs exceed standard TS  by technical parameters and they are cheaper.
SPATIAL RESOLUTION OF IF
In order to test the precision large mirrors, IF with high spatial resolution are required. In practice, this parameter is usually determined by number of interference fringes recorded by the CCD camera. A number of modern IF allows recording more than 1 000 fringes , and this fact makes it possible to perform AS test with quite large deviation from the nearest sphere. In this case the diaphragm diameter d (see Fig. 1a) must be about ~8 mm. However work with such great number of fringes violates the Fizeau principle (common path of test and reference beams), which, on one hand, results in the decrease of measurement accuracy (to λ/4 ) and requires calibration but, on the other hand, increases the device cost due to higher requirements to the quality of optical system.
Reasonable compromise includes the use of replaceable diaphragms switched by operator command. In one of the modifications of FTI-100 model, switching of replaceable diaphragms is implemented from d = 1.2 to 4.8 mm. In Fig. 3 the example of operational window of interferometer FTI-100 is shown during the plate Ш 100 mm test. Recorded interferogram (in Fig. 3a number of fringes is equal to 110), the plot of fringes intensity distribution by Y axis, table of aberrations are located in left part of the window; the right part contains phase map (2 – or 3-dimensional). The magnified (10Ч) interferogram of central area (Ш 10 mm) of plate under test is shown in Fig. 3b. The window of additional CCD camera (VC2, in Fig. 1) is shown in Fig. 3c with the magnitude 8Ч. Autocollimating spots of test and reference beams, which generate the interferogram, are well seen. The center of grid corresponds to the diaphragm center, and its outer ring – to angular deviation of 3 angular minutes.
Universal (Durango , IntelliWave , Reveal  etc.) software (SW) and specialized SW integrated with the device are used for interferogram processing (see Table 1). The latter variant is more convenient because it allows implementing the series of measurements, displaying of phase map on a real-time basis, automatic adjustment and all standard functions of device calibration and setting.
Interferometer has communication with computer through wire or wireless connection. There are several possibilities. In the first variant, the control unit is located in computer on special expansion board (PCI, PCI Express); CCD cameras are connected separately by the interfaces of USB 3.0. or Camera Link type. In the second case, the connection of interferometer and controlling computer through the interface USB 2.0/3.0 has a number of advantages: high speed of data transmission and capability of support of PnP technology, which makes the procedure of device connection easier. However, USB has significant disadvantages, which consist in the restriction of cable length (it is important if interferometer is located in "clean room" or inaccessible place) and low stability. In the third variant of communication, which is progressive, the connection of interferometer to computer is performed by the network interface Gigabit Ethernet (GigE) of 802.3ab specification. It provides required speed of data transmission, and at the same time it does not have disadvantages connected with the restriction of cable length and stability. Interferometers operating with GigE interface can be installed at the distance up to 100 m from operator. Besides, the interferometers with GigE can be included into the corporate network, which makes it possible to organize the multi-user access. Thus, GigE interface is the most universal and it can be easily applied under production conditions. Interferometers of FTI-100 type are equipped with GigE interface.
OF FUNCTIONAL CAPABILITIES
By its intended purpose, interferometer is intended for the measurement of deviation of surface shape in comparison with standard. However, the optical layout of IF (see Fig. 1a) has much in common with the autocollimator, which is intended for the measurement of inclination angle of controlled surface. Combination of these functions in one device allows enhancing its functional capabilities considerably. In interferometer FTI-100 the image from additional CCD camera (see Fig. 3c) is analyzed by controlling computer, which performs the search of centers of autocollimating spots and calculates their shift from the initial position by two coordinates (dx and dy, see Fig. 3c). In other words, the function of photoelectric autocollimator, which allows performing adjustment of CGH surface, is implemented in interferometer . At the stage of adjustment, the additional CCD camera records the image in the form of two spots (from reference and from surface under test) in diaphragm plane (see Fig. 3c). In case of accurate setting of interferometer they must be projected precisely at the diaphragm center and generate interferogram with "infinite" fringe. Image of these light spots is observed by operator and also it is sent to the unit of image processing where the coordinates of their centers and current error of pickup are calculated.
PROCESSING OF INTERFEROGRAMS
For calculation of wavefront shape and Zernike coefficients the method of phase shift (5 interferograms), method of SA based on Fourier transformation and method of tracking of interference fringes are used (at the option of operator) in interferometer FTI-100 [29, 30]. At the same time, PS algorithm can be self-calibrating . In order to increase the accuracy of restoration of wavefront shape in case of the presence of air turbulence and vibrations, automatic carrying out of the series of measurements with result averaging is provided.
Accuracy of the measurements is usually limited by the quality of TF or TS, lenses or holograms. SW of the interferometer FTI-100 make it possible to subtract calibration data from measurement results, and this fact considerably enhances the measurement accuracy. The main parameter, which characterizes the device, includes the reproducibility of results in the series of several tens of measurements.
Examples of interferograms and maps of surface of controlled plate with the diameter of 100 mm obtained by Diopto program using three methods are shown in Fig. 4. It is seen that PS method gives better detailing of surface shape (Fig. 4b); as a result the value PV is higher than the values obtained by other methods (see Fig. 4d, e), although their root-mean-square (rms) values coincide with the error λ/1000. Advantage of PS method demonstrates the example of interferogram and maps of surface shown in Fig. 5. It is apparent that PS method reproduces the rectangular shape of phase plate profile, whereas the method of spectral analysis smoothes out its shape as shown in plots (see insertions in Fig. 5b, c).
Interferometers are irreplaceable instruments during the production and control of precision optical and laser systems. For the efficient application of interferometers it is necessary to know and understand their peculiarities, application ranges and areas of their improvement. In this paper the configuration of Fizeau interferometer is considered in details, the review of the most common models is given, the characteristics and peculiarities of their layouts and development trends are discussed. The main focus is concentrated on the application of CGH in IF by the example of FTI-100. The new type of combined CGH is presented, which allows reducing the errors of AS test at the expense of elimination of the influence of substrate inhomogeneities. The results of development of IF with enhanced functional characteristics for high-accuracy optical surfaces test are given. Suggested IF combines the functions of interferometer and photoelectric autocollimator, which makes it possible to perform CGH alignment with high level of accuracy and automate the setting process. Original software is developed for device control and interferogram analysis.
Authors show appreciation to V.G.Maksimov, V.A.Tartakovsky, S.A.Chudinov for the development of software module POINT of Diopto program for interferometer control, A.E.Kachkin and B.V.Drachkov for design and fabrication of interferometer FTI-100.