
In the terahertz range, radiation detectors are key elements in optical systems requiring image recording. In this part we present holography as an imaging technique for amplitude and phase objects. An image appears after numerical reconstruction of a digital hologram described in “Classical holography imaging” section below. Here we leave out of our scope singlechannel detectors, hotelectron detectors^{26}, and semiconductor detectors^{27}, which can be applied to recording of holograms by scanning techniques^{20}. All matrix detectors we employed to record terahertz images in different experiments are based on the detection of the thermal effect of radiation, but with different techniques of registration of the thermal imprint.
In more than 15 years of our research in that area^{28, 29}, which can be called holography in the broad sense of the word, five imaging systems have been created or adapted to the terahertz range by our effort^{30}. None of them is universal, combining high sensitivity, high spatial and temporal resolutions, as well as a large working area, which is important for achieving a high Nyquist frequency, particularly in the case of longwavelength radiation. For this reason, the choice of recorder is determined by requirements of each particular experiment.
The recorders are shown in Fig. 1, and their characteristics are summarized in Table 1. The first three devices are not directly sensitive to terahertz radiation, and they could only register its heat trace. The very first primitive Fresnel holograms, diffraction patterns of amplitude masks with a reference hole for the reference beam^{28}, were captured with the use of a thermal imager (TI) sensitive to nearinfrared radiation. Due to the low spatial and temporal resolution of the holograms recorded in this way, as well as the strong thermal noise at the laboratory at a wavelength of 3 μm, later we used other methods.
Fig. 1 Imaging devices used in terahertz range. Blue bars: ranges of spectral sensitivity of devices. Red bars: operating wavelength of final recorders.
Device TSI TSPP TI MBA Pyrocam IV Mode active active passive direct direct ∆x (mm) 0.3 0.25/1.5 2 0.051* 0.08* ∆t (s) 0.3 0.2/1.3 0.02 0.02 Repetition rate (fps) − 4/1 25 20 25 Area (mm^{2}) 50 × 50 75 × 75 − 16.32 × 12.24 25.6 × 25.6 Sensitivity medium low medium very high high Table 1. Characteristics of imaging devices at wavelength of 140 μm (asterisk denotes pixel size)
The other two devices, in which radiation of the visible range was used for registration of terahertz images, had a higher spatial resolution. The large area of holograms obtained with their help improves the quality of their reconstruction. In a thermal sensitive interferometer (TSI), a video camera records the dynamics of an interference pattern in a planeparallel glass plate illuminated by a collimated visible diode laser beam. The investigated terahertz radiation illuminates the plate, which is not transparent in the terahertz range, from the reverse side. The change in the refractive index in the thin glass layer due to the heating increases locally the optical path length. Since it is known a priori that when the plate is exposed to terahertz radiation after opening of the shutter, the phase incursion at each point can only be positive, the interferogram can be easily converted numerically into the intensity distribution of the terahertz beam. The thermooptical properties of BK7 glass are standard and well known. In Ref. 31, we showed that such an interferometer is an absolute power meter. It does not require any calibration, because a shift by one fringe depends only on thermaloptical characteristics of the plate, and for K7 glass it corresponds to a locally absorbed power of 5.1 J/cm^{2}. Unlike all other visualizers, this device is used in the single measurement mode. Remeasurement is possible only after the plate has cooled down. The possible size of the image is determined by the size of the illuminated area, which in our case was equal to 60 mm.
Another largearea recorders, eight Macken Instruments thermosensitive phosphor plates (TSPPs), were capable of recording holograms with an area of 75 by 75 mm. The principle of their operation is based on thermal quenching of luminescence excited by a mercury lamp. We have shown^{32} that the response to heating is linear up to a quenching value of about 50%. Table 1 shows the characteristics of plates No. 7 and No. 8.
For the first time, matrix receivers capable of directly detecting terahertz radiation in real time were used in the mid2000s^{33, 34}. Those were microbolometer arrays (MBAs) with vanadium oxide sensors designed for the midinfrared range. Their sensitivity threshold in the THz range was 15 times less than that in the IR range. Nevertheless, it was rather high (
$ 10^{3} $ W/cm^{2} or 33 nW/pixel). It was soon discovered that they were polarizationsensitive, and the antenna effect in the lead wires turned out to be a mechanism for absorbing terahertz radiation^{35}. The antenna effect is now employed in newly designed microbolometer arrays^{36}. The resolution of array receivers is restricted by the wavelength limit since the size of their sensitive elements is smaller than the wavelength. A relative disadvantage of these cameras, for example, when shooting holograms, is their small physical size. Recently, commercially available Pyrocam IV pyroelectric camera with a$ 320 \times 320 $ matrix, but with sensitivity lower than the MBA sensitivity, have appeared. Thus, at present, there is some possibility of choosing imaging devices in the terahertz range. Examples of holograms recorded by different imaging devices and reconstructed images are shown in Fig. 2.Fig. 2 Holograms (top row) recorded a by thermal imager (two circular apertures with diameter of 6 mm and spacing of 14 mm), b thermal sensitive phosphor plate, c microbolometer matrix, and respective reconstructed images (bottom row); objects (metal masks). Figure a is adapted with permission from Ref. 28; Figures b and c are adapted with permission from Ref. 29.

Because of the lack of media that store the intensity distribution in the THz range, holograms can be recorded only digitally, and thus they can be reconstructed either digitally or by means of a second light source (e.g. visible range laser source). There are several approaches to calculation of back propagation, overviewed in Ref. 29. We consider back propagation as the most adequate technique for image reconstruction. In the terahertz range, the pixel size to the wavelength ratio limits the applicability of reconstruction techniques. Here, given our experience with sampling limitations, we will show reconstruction using only the RayleighSommerfeld convolution (RSC) approach since it works the best with our geometries. Let us briefly recall that to perform the reconstruction numerically, one needs to multiply the hologram transmission function
$ H(\xi,\eta) $ by the complex conjugate reference wave$ E_{R}(\xi,\eta) $ in the hologram plane$ \{\xi,\eta\} $ (Fig. 3a). We used for the reconstruction a plane wave with uniform intensity,Fig. 3 Images of example object showing finest achieved resolution for inline and offaxis geometries. Recording and reconstruction schemes a, b. Holograms of metal mask c, e and reconstructed images d, f with RSC approach, for inline and offaxis scheme respectively.
$$ E_{R}(\xi,\eta)= E_{R}^{*}(\xi,\eta)=1. $$ (1) The output wave propagates through free space in the backward direction and forms a wave field in the object plane
$ \{x,y\} $ , which is described by the RayleighSommerfeld diffraction integral$$ E(x,y)= \frac{i}{\lambda} \iint H(\xi,\eta)\frac{\exp(ikr)}{r}\cos(\mathbf{n,r})d\xi d\eta, $$ (2) where
$ H(\xi,\eta) $ is the hologram transmission function, and λ is the wavelength. The distance$ r(\xi,\eta,x,y) $ is determined as follows:$$ r=z^{2}+(\xix)^{2}+(\etay)^{2}. $$ (3) $$ \cos(\mathbf{n,r})=z/r, $$ (4) is the reconstructed wave. Solution of Eq. 2 enables derivation of the intensity and phase distributions in the object plane using the following expressions:
$$ I(x,y) = \lvert E(x,y) \rvert^{2}, $$ (5) $$ \varphi(x,y) = \arg \frac{Im [E(x,y)]}{Re [E(x,y)]}. $$ (6) Eq. 2 can be solved directly without any approximations, or, if the experimental configuration allows, it can be simplified and solved by other methods, which include one or several (Fast) Fourier transforms. For the Fraunhofer diffraction zone (
$ z>k[(\Delta x)^{2}+(\Delta y)^{2}]_{max}/2 $ ), the hologram can be reconstructed with a single Fourier transform. Since in a real holographic experiment, the terahertz hologram is almost always located in the near zone, hereinafter we will consider only the Fresnel diffraction.By virtue of the convolution theorem and Eq. 2,
$ E(x,y) $ can be written down as$$ E(x,y) = {\cal{F}}^{1}\{ {\cal{F}}[H(\xi, \eta)]\cdot {\cal{F}}[h(\xi, \eta))] \}, $$ (7) which is referred below to as the RSC, where
$$ h(\xi, \eta) = \frac{iz\exp(ik\sqrt{\xi^{2}+\eta^{2}+z^{2}})}{\lambda (\xi^{2}+\eta^{2}+z^{2})} $$ (8) is the impulse response function.
Reconstruction of holographic images without aliasing requires fulfillment of the conditions of the sampling theorem. In the case of the RSC, such conditions, depending on four dimensionless parameters, were derived in Ref. 29.
The RSC approach also works for offaxis holograms recorded in the scheme of the MachZehnder interferometer. To compare the offaxis and inline holography resolutions, we created an image of a target with horizontal and vertical slits of 0.6 mm (Fig. 3). For higher spatial resolution, the offaxis hologram was made of five frames by scanning in a plane perpendicular to the direction of beam propagation. It can be seen that this resolution is limited and unattainable in the absence of an expansion of the hologram recording field.
To conclude, inline holography systems have a larger effective field of view and a higher imaging resolution as compared with offaxis ones^{37} for the terahertz range. However, there are systems with a strong zero order diffraction, like imaging in attenuated total reflection systems, which require using an offaxis recording technique^{38, 39}. Besides, it should be mentioned that information about phase in an inline scheme in general can be obtained with additional phase retrieval algorithms, while offaxis holography easily provides it. At the same time, specifically to the THz frequency range, where, as it was mentioned earlier (Table 1), the pixel size can be less than the wavelength, the angles between the reference and object waves can be larger. Thus, it is possible to form subwavelength interference fringes, which will not degrade the resolution during the reconstruction, similarly, as it was done in numerical postprocessing in work Ref. 10.

The appearance of powerful terahertz gyrotrons and their modifications opens up the possibility of creating terahertz lidar systems and communication systems in free space^{64, 65}. Investigation of the propagation of highpower terahertz beams with different transverse structures in inhomogeneous media at the Novosibirsk variablewavelength free electron laser makes it possible to carry out fullscale modeling and optimization of such systems. For sources like free electron laser it, is nearly impossible to tune to higher modes inside an optical resonator.
Computersynthesized holograms were proposed to form singlemode GaussLaguerre and GaussHermite beams from an illuminating laser beam in the visible or infrared range^{66, 67}. For reduction of complex transmission function to a pure phase or purely amplitude form, coding methods based on the introduction of a modulated carrier into the phase (amplitude) of a digital hologram were applied. However, this approach led to a low diffraction efficiency of the hologram, which was associated with the emergence of a large number of higher diffraction orders. As shown in Ref. 42, in the case of loworder GaussHermite modes, a transparent element the phase function of which is the portrait of the mode to form enables creation of a beam with a required mode content of more than 80% when the element is illuminated with a Gaussian beam.
This approach was used in Ref. 68 for production of elements intended for the formation of singlemode GaussHermite beams from an illuminating beam of a highpower terahertz laser. Since the phase function of the GaussHermite mode over the entire crosssection area of the beam takes one of two values, that differ by π (Fig. 6), the corresponding element will have a binary microrelief, created in Ref. 68 by the technology of single reactive ion etching (Bosch process) of the surface of a silicon substrate. This method was previously used in Ref. 52.
Fig. 6 Phase functions of DOEs generating GaussHermite (1,0) beam a, GaussHermite (1,1) beam b, beam consisting of the sum of GaussLaguerre modes (2,2) and (2,2) with equal weights c (black colour corresponds to phase of 0 and white colour corresponds to π); intensity distributions d−f in crosssections of beams generated by DOEs with phase functions a−c respectively, as result of computer simulation; intensity distributions g−i in crosssections of beams generated by DOEs with phase functions a−c respectively, as result of natural experiment.
Fig. 6a shows the phase of the element manufactured in Ref. 68, intended for the formation of the GaussHermite mode (1,0) (white color: phase value of π, black color : of 0). The intensity of the beam formed by the manufactured element in the focal plane of a TPX lens installed directly behind the element is shown in Fig. 6d for simulation and in Fig. 6g for the experiment. Figs. 6b, e, and h show similar results of a study of a silicon binary element^{68}, which forms a singlemode GaussHermite beam (1,1). Fig. 6b shows the phase function of the element, and Fig. 6h depicts the intensity of the beam formed by the manufactured element^{68}, at the focal distance.
It is known that the use of diffractive optical elements (computersynthesized holograms) makes it possible to obtain beams of gas, semiconductor and solidstate lasers with a given composition of transverse modes at a wavelength corresponding to one of the longitudinal laser modes^{42}. DOEs, the manufacturing methods of which are described in Ref. 68, enable formation of beams of coherent radiation with a given transversemode composition in the terahertz range. An important property of NovoFEL for various experiments is the ability to tune its wavelength over a wide range^{69}, whereas DOEs with continuous relief and binary DOEs are calculated to operate at a given wavelength. In our studies, we have successfully used various phase elements calculated for the wavelength
$ \lambda=141 $ μm in experiments performed at wavelengths$ \lambda/(2m1) $ , in particular, at$ \lambda/3=47 $ μm. We also experimentally demonstrated that diffractive elements, for example, helical binary axicons, are capable of forming a given transverse mode composition even when the wavelength deviates from the calculated one by 25−30%. In this case, the main unwanted mode is the zeroorder diffraction, which can be easily filtered out. 
Light beams with a topological charge are considered in reviews^{70, 71}. In Ref. 42, computersynthesized holograms were applied to form beams with a topological charge (“vortex beams”) of the visible and infrared ranges. For calculation of binary diffractive optical elements that form a terahertz Bessel beam with a topological charge, the same approach has been applied as in Refs. 72−74. The phase function of the element is described by the formula
$$ \Phi(r,\phi)=(\pi/2){\rm{sign}}(\sin(l\phi\kappa r)), $$ (9) where
$p=2\pi/\kappa $ is the period.The element forms a beam, which amplitude is described by the Bessel function of the first kind:
$$ E(r,\phi, z, t) =E_0 J_{\lvert l \rvert} (k_r r)\exp [{i(l\phi + k_{z}z\omega t)]}, $$ (10) where
$ E_{0} $ is the maximum amplitude of oscillations, ϕ is the azimuthal angle, ω is the cyclic frequency of oscillations, k is the wavenumber, z is the distance from the source to the object, and l is a positive or negative integer, called the topological charge, which determines the degree of “twisting” of the beam.Longitudinal and cross sections of beams with orbital angular momentum formed from a NovoFEL Gaussian beam by binary spiral axicons are shown in Fig. 7. The phase relief of the axicons is shown in the insets (black colour corresponds to a phase of 0, and white colour corresponds to π). Crosssections of the beams can be described well by the square of the Bessel function of the first kind
$ \left( J_{\lvert l \rvert}(k_{r}r) \right)^{2} $ with the topological charges$ l = 1 $ and$ l = 2 $ . The intensity distribution in the crosssection maintains within a distance of about 180 mm, after which, due to the limited aperture of the axicons, it begins to diverge. Therefore, the beams can be considered “diffractionfree”. The formed beam had an annular structure with the diameters of the rings increasing with the topological charge. The value and sign of the beam topological charge can be detected with the MachZehnder interferometer shown in the bottom of the figure. This scheme is, in fact, a holographic one.
Holography with highpower CW coherent terahertz source: optical components, imaging, and applications
 Light: Advanced Manufacturing 3, Article number: (2022)
 Received: 09 September 2021
 Revised: 25 April 2022
 Accepted: 25 April 2022 Published online: 09 June 2022
doi: https://doi.org/10.37188/lam.2022.031
Abstract:
This paper presents the results of 15 years of studies in the field of terahertz holography at the Novosibirsk free electron laser. They cover two areas: research on obtaining holographic images in the terahertz range and the use of diffractive optical elements to form highpower terahertz radiation fields with specified characteristics (intensity, phase, and polarization), using wellstudied and widely applied in the optical range methods of optical (analog), digital, and computergenerated holography. All experiments were performed with the application of highpower coherent monochromatic frequencytunable radiation from the Novosibirsk free electron laser. The features of hologram registration in the terahertz range are described. Methods, technologies, and optical materials for terahertz holographic elements are discussed. A wide range of promising applications of highpower terahertz fields with a given spatial structure is considered. The results of the study of terahertz holograms recorded as digital holograms, as well as radiationresistive optical elements realized as computersynthesized holograms, are presented.
Research Summary
Holography expanded to submillimeter range
Holography has become part of our daily life. ID cards, CDs, medical imaging, and augmented reality are examples of the widespread use of holographic imaging. The first holograms were recorded as interference patterns on photo plates illuminated by direct laser radiation and light scattered from objects. Now, computers can instantly calculate holograms of any given objects (even nonexistent ones) and transmit them in real time to spatial light modulators, illumination of which enables formation of the required image. However, previously developed methods are inapplicable in the case of submillimeter (terahertz) radiation. Yulia Choporova and her colleagues from Novosibirsk and Samara Universities and BINP SB RAS report on the development of analog, digital, and computer holography using terahertz radiation from the Novosibirsk free electron laser and give examples of its application in photonics, plasmonics, and telecommunications.
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