Terahertz conversion with quartz waveplate sets

Andrey K. Kaveev,1,* Grigory I. Kropotov,1 Ekaterina V. Tsygankova,1 Ivan A. Tzibizov,1 Sergey D. Ganichev,2 Sergey N. Danilov,2 Peter Olbrich,2 Christina Zoth,2 Elizaveta G. Kaveeva,3 Alexander I. Zhdanov,4,7 Andrey A. Ivanov,4 Ramil Z. Deyanov,5 and Britta Redlich6 1TYDEX J. S. Co., 16 Domostroitelnaya str., St. Petersburg 194292, Russia 2Terahertz Center, University of Regensburg, Regensburg D-93040, Germany 3St. Petersburg State Polytechnic University, 26 Polytechnicheskaya str., St. Petersburg 194021, Russia 4Samara State Aerospace University, 34 Moskovskoye Schosse, Samara 443086, Russia 5Lomonosov Moscow State University, GSP-1 Leninskie gory, Moscow 119991, Russia 6FOM Institute Rijnhuizen, P.O. Box 1207, Nieuwegein NL-3430 BE, The Netherlands 7Current address: Samara State Technical University, 244 Molodogvardeyskaya str., Samara 443100, Russia *Corresponding author: [email protected]

Received 12 September 2012; accepted 30 November 2012; posted 11 December 2012 (Doc. ID 175976); published 21 January 2013

We present the results of calculation and experimental testing of an achromatic polarization converter and a composite terahertz waveplate (WP), which are represented by sets of plane-parallel birefringent plates with in-plane axis. The calculations took into account the effect of interference, which was especially prominent when plates were separated by an air gap. The possibility of development of a spectrum analyzer design based on a set of WPs is also discussed. © 2013 Optical Society of America OCIS codes: 040.2235, 230.5440.

1. Introduction substances, due to the fact that many vibrational The terahertz (THz) frequency range (300 GHz– and rotational levels of large organic molecules occur 10 THz) is quite a significant portion of the electro- in the THz range), security and alarm systems, magnetic spectrum lying between microwave and explosives, and weapon and drug detection [1–5]. infrared ranges. Unlike the latter, the THz range Besides that, THz radiation can be very promising was virtually unexplored until recent times, due to in tomographic and medical applications [4,6]. the absence of powerful THz sources and detectors Development of femtosecond solid-state lasers and operating in this wavelength range. Promising appli- microelectronics during the last 15–20 years led to a cations of THz radiation include ultrafast secure breakthrough in THz research. Several new methods data exchange, internal and external communica- of signal generation were developed. THz radiation tions in integrated circuits, spectroscopy (namely, may be detected using various methods. The current determining the chemical composition of complex state of scientific research and the existing technical background allow one to anticipate the development 1559-128X/13/040B60-10$15.00/0 of commercially viable compact high-power sources © 2013 Optical Society of America and detectors of THz radiation [7–9].

B60 APPLIED / Vol. 52, No. 4 / 1 February 2013 angles and plate thicknesses, one may achieve the achromatic effect in different wavelength ranges and also for different retardations (for example, λ∕4 and λ∕2).

1. Calculation There are several papers [12–16] related to the cal- culation methods of quartz and sapphire AWPs. Some corrections were described in basic methods of WP calculations, because they are not suitable for the case when the measuring system has high resolution (as compared to the FWHM of interference peaks). So there were some modifications of methods Fig. 1. Transmission spectrum of crystalline quartz, thickness that take into account the interference effect. In 0.4 mm, x-oriented. Tydex J. S. Co. we have applied the methods for real AWP calculations. Also, we have carried out the mea- surements of WP, produced on the basis of the simu- Rapid progress in THz optoelectronics necessitates lations. These experiments confirm the applicability the development of production of optical components of the methods described. Thus the production of for the THz spectral range as demanded by the spe- these objects is achieved. cifics of the instruments operating in the aforemen- According to Jones formalism [11,17] the system tioned range. The material traditionally used for of several retardation plates is optically equal to THz instrumentation is crystalline quartz. Its trans- system containing only two elements—the so called mission spectrum in the spectral range of interest is “retarder” and “rotator” (Fig. 2). shown in Fig. 1. The retarder provides required phase shift (for Besides that, crystalline quartz is a birefringent example, π or π∕2). The rotator turns the polarization material [10]; this fact makes possible its use in THz plane at angle ω. There are two types of the optics to create polarization-converting components. polarization converter depending on the ω value: In this paper, we present theoretical calculations and (1) ω is close to zero within the operating wavelength modeling as well as experimental studies of several range. In this case, it is a common AWP, and its such components based on crystalline quartz. It operating principle is the same as that of the mono- should be noted that instead of quartz, any other chromatic WP. For π—retardation, the polarization birefringent material transparent in the THz range plane transmitted through AWP radiation is situ- may be used, such as sapphire or boron nitride. The ated at 2θ to the axis, where θ is an angle birefringence axis must lie in the plane of the plates, of “effective optical axis” (EOA) of AWP. (2) ω depends which constitutes a technological limitation in the on wavelength. In this case the object is not common latter case. “AWP” and may be called a broadband polarization converter (BBPC)—a special case of AWP. For exam- 2. Results ple, in the case of circular-to-linear polarization transformation, the radiation transmitted through A. Achromatic Polarization Converters the converter has its polarization plane oriented at β ω 45 In this work we have modeled, produced, and tested angle ° to the polarizer axis. In the case the wide wavelength-range THz polarization conver- of linear-to-linear polarization transformation, β ω 2θ ter. In particular, it is a well-known achromatic . In this work we have calculated AWPs waveplate (AWP) [11]. This object consists of a set and BBPCs for λ∕4 and λ∕2 cases in different of plane-parallel birefringent plates, which are trans- THz ranges. Taking into account the interference parent in the THz wavelength range with optical axes effect, we have used [13] modified Jones matrix 4 × 4, of birefringence lying in plane of the plate. Unlike written for each plate:

2 3 cos kedjne sin ked 00 1 ⌢ 6 j sin ked cos ked 007 6 ne 7 Q 4 5; (1) 00cos k0djno sin kod 1 00j sin kod cos k0d no

monochromatic WP, which provides necessary phase where ke 2πne∕λ and ko 2πno∕λ, d is the retardation for a specific wavelength, AWP provides it thickness of the quartz plate, ne and n0 are respective in a wide wavelength range. By varying azimuth refraction coefficients for extraordinary and ordinary

1 February 2013 / Vol. 52, No. 4 / APPLIED OPTICS B61 rays, and j is the imaginary unit. The resulting are near zero, modules of diagonal elements are Jones matrix for the set of N plane-parallel plates about 1, and the phase difference between diagonal is a product of N matrices given by Eq. (1) for each elements is equal to the specified retardation (here, plate: δi 2πne − nodi∕λ, φi is the EOA angle of the plate relative to the polarizer axis). YN Value set δi; φi corresponding to the ith WP can ˆ ˆ ˆ ˆ −1 M FiQiFi ; (2) be calculated using the simulated annealing algo- i1 rithm [15,18]. When calculating these values as where noted above, depending on the minimization para- meters, there are two possible cases: ω ≈ const (for 0 1 ω ≠ ω φ 0 − φ 0 instance, 0), and const. The value has a mean- cos sin “ ” B 0 φ 0 − φ C ing of rotational angle of the effective rotator. Fˆ B cos sin C Degree of polarization, i.e., the ratio of linearly po- @ φ 0 φ 0 A 3 sin cos larized light intensity to total intensity, can be calcu- 0 sin φ 0 cos φ lated as a function of wavelength. This degree is zero for completely circularly polarized light, whereas in is a 4 × 4 rotation matrix describing the angular the case of linear-to- transfor- orientation of the ith plate relative to the plane of mation (linear polarization along the x axis), the polarization of incident radiation. The matrix Mˆ in- degree can be calculated as follows: terrelates the electrical and magnetic field vectors of ¯ ¯ the incident, propagated, and reflected waves. The jE2 · E2 − E1 · E1j matrix that determinates the polarization of the in- I ¯ ¯ ; (6) E2 · E2 E1 · E1 cident wave given the (known) polarization of the propagated wave is as follows: where

−1 −1 E1 P11 cosηP21 sinη;E2 M11M12M21M22 M13M14M23M24 Pˆ 2 2 : (4) P−1 η − P−1 η; η M31M32M41M42 M33M34M43M44 21 cos 11 sin 2 2 P−1P¯ −1 P−1P¯ −1 0 5 11 21 21 11 : . arctan −1 ¯ −1 −1 ¯ −1 (7) P11P11 − P21P21 To calculate the polarization of the propagated wave from the polarization of the incident wave, ˆ −1 one may use the matrix P . That matrix is a kind Sample dependence for a BBPC is depicted of counterpart of the plain [11] 2 × 2 Jones matrix. Ac- in Fig. 3. counting for interference effect in the system of WPs The EOA angle for a λ∕4 polarization converter shows that the system behaves differently when con- is related to the elements of the Jones matrix as verting linear polarization to circular and vice versa. follows:

P−1e−jν P−1e−jν P−1e−jν P−1e−jν θ 0 5 Im 12 Re 11 Re 12 Im 11 ; . arctg −1 −jν −1 −jν −1 −jν −1 −jν (8) ImP11e ReP11e − ReP12e ImP12e

ν 0 5 −P−1P−1 ν 0 5 P−1P−1— The described system exhibits the properties of an where 1 . arg 12 21 , 2 . arg 11 22 AWP with fixed EOA alignment along polarizer axis phase. The exponential factor in this equation is due x to the fact that the counterpart plain Jones matrix in the wavelength range where nondiagonal ele- ˆ −1 ments of a plain 2 × 2 Jones matrix P is no longer described by Eq. (5). There are

0 1 N δi δi δi Y cos j cos2φi sin j sin2φi sin ˆ ˆ ˆ @ 2 2 2 A J Ji; Ji ; (5) δi δi δi i1 j sin2φi sin 2 cos 2 − j cos2φi sin 2

B62 APPLIED OPTICS / Vol. 52, No. 4 / 1 February 2013 Fig. 2. (Color online) Schematic diagram of a set of plane-parallel birefringent media with in-plane birefringence axes. two causes of this, namely, correction for interference and factoring in full phase incursion in the polariza- Fig. 4. (Color online) Phase versus wavelength for BBPC tion converter. To use equations for δ, θ, and ω, which designed for 60–300 μm range. are valid for a plain Jones matrix (5), for approximate −1 calculations, the matrix Pˆ must be multiplied by Thus, a generic nonabsorbing achromatic polariza- exp−jν. Phase values calculated with use of diago- tion converter is described by three parameters, δ, θ, nal and nondiagonal matrix elements must be and ω. The latter must be taken into account, for ex- checked to verify that they are close to each other. ample, when converting circular to linear polariza- Figure 4 depicts sample graph of phase ν versus tion using a quarter-wave converter. Orientation of the wavelength calculated for the same BBPC as the polarization plane of the light output from the ω shown in Fig. 3. In this case the graphs ν1 for ν2 converter will depend on . The polarization plane are similar. of the light after the achromatic converter will be Angle ω may be calculated as follows: angled at ω 45° relative to the polarizer axis. Use of the matrix (1) instead of (5) is feasible when ReP−1e−jν the interference must be taken into account due to ω arctg − 12 : (9) ReP−1e−jν practical considerations, for example, when the spec- 11 tral bandwidth of the source is less than the typical Finally, retardation of the system can be calculated width of the diffraction maximums. Wavelength Pˆ −1 bandwidth can be estimated from a simple formula, as shown below. Given the matrix with near-zero Δλ λ2Δf ∕c Δf nondiagonal elements, , where is the frequency bandwidth. It should be compared to the interference period ob- −1 −1 tained from δλ curve for the same wavelength. δ argP22 − arg P11: (10) Whenever Δλ exceeds interference period, the matrix In general, the retardation can be calculated from (5) is generally sufficient for use. −1 a subset of Pˆ matrix elements. For example, Figure 5 depicts sample dependencies δλ, φλ, and ωλ calculated for 2 × 2 and 4 × 4 Jones matrices ω ≈ ω ≠ ImP−1e−jν2 ImP−1e−jν2 0.5 (for the cases const and const) for a BBPC δ 2 arctg 11 12 : with two-sided antireflection coating. It can be seen P−1e−jν2 P−1e−jν2 Re 11 Re 12 that the general shapes of the curves for different (11) Jones matrices are the same, but the interference effect, which is most prominent for φλ and δλ, leads to “noisiness” of the curves at specific wave- lengths. The same effect is less eminent for ωλ.It should be noted that the interference effect would be much more pronounced without antireflection coating. In the case of AWP, the δλ and φλ curves will be similar, whereas the ωλ curve will be near constant, which can be seen, for example, in Fig. 5(d): throughout the entire range, the angle variation is less than 2°. An important conclusion of the analysis described above is that two-sided antireflection coating is highly recommended for quarter-wave achromatic polarization converters to suppress the interference effect. The antireflection coating is accounted for Fig. 3. Linearly polarized to total intensity ratio as a function of in the analysis above by multiplying the expression wavelength for the light converted to circular polarization using a (2) both on the left and on the right by matrix (1) BBPC designed for the 60–300 μm range. describing such a coating.

1 February 2013 / Vol. 52, No. 4 / APPLIED OPTICS B63 Fig. 5. (Color online) (a) EOA angle φ versus the wavelength in the cases of 2 × 2 (bold curve, no account for interference), and 4 × 4 (fine curve, with interference) Jones matrix. The calculation assumes BBPC (ω ≠ const) designed for the 60–300 μm range. (b) “Effective rotator” rotation angle versus wavelength ωλ for the same BBPC. (c) Retardation versus wavelength δλ for the same BBPC. (d) Sample of a dependence ωλ for an AWP designed for the same spectral range.

2. Experimental Results orientation of the analyzer. It can be seen that the Experimental approval of the method has been variation of the transmittance values for each ori- carried out with use of Fourier spectrometer Vertex entation is within the signal-to-noise ratio (SNR) 70. We have used quarter-wave BBPC for then 60– of the Fourier spectrometer, which was in this case 300 μm wavelength range. The results are shown is about 1∶100. This fact indicates that the transmit- in Fig. 6. tance of the system is not dependent on angular The figure depicts a set of transmission spectra of position of the analyzer, or, in other words, linear po- an AWP situated between two linear . Each larization of light is efficiently converted to circular spectrum in the set corresponds to a specific angular in the specified spectral range 60–300 μm.

Fig. 6. (Color online) Dependence of linearly polarized light transmission for λ∕4 BBPC on wavelength (a), same for different angles of analyzer [rotated linear polarizer, ordinate axis in (b)]. Each sequence in (b) corresponds to its own wavelength [upper right side of (b)] taken from (a). The curves in (a) are very similar. The small misfit of the curves is related to the relatively low SNR of the spectrometer.

B64 APPLIED OPTICS / Vol. 52, No. 4 / 1 February 2013 slightly higher light absorption, AWP generates bet- ter circular polarization. Some residual ellipticity (about 10%) may be ascribed to measurement error due to imperfect polarizer, etc. Molecular lasers have a discrete generation spec- trum with a fairly narrow generation line. AWP stu- dies throughout the entire applicable wavelength range were carried out using a free-electron laser at FOM Institute Rijnhuizen, The Netherlands. [22,23] The experimental setup was similar. Figure 8 shows spectral dependencies of the signal at different anal- yzer angles. For illustrative purposes, the results are normalized to signal level at analyzer angle β 0.It can be seen that relative signal deviation at various analyzer angles (including the effect of analyzer im- perfection) is about 10%. The measurements were Fig. 7. (Color online) Laser radiation intensity versus analyzer carried out in increments of 15°, but for the sake of rotation angle without AWP (circles) and with quarter-wave visualization, the results for intermediate β angles AWP (rhombi). Solid curves represent theoretical analyzer are not shown. They lie within the same bounds. (eight-shape) and AWP (O-shape) characteristics. Triangles repre- All experimental results are in agreement with the sent signal diagram for standard quarter-wave plate with rated mathematical model used as a basis of design calcu- wavelength 148 μm. lation of the achromatic polarization converter. Optical properties of the discussed AWP were also B. Composite Waveplates studied using optically pumped THz molecular la- Like achromatic polarization converters, composite sers: a high-power pulsed NH3 laser operating at waveplates (CWPs) are special assemblies of identi- 90, 148, and 280 μm, as well as continuous-wave μ – cal plates made of the same material. The paper [24] methanol laser operating at 118 m[19 21]. Verti- establishes that to achieve controlled adjustment of a cally polarized laser radiation passing through the retardation plate for a specific wavelength and arbi- quarter-wave AWP was measured as a function of trary retardation, a set of three identical WPs is the rotation angle of a linear analyzer downstream sufficient, with the outermost plates having parallel of the AWP. A quarter-wave plate situated at a spe- birefringence axes. Optical axes of the outermost cific angle between the direction of polarization of la- plates are aligned, whereas the angle between them ser radiation and the optical axis transforms linear and the optical axis of the middle plate must be cal- polarization to circular. Typical measured signal as a culated using a certain method. After that, the function of analyzer angle (rotated clockwise) is assembly is set at a specific lateral angle to the elec- shown in Fig. 7. Circles represent the linear analyzer μ tric field vector of the incident beam. This angle can signal in the absence of AWP (wavelength 148 m). be calculated by the same method as the angle be- The gray curve is an approximation of the analyzer 2ν ν tween optical axes of the plates in the assembly. If signal by theoretical function cos , where is the the assembly allows one to adjust orientation angles analyzer rotation angle. Rhombi are the signal in (i.e., the plates are not in optical contact and not the presence of AWP. It can be seen that AWP exhi- glued), the set may be fairly well adjusted for a series bits almost no deviation from perfect behavior. The of arbitrary wavelengths and operate as a quarter- same figure shows for comparison the signal diagram wave or half-wave plate. Rather than being achro- for a standard monochromatic quarter-wave plate matic or superachromatic, the composite retardation (shown in triangles). The graph shows that, despite plates are controllable WPs.

1. Calculation To evaluate CWP for specific retardation at a specific wavelength, one must first calculate [19] the rotation angle α of the middle plate using the equation 0 1 δ δ0 cosδ cos 2 − cos 2 α 0.5 arccos@ A; (12) δ sinδ sin 2

where δ 2πnod∕λ, and δ0 is the required retarda- Fig. 8. (Color online) Spectral dependence of intensity of tion of the CWP. Reasonable plate thickness is radiation generated by free-electron laser and passing though selected to achieve CWP operation in the required quarter-wave AWP at various analyzer angles. wavelength range. The CWP’s EOA rotation angle

1 February 2013 / Vol. 52, No. 4 / APPLIED OPTICS B65 Fig. 9. (Color online) (a) Wavelength dependence of the middle plate rotation angle (α) and the assembly’s EOA angle (β) calculated for CWP operating in the range of 60–300 μm. (b) Comparison of retardations of the said CWP and monochromatic WP rated for 118 μm.

β can be determined by substituting the calculated α For example, let us consider linear-to-circular value into Eq. (8) using matrix elements (5). Then the transformation of polarization. Similar to the case angles are corrected for interference. Figure 9(a) of the achromatic converter, the wavelength depen- shows sample results of the calculation of these dence of the intensity ratio I of linearly polarized light angles for a λ∕4 CWP designed for operation in the to total intensity is given by Eq. (6). To find the 60–300 μm range, obtained using 2 × 2 Jones minimum of this value by varying the angle pair matrices (i.e., without correction for interference). (α, β) (i.e., respectively, the angular position of the Figure 9(b) compares retardation behavior of the middle plate relative to the outmost ones and overall aforementioned CWP at angles corresponding to unit rotation), one can use the following search algo- λ∕4 retardation at 118 μm with the same for mono- rithm. All parameters of Eq. (6) depend on these an- chromatic WP rated for 118 μm wavelength. It can gles; therefore, a bivariate dependence Iα; β can be be seen that at the indicated wavelength the CWP constructed. Then the angle pair corresponding to the and common WP exhibit similar behavior. minimum I value can be found. Interference correc- Since the reconfigurable WP requires rotation of tions to α and β introduced by substituting the middle plate relative to the outer ones, the design matrices (1) instead of (6) may amount to several of the unit must provide an air gap between them. To angle degrees. evaluate such a system using Jones matrices, besides A prototype CWP is currently undergoing experi- type (1) matrices one needs to introduce two matrices mental validation. We do not report the results given by yet because the validation is not complete. But preli- minary results are in agreement with the theory described above. 2 3 cos 2π h j sin 2π h 00 λ0 λ0 C. Concept of a Quartz Spectrum Analyzer 6 h h 7 ⌢ j sin 2π cos 2π 00 6 λ0 λ0 7 One more interesting application can be thought of Q0 6 7; 4 00cos 2π h j sin 2π h 5 for the systems of birefringent plates with birefrin- λ0 λ0 00j sin 2π h cos 2π h gence axis lying in the plane of the plates. Namely, λ0 λ0 a set of three plates situated between two crossed (13) polarizers can be used as a THz spectrum analyzer. The general spectroscopic problem can be stated as follows [25]. Assume that a beam of light with un- to describe the air gap. Here, h is the air-gap width, known spectral content illuminates a system consist- and λ0 is the operational wavelength of the CWP. It ing of a nonselective (integral) radiation receiver should be noted that the air-gap requirement sub- and a spectroscopic attachment. The attachment stantially increases the overall complexity of the can be described using generic instrument function unit’s manufacture, because all the plates must Kλ; γ, where λ is the wavelength of the incident ra- be as parallel as possible, and the gap width must diation, and γ is a generalized coordinate describing be kept at minimum (several micrometers). In the control of the spectrometer. Some examples of such case of wedge-shaped gap the system may also systems are the Fabry–Perot interferometer with in- be evaluated using the Jones method as long as strument function [26] Kλ; γ ∼ 1 cos2πγ∕λ,the the wedge’s parameters are known, but it is unlikely Michelson interferometer [27], etc. To solve the spec- in a practical situation. In such a case the calculation troscopic problem means to deduce the spectral dis- includes integration over small areas of the plates tribution of the incident radiation from the response corresponding to actual gap width in each point. of the system with known instrument function. This

B66 APPLIED OPTICS / Vol. 52, No. 4 / 1 February 2013 Fig. 10. (Color online) (a) Sample Fredholm kernel (horizontal and vertical axes correspond to λ and γ) and (b) sample response, i.e., Fγ for known Gaussian Iλ. problem is ill-conditioned and assumes solution of of the nondiagonal element of the product of Jones the Fredholm equation of the first kind, matrices, which are, in the simplest case, given by Z Eq. (5). Interference corrections and accounts for λ2 the air gap required for plate rotation can be intro- Kλ; γIλdλ Fγ: (14) duced using a method described in Section 2.B, i.e., λ1 by multiplying matrices (1) and (11). Sample kernel, Incident signal spectrum Iλ must be derived from Iλ and Fγ, functions are shown in Fig. 10. In this the dependence Fγ, which is measured experimen- case the parameter γ is a generalized coordinate de- tally by the instrument. Variation limits λ and γ are scribing the angular position of the middle plate re- known. Fredholm kernel is the instrument function. lative to the outer plates, as well as angular position Such problem can be solved by numerical methods, of the whole system. It can be seen that both the ker- such as regularization, etc. nel and Fγ are very complex, whereas Iλ is quite A set of quartz plates can be used as the “spectro- simple. scopic attachment.” It was shown that to obtain the Presently the authors are working on obtaining best solution of the problem, three plates are suffi- numerical solutions of this problem using known cient, with middle-to-outer plate thickness ratio (model) functions Fγ. Quantification of the equation about 0.8–1. Such conditions produce maximum rank leads to densely populated, ill-conditioned square of the matrix corresponding to the Fredholm kernel. matrices unstable in regard to right-side errors. The analytical expression for the Fredholm kernel Thus, the solutions obtained using singular value of such a system is very cumbersome. The kernel is decomposition and iterative methods (such as Seidel, the transmittance of a system of quartz plates be- Jacobi, Craig, and Lebedev) are against the physical tween two crossed polarizers as a function of rotation meaning of the problem despite producing near- angles of the plates. It is equal to the squared module zero residuals Rfactor ‖Ax − b‖∕‖b‖. One of the

Fig. 11. (Color online) (a) Original and reconstructed Iλ represented by several overlapping peaks. Without right-side disturbance the function is reconstructed with adequate precision. (b) Solution obtained using Tikhonov regularization method at 1% disturbance of the right-side vector, and undisturbed solution.

1 February 2013 / Vol. 52, No. 4 / APPLIED OPTICS B67 possible methods to solve this problem is the global δ × σ2 A α2 max ; (19) minimization approach [28] applied to the function ‖b~‖ δ Iλ, which is represented by the superposition 2 of symmetrical and asymmetrical peaks [see assuming that the matrix A is precisely defined, and Fig. 11(a)]. Such an approach is physically meaning- all disturbances are the disturbances of the right- ~ ful in the context of spectrum analysis. Physically side vector b, where ‖b − b‖ ≤ δ, and σmaxA is the valid solutions were obtained by means of parame- maximum singular value of the matrix A. The terization of asymmetric peaks using Edgeworth Tikhonov functional was minimized using extended series [29]: regularized systems method [31]. The authors were able to deduce physically valid X∞ β solutions, but their accuracy is still inadequate for AxA x −1k k Akx; G k! G (15) practical purposes. The work on this problem con- k3 tinues. Figure 11(b) shows the regularized solution A ;Ak of the problem (fine curve) with 1% disturbance of where G G are normal the density function and the right side. Alternative methods were considered its derivatives, and βk are quasi-momentums ex- κ to select the regularization parameter, such as the pressed through cumulants n in a nonlinear way L-curve method, the cross-validation method, and [29]. The Edgeworth series parameters were the coef- n∕2 3∕2 its generalization for stationary iterative methods. ficients γn κn∕κ2 , such as γ3 κ3∕κ2 (peak γ κ ∕κ2 However, these attempts were unsuccessful. Devel- asymmetry coefficient), and 4 4 2 (excess coeffi- opment of a quartz spectrum analyzer continues. cient that describes deviation of the distribution from “ ” “ ” normal to pointer or flatter vertex). Then we 3. Conclusions considered a finite Edgeworth series with cumulant coefficients γ1; γ2; …; γ10. Optimal values of the vari- We have demonstrated by experiments that sets ates γ1; γ2; …; γ10 were obtained using the least- of plane-parallel birefringent plates with in-plane squares method: birefringence axis are instrumental for THz polarization optics. We present results of calculations X and experimental validation of an achromatic polar- 2 F Yexperim − Ytheor → min : (16) ization converter and a CWP for THz spectral range, i i taking into account the interference effect and air gaps between the plates. A mathematical model is presented demonstrating the possibility of develop- The functional F was shown to be multiextremal, ment of a spectrum analyzer design based on a set which led to crucial importance of the global func- of WPs. tional minimization method [28]. However, the issue of solution instability in regard This work was supported by the Russian Founda- to disturbances of Fγ remains unsolved. This issue tion for Assistance to Small Innovative Enterprises by all means can emerge in actual spectrum analysis (FASIE), Contract No. 9776r. The support by the experiment. The matrix of the problem is rank- DFG (SPP. 1459) is gratefully acknowledged. We also deficient. It leads to substantial difficulties even gratefully acknowledge the support by the Stichting when solving the problem without noise distortion voor Fundamenteel Onderzoek der Materie (FOM) in of the right-side vector [see bold curve in Fig. 11(b)]. providing the required beam time on FELIX. In the case when the source data are distorted by noise, one possible approach is the Tikhonov regular- References ization method, which considers a problem of mini- 1. J. Federici, B. Schulkin, F. Huang, D. Gary, R. Barat, F. mization of the following parametrical functional: Oliveira, and D. Zimdars, “THz imaging and sensing for secur- ity applications—explosives, weapons and drugs,” Semicond. 20 – ~ ~ 2 2 2 Sci. 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