RF STATIONARY WAVES INTEGRATED FOURIER TRANSFORM SPECTROMETER S. Hemour1, F. Podevin1, D. Rauly1 and P. Xavier1 1 IMEP, UMR INPG-UJF-CNRS 5130 BP 257 38 016 Grenoble Cedex, France Corresponding author : [email protected] Fax number : 0 +33 4 56 52 95 01 ABSTRACT: We describe here a new type of analog correlators based on stationnary waves recovery and spatial Fourier transformation. The theoretical basis of the Stationary Waves Integrated Fourier Transfrom Spectrometer (SWIFTS) is first introduced. Based on robust physical principles validated by circuits type simulations, the spectrometer could be an interesting competitor with actual RF analog correlators.

observation of interferences fringes. At that time one major inconvenient not to develop this method was due to the poor computer calculation abilities for the Fourier transformation. Of course, this is not a problem anymore. It should be noticed that this solution is also identical to the process developped and described by Lippmann at the end of the ninetenth century for his colour interferential photography [6]. towards detector

towards detector

Microstrip line Cell 1

towards detector

towards detector

Cell N

Cell 2 L Incident wave Reflected wave

Mirror: Open or Short

Instantaneous power x

Key words: RF spectrometers, analog correlators, periodic structures.

λg/2

I. INTRODUCTION In the domain of astronomical observations, it is sometimes necessary to use wide-band spectrometers with only a moderate frequency resolution. This is the case, for instance, with the extragalatic submillimeter lines observation or for the observation of molecular and atomic lines from high-z objects (z=2-4 era of our galaxy formation). This is the case also for the cosmic microwave background interferometry or lastly the remote-sensing of the Earth’s atmosphere [1]. By considering more precisely the spectrometers family, one could separate them into two distinct classes, narrow-band with a high spectral resolution, wideband with poorer resolution. Heterodyne instrumentation for mm-waves applications has demonstrated for a long time that a high resolution could be reached. Spectra is directly measured in the frequency domain. As an example, the HIFI instrument provides a high frequency resolution varying between 140 and 280 kHz for the 4801250 GHz frequency range) [2]. For such instrumentation and coverage, balanced mixers with Schottky or cooled SIS diodes are often used since they provide good noise immunity [3]. To enhance the broad-band aspect of those instruments analog auto- and cross-correlation lag spectrometers have been developped such as the Wide-band Analog Spectrometer (WASP) from Harris and Zmuidzinas [4]. The original idea was proposed by Blum in 1960 for RF spectroscopy [5]. It is based on time domain spectrum recovery thanks to correlation function measurement. This RF correlation solution itself was inspired by optical considerations from the

Figure 1: Schematic view of the SWIFTS. In the WASP concept, for time discretization into N lags τ, the two correlated and shifted signals are carried to the N multipliers thanks to two distinct parallel lines. Here, our proposal consists in having the two signals meeting directly on the same microstrip line, as described on Figure 1. In fact, by ending a microstrip line with a short or an open, acting as a mirror, the incident and the reflected waves interfer between each other. As a consequence, whatever the initial frequency is, a spatial stationnary wave is generated along the ribbon. When carrefully positionning quadratic detectors all along it, the stationnary wave is sampled. The spectrum is obtained, then, thanks to a discrete “spatial” Fourier transform [7]. Clearly, the behavior is the same as for the analog correlators. Never-the-less, the SWIFTS offers a simpler configuration together with a gain in terms of dimensions. Any spectrometer in mm or optical wavelength may take benefits from this process. In optics, when compared with conventional optical interferometre, the SWIFTS appears to be strongly smaller. As an example, for a resolution identical to the WaPHIR instrument (1000), the volume could be reduced by a factor of 40. In this paper, we try to demonstrate the feasability of a SWIFTS in the RF domain. We focuse on one of the water absorption lines between 19 and 25 GHz. In a first part, we develop the theoretical basis thanks to which the topology and characteristics of the SWIFTS can be designed. Such a device may be seen as a

periodic structure of N unit cells. Thus, a second part is dedicated to the optimisation of one unit cell between 19 and 25 GHz thanks to circuit type analysis. Finally, we check the SWIFTS behaviour itself before concluding and giving some perspectives to our work. II. BASIC PRINCIPLES A. Stationnary power equation along the SWIFTS

with a factor

K e and

coupled with

K e whereas Hr is coupled with - K h .

This is due to the conventional current distribution along the x and y axis, towards the x and y positive values respectively. Considering the schematic view of Figure 3, a relation between α and K exists that is given by equation 2. e -2.α.L = (1− K )

SWIFTS theory relies on the assumption that sampling the stationnary wave does not affect the signal on the microstrip line. Obviously this is not the case and a compromise has to be chosen between sampling a huge quantity of signal which deteriorates stationnarity, or sampling a very small amount of it which leads to very low sensitivity. Let us consider a single cell on Figure 2. The sampling towards the detector has been represented as a microstrip line perpendicular to the main line. The power coupling factor between main and secondary lines is K. We suppose that both microstrip lines have no dielectric nor conductive losses. a unit cell

L Hi

− K h .H r

Ei

K h .H i

Hr

Er

mirror : open

100% P0

Both Ei and Hi are coupled to the secondary line

Cell 1

Cell N

K.P0

K.P0

Figure 3: Modeling of the main line as a lossy line.

The coupled mean power during a time period is determined by the Poynting vector as in equation 3, where E and H are respectively the total electric and magnetic fields on the secondary line. r 1r r P = E ∧ H* 2

Eq. 3

which is identical to the following equation: r 1 P= 2

Figure 2: Field coupling towards the detector for a unit cell.

Γ = α + jβ

(1-K)NP0

(1-K)P0

e-2.α.L.P0

K e .E i

Ei and Hi are respectively the incident electric and magnetic fields on the main line while Er and Hr are the reflective electric and magnetic fields. They can be expressed as in equation 1, where Γ is the complex wave constant, with an attenuation constant α and a propagation constant β. α is due to the coupling effect towards the detector. Because of the sampling, the wave along the main line is attenuated periodically which is modelled as a linear attenuation. r r .E i = E 0 .e − Γx r r .E r = E 0 .e + Γx e − 2.Γ . L r r .H i = H 0 .e −Γx Eq. 1 r r .H r = −H 0 .e + Γx e − 2.Γ . L

L

y

K e .E r towards detector

Eq. 2

N

x

x y

K h respectively. Er is also

( K .Er + e

i

) (K

r K e .E r ∧

h

)

r r .H i − K h .H r *

Eq. 4

We suppose a monochromatic waves to simplify the demonstration. Power propagation is oriented towards the detector. Consequently the expression becomes :

(

r 1 P = .K . E 0 .e − Γx + E 0 .e + Γx e − 2.Γ .L 2 r . H 0 .e − Γx + H 0 .e + Γx e −2.Γ .L * . e y

(

)

) Eq. 5

So, and after further simplifications in the formula, equation 6 is obtained. P is an image of the stationnary power along the main line. It can be divided into two parts : an useful signal which contains the spectral information thanks to the constant β a continuous part varying as a hyperbolic cosine which should be minimised in order to reach the highest instrument efficiency.

  P = K .E 0 .H 0 .e − 2.L .α . cosh[2.α( L − x )] + cos[2.β( L − x )] 3  1442443 144244 usefull signal  no spectral inf ormation  Eq. 6

B. Maximisation of the SWIFTS efficiency

Let us consider the particular detector of cell k, located at the position xk. Signal processing consists first in substracting the non usefull part from the parenthesis : by adding the following expression (− cosh[2.α(L − x )] + cosh[2.α( L − L)]) that is to say (− cosh[2.α( L − x )] + 1) , only the useful part of the signal stays. It varies as a cosine centered around 1. 1  N P' ( x k ) = 2.K . E 0 .H 0 .(1 − K ) .(1 + cos[2.β( L − x )]) 2 14243 P0

Eq. 7

The complete power detected by the SWIFTS is obtained by summing the various P’(xk) for k varying from 1 to N. If the detectors are regularly spaced with xk=kL/N, and N chosen such as covering an integer number of spatial periods, at a given frequency f=f0, for which β=β0, it is possible to get N ∑ cos[2.β0 (L − x k )] = 0 . At frequency f0, the SWIFTS k =1

internal efficiency ηi is then given by equation 8:

ηi = 2.K.(1 − K ) .N N

Eq. 8

appears to be an interesting compromise between simulation and experimentation facilities with a rather good efficiency of the instrument. For 128 lags, it seems that a –25 dB tap delivers the propagating signal to the multiplier. Our demonstration shows that the optimum value is 0.78 % ie –21 dB for 128 detectors. A value of –25 dB decreases the efficiency downto about 53 %. It appears that the optimisation of only one cell is sufficient to ensure a performant behaviour of the SWIFTS. This is developped in the next part. III. UNIT CELL OPTIMISATION BY CIRCUIT ANALYSIS The optimisations were performed considering a microstrip device using a RT-duroid 5880 substrate (εr = 2.2) with a thickness of 787 µm. 50 Ω lines are 2.56 mm wide at 22 GHz. Detectors are zero bias GaAs Schotty diodes from Avago Technologies (HSCH-9161), [8]. Such diodes are designed for quadratic detection upto 110 GHz and are usually used for astronomical applications, [9], [10]. On Figure 5, a unit cell has been represented. Optimisation was performed thanks to Advanced Design system (ADS) by Agilent Technologies using harmonic balanced computations. According to previous considerations, the goals consist in matching the cell in the whole band and try to maintain K constant equal to 4.76 % in this band. Lcell / 2

Lcell / 2

Figure 4 shows the internal efficiency as a function of K for various N values. For a given N, the optimum efficiency is reached for K=1/(1+N). The maximal efficiency of such a spectrometer is 73,6% for N infinite. Internal efficiency ηi (%)

80 70

N= 1000 500 200 128 50

gap capacitance

20 10

60 50 HSCH9161

40 30 20

matching network

filtering pad

10 0 0,01

highimpedance line towards detector

0,78

4,76

0,1 1 10 Coupled power factor K (%)

100

Figure 4: SWIFTS internal efficiency versus the percentage of

Figure 5: One cell design

coupled power towards each detector.

For 20 detectors, the efficiency reaches 71,8 % with a coupling of 4,76 %, ie –13,2 dB. This value is not so far from the maximum 73,6 % and 20 detectors

On Figure 5, coupling is performed thanks to a microstrip gap of 40 µm in the middle of a 200 µm wide high impedance line. The large pad filters the DC component after detection in order to read

VDC_out, image of the coupled RF power PRF_in brought to the diode. The stubs are designed to match the diode impedance. Due to the short space between two unit cells, as shown on figure 7, they need to be curved. 0

0,00

-10

The previous studied cell has been cascaded 20 times (fig. 7) to form the SWIFTS. Coupling between two adjacent cells is avoided thanks to the comb topology. Distance between both cells is exactly a quarter wavelength at 25 GHz. The open length has been electrically adjusted so that diodes could detect the stationary wave power local extrema at 25 GHz.

-0,20

-20

-0,40

-30

-0,60

S11

S21 (dB)

S11 (dB)

S21

open length -40

-0,80

-50

-1,00 19

20

21

22

23

24

25

Frequency (GHz)

8

-0,20

S21

6 4.76 4

-0,40 -0,60

2

Figure 7: The complete SWIFTS.

-0,80 K

0

-1,00 19

20

21

22

23

24

25

Frequency (GHz)

(b) Figure 6: Results from 19 GHz to 25 GHz, (a) S11 or cell matching with S21 or transmission factor (1-K), (b) Coupling factor K from main line to secondary line towards detector with S21.

Figure 6 shows the results obtained after circuit analysis optimisation. One can see that matching is better than –20 dB on the whole band. On Figure 6.b, it can be seen that the K value of 4.76 % is reached for 4 discrete frequencies. The SWIFTS behaviour as a spectrometer of internal efficiency of 71.6 % will be checked at the precise frequency of 25 GHz. IV. VALIDATION OF THE SWIFTS BEHAVIOR AT 25 GHZ

Figure 8 represents the DC voltage probed on each filtering pad as a function of the distance from the mirror of the considered detecting point. The bold black curve clearly shows the superposition of a cosine with a hyperbolic cosine. The bold grey curve is the signal obtained after correction by substracting the hyperbolic part. This part varies such as cosh[2.α( L − x )] . (L-x) is the distance from the mirror. α is calculated from equation 2 with K = 4.76 % and is equal to 11 Np/m. After signal processing the initial spectrum (a single line at 25 GHz in that precise case) may be recovered from the grey curve. 50

Available power = 0.1mW

40 VDC_out (mV)

0,00

S21 (dB)

K (%)

(a) 10

30

Uncorrected Corrected

20 10 0 0

Mirror

5

10

15

20

25

30

35

40

45

50

Distance from the mirror (mm) -x

Figure 8: DC voltage at the N diodes output for an available power of 0.1 mW at 25 GHz, after optimisation of the open length between the last diode and the mirror. Dots represent the sampled data. Curves are smoothly extrapolated. V. CONCLUSION AND PROSPECTS We have demonstrated the feasibility of a new type of analog correlators which appears to be more convenient than actual WASP’s. This preliminary study has to be confirmed by experimental

measurements. In a second time, a SWIFTS with 100 diodes will be implemented in a fully integrated process. We are expecting for such a device a coupling factor K constant on the whole band in order to avoid strong discrepancy before signal processing. First simulations have shown that a factor of 1 % is easier to obtain on a wide band than higher values. Finally, sampling has to be improved. Non-regular sampling could provide better efficiency of the instrument. REFERENCES [1] A.I. Harris, “Heterodyne spectrometers with very wide bandwidths”, Proceedings of the SPIE, Millimeter and Submillimeter Detectors for Astronomy, T. G. Phillips, J. Zmuidzinas, Editors, Vol. 4855, Feb. 2003, pp. 279-289. [2] G.L. Pilbratt, ‘Herschel mission: status and observing opportunities’, Proceedings of the SPIE-The International Society for Optical Engineering, Vol. 5487(1): 2004, pp 401412. [3] J. W. Kooi, A. Kovacs, B. Bumble, G. Chattopadhyay, M. L. Edgar, S. Kaye, R. Leduc, J. Zmuidzinas, and T. G. Phillips, “Heterodyne instrumentation upgrade at the Caltech Submillimeter Observatory”, Proceedings of the SPIE,

Millimeter and Submillimeter Detectors for Astronomy, T. G. Phillips, J. Zmuidzinas, Editors, Vol. 4855, Feb. 2003, pp. 265-278. [4] A.I. Harris and J. Zmuidzinas, “A wide-band lag correlator for heterodyne spectroscopy of broad astronomical and atmospheric spectral lines”, Rev. Sci. Inst, Vol. 72, 2001, pp 1531-1538. [5] E.-J. Blum, “Les mesures spectrales en radioastronomie”, Compte-Rendu de l’Académie des Sciences, séance du 16 mai 1960, pp 3279-3281. [6] G. Lippmann, “La photographie des couleurs”, Revue générale des sciences pures et appliquées, 1891, pp 161-172. [7] E. Le Coarer, P. Benech, Patent 04/52992, 15 Dec 2004. [8] Avago Technologies, “Agilent HSCH-9161 GaAs detector diode sensitivity measurements”, Product Note #12, March 1999. [9] N. Roddis, D. Kettle, F. Winder, B. Aja, E. Artal, M. L. de la Fuente, J. P. Pascual, A. Mediavilla, L. Pradell and P. de Paco, “Differential Radiometer at 30 GHz for the Planck Emission”, 3rd ESA Workshop on Millimetre Wave Technology and Applications, 2003, pp 81-86. [10] E. Artal, B. Aja, M. L. de la Fuente, N. Roddis, D. Kettle, F. Winder, L. Pradell and P. de Paco, “Radiometers at 30 and 44 GHz for the Plank Emission”, Microwave Technology and Techniques Workshop. European Space Agency – CNES, Oct. 2002, pp 41-48.