Interference imagery: Variable lateral shear

TSPDR structures. As shown in Table 1, the results are consistent with experimental values. 4. CONCLUSION

The proposed applicator using TSPDL offers significant improvement in the capability of controlling microwave energy coupling to dielectric sheets over the conventional TSPDR. For obtaining uniform power dissipation across the dielectric sheet, TSPDL provides an extra degree of freedom over the

:TSPDL

\

- - - -:TSPDR

1.3 -- \ 1.1

0.9

0.7 0.5 0.3

4 0

I 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

z/L

1

Figure 4 Microwave power dissipation across the dielectric sheet in TSPDL and TSPDR with tapering the gap between two L septa or ridges. a = 86 mm, b/a = 0.5, s/a = 0.15, d / a = 0.25, t/a = 0.01, w/a = 0.02, L = 1000 mm, E” = 0.63, f = 2.45 GHz

conventional TSPDR in the form of tapering the width of the

L septa. The material processing capability of the microwave applicator using TSPDL technique is twice as effective as that offered by TSPDR. Hence, the applicator using TSPDL can process dielectric sheets that have a higher loss factor and/or larger physical dimensions. The Ritz-Galerkin method combined with perturbation theory is simple and straightforward and can predict results with engineering accuracy. ACKNOWLEDGMENT This work was partially supported by James Hardie & Coy. Pty Ltd.

REFERENCES 1. A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus, London, 1983, pp. 34-35 and 122-124. 2. M. El-sayed El-Deek and M. K. Hashem Adel, “Ridged Waveguide Applicators for Uniform Microwave Heating of Sheet Materials,” Journal of Microwave Power, Vol. 19, No. 2, 1984, pp. 111-117. 3. J. P. Monotgomery, “On the Complete Eigenvalue Solution of Ridged Waveguide,” IEEE Trans. Microwave Theory Tech., Vol. M’IT-19, No. 6, June 1971, pp. 547-555. 4. D. Dasgupta and P. K. Saha, “Rectangular Waveguide with Two Double Ridges,” IEEE Trans. Microwave Theory Tech., Vol. MlT31, NO.11, NOV.1983, pp. 938-941. 5. H. Z. Zhang, G. E. Beard, A. S. Mohan, and W. R. Belcher, “Two Symmetrically Placed Double L-septa Waveguide,” Electron. Lett., 28 Oct. 1993, pp. 1956-1957. 6. J. L. Altman, Microwave Circuit, D. Van Norstrand, New York, 1964, pp. 172-176. Received 5-9-94 Microwave and Optical Technology Letters, 7/15, 717-720 0 1994 John Wiley & Sons, Inc. CCC 0895-2477/94

I

0.8

.dF/a=0.075 0.7

0.6

Eduardo TepichCn and Gustavo Ramirez National Institute of Astrophysics, Optics and Electronics Apdo. Postal 216, Puebla 72000, Puebla, Mexico

0.5

KEY TERMS Gratings, Diffractiue Optics, Computer-GeneratedHolography, Interferometers, Signal Processing

0.4 0.3

INTERFERENCE IMAGERY: VARIABLE LATERAL SHEAR

4 0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Figure 5 Microwave power dissipation across the dielectric sheet in TSPDL with tapering the width of the L septa. a = 86 mm, b / a = 0.5, s/a = 0.15, I/a = 0.1, t / a = 0.01, w/a = 0.02, L = 1000 mm, E “ = 0.63, f = 2.45 GHz

ABSTRACT We present an optical setup that produces simultaneously spatially filtered images and lateral-shearinterferometry. Our device uses two gratings. One grating is a computer-generatedhologram that acts as a spatial filter. 0 1994 John Wiley & Sons, Inc. 1. INTRODUCTION

TABLE 1. Comparison of the Values Obtained from Calculation and Experiment. a = 86 mm, b / a = 0.5, t = 3 mm, I = 4 mm, d = 2 0 mm, w = 2.25 mm, E = 2.3 -j0.27, f = 2.45 GHz

Methods Ritz-Galerkin & perturbation Measurement

720

Attenuation in TSPDL (dB/m)

Attenuation in TSPDR (dB/m)

1.07 1.1

1.42 1.5

Robotics and machine vision substitute for human vision in imaging tasks that are dangerous, dull, or complex. An example of the latter type of applications is the visualization of transparent structures. Phase structures appear as input objects in different fields of optics such as in fluids, optical testing, or in thermal expansions. Even when this type of phase structure remains invisible to the unaided human eye, they are important. Consequently, it is necessary to employ either spatial filtering methods or interferometric techniques to render them visible.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 7, No. 15, October 20 1994

On the one hand, variable lateral shear interferometers [l, 21 provide a flexible tool for detecting phase structures. On the other hand, spatially filtered imagery such as that obtained using phase contrast or Schlieren techniques [3] offer the possibility of visualizing certain important features of the phase structure directly on the image irradiance. The aim of this article is to present and experimentally verify an optical technique for simultaneously producing spatially filtered images and variable lateral-shear interferometry. Hence, our proposed system can be used as a smart sensor to visualize phase structures as variable shear interferograms of preprocessed images. We are interested in an all-diffractive-elements system. Therefore we need a grating capable of modifying the amplitude and phase information of the Fourier spectrum of the object under test, in order to produce the desired spatially filtered images. This complex operation can be achieved with a computer-generated hologram that acts as a complex-amplitude spatial filter. Our technique uses two gratings. One of them acts as the spatial filter, or computer-generated hologram, for creating spatially filtered images that are later superimposed with variable lateral shear to create interferograms. In Section 2, we discuss the basic theory of our optical system. In Section 3, we present some experimental verifications. 2. OPTICAL SYSTEM

In Figure 1 we show schematically our optical setup. We denote as GI a computer-generated hologram in the form of a phase detour grating 141; G, denotes a low-frequency Ronchi grating. The amplitude transmittance of the phase detour grating G, can be represented as

i'

OBJECT

Figure 1 Schematic diagram of the proposed optical setup in which G , is a computer-generated hologram and G , a low-frequency Ronchi grating

where h is the wavelength of the incident radiation, f the focal distance, and X the maximum x width of the object. For this particular case, the laterally displaced preprocessed images described in Eq. (2) do not overlap. We select, then, two of these preprocessed images, say of order + M , which is denoted as Imfil+M,and of order - M , denoted as Im,l-M, as an input for the second 4f optical processor; as is shown again in Figure 1. Therefore, the amplitude distribution at the back focal plane of L, is now given by

W

GI(v1, pl)= fil(vl, pI)

6(vI

m=

- mu,),

(1)

-w

where, fil(v,, p l ) is the filter function and v,, the sampling interval. In Eq. (1) we denote as V, and pI the spatial frequency coordinates. For a given object, O(x,, yl>, lxll IX/2 and lyll IY/2, placed in the front focal lens of L, in Figure 1, it is straightforward to show that the image's complex amplitude distribution, U(x,, y,), at the back focal plane of L, is U(x,,y,)

=

Imfil(xz,yz)* m=

-w

where

As mentioned before, for this second 4f optical processor we used as a spatial filter a Ronchi-type diffraction grating G,; the amplitude transmittance of which can be represented by its Fourier series as W

C,, exp(i2ax3/L),

GZ(x3,y 3 ) = n=

(6)

-w

where L-' is the fundamental spatial frequency of G,. In order to add two extra degrees of freedom to our optical setup, G, is placed at a distance e before the back focal plane of L, and is in-plane rotated by an angle y , as is shown in Figure 2. It can be shown that, in this case, the amplitude distribution, W ( x ,y ) at the back focal plane of L, is W ( x ,y )

In Eq. (2) the asterisk means a convolution operation, and in Eq. (3), 2T1[ ] represents the 2D inverse Fourier transformation of the product of tbe Fourier spectrum of the object O(x,,y,), denoted as O(v,, p,), and the filter function fil(vI, pI),with uI and pl the spatial frequency coordinates. From Eq. (2), we notice that at the output of the first 4f optical processor of our setup, we obtain lateral replicas of the filtered image Im,(x,, y,), of the object Oh,, y I ) . In our experiment, v o is chosen according to v0X 2 hf,

W

C , exp( -i3ah en2/L2)

=

n=

-w

x exp(i2ane(x cos y

+ y sin y)/fL)

nhfcosy

hf

- M-,y VO

-

nhfsiny L

(4)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 7, No. 15, October 20 1994

(7)

721

small value of y, we can get the first derivative of the preprocessed image [2]. On the other hand, the tilt was introduced by placing G, a certain distance e before the back focal plane of L , [6].From Eq. ( 8 ) it is easy to show that the reference fringes produced by this tilt are proportional to

Figure 2 Second 4f optical processor of the optical setup in Figure 1. The grating G , is placed at a distance e before the back focal plane of L, (tilt), and is in-plane rotated by an angle y (shear)

It can be noticed from Eq. (9) that by displacing the grating G, along the optical axis (change the value of el, the amount of reference fringes together with their orientation can be easily modified. This feature allows us to set either the so-called a-type fringes or p-type fringes 17, 81. Finally, it should be noted that for e = 0 and y = 0, the fringe spacing will become infinite. We show next some experimental verifications. 3. EXPERIMENTAL RESULTS

We verify our proposed technique as follows. The first grating, GI, was a phase detour grating that implements the first derivative of the object along the x axis; see Reference [4]. As an input object we employ a bleached photograph of a set of five letters, as is shown in Figure 3(a). The image after the first optical processor is displayed in Figure 3(b), where

=

C, exp(i27re(x cosy

+ y sin y)/fL)

XIfil+l

In Eq. (8) we dropped the first exponential term of Eq. (7) because it will not affect the resultant irradiance distribution of the fringe pattern. As is common in this type of interferometer, the fringe pattern is a combination of lateral shear and tilt [6, 71. It is clear from Eq. (8) that the amount of lateral shear, due to the in-plane rotation of G,, is given by 2 h f l - l sin y. This amount can be modified by changing the value of y; therefore for a

722

Figure 3 Input object, and its first derivative along the n axis

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 7, No. 15, October 20 1994

one can observe that the first diffraction order exhibits a lateral edge enhancement, produced by the first derivative operation. In Figure 4(a) we show the interferograms obtained at the output of the second 4f optical processor, for the combination of diffraction orders (- 1,O) and (1, - 2) at the left-hand side; (- 1 , l ) and (1, - 1) at the center; (1,O); and (- 1, - 2), at the right-hand side. As explained before, the maximum contrast appears for the (- 1 , l ) and (1, - 1) pattern. For this figure we set y = 0 (zero shear) and e # 0 (nonzero tilt). Therefore we obtain vertical reference fringes. In Figure 4(b) we show the interferogram obtained for the same amount of tilt as in Figure 4(a), but now the shearing distance is different from zero. 4. CONCLUSIONS

The above experiment shows that double grating interferometry can be usefully extended to implement interference imagery with variable shear; by substituting standard diffraction gratings with computer-generated spatial filters, in the form of phase-detour gratings. For small values of the shear, we can obtain the first derivative of the preprocessed images, adding more flexibility to our technique.

ACKNOWLEDGMENTS We are indebted to Jorge Ojeda-Castaiieda for suggesting this project, as well as for useful comments, and to Alejandro Landa for his assistance. REFERENCES 1. W. H. Steel, Interferometry Cambridge University Press, Cambridge, 1983, p. 169. 2. A. W. Lohmann and 0. Bryngdahl, “A Lateral Wave Front Shearing Interferometer with Variable Shear,” Appl. Opt., Vol. 5, 1967, p. 1943. 3. J. Ojeda-Castafieda, in D. Malacara (Ed)., Optical Shop Testing, John Wiley & Sons, New York, 1992, p. 265. 4. A. W. Lohmann and D. P. Paris, “Computer Generated Spatial Filters for Coherent Optical Data Processing,” Appl. Opt., Vol. 7, 1968, p. 651. 5. K. Patorski, “Grating Shearing Interferometer with Variable Shear and Fringe Orientation,” Appl. Opt., Vol. 25,1986, p. 4192. 6. J. D. Briers, “Prism Shearing Interferometer,” Opt. Technol.,Vol. 1, 1969, p. 196. 7. J. D. Briers, “Ronchi test formulae,” Opt. Laser Technol.,Vol. 11, 1979, p. 189. 8. M. P. R i m e r and J. C. Wyant, “Evaluation of Large Aberrations Using a Lateral-Shear Interferometer Having Variable Shear,” Appl. Opt., Vol. 14, 1975, p. 142.

Received 2-4-94: revised 5-18-94 Microwave and Optical Technology Letters, 7/15, 720-723 0 1994 John Wiley & Sons, Inc.

CCC 0895-2477/94

MICROWAVE CORRELATION TEMPERATURE OF AN ISOTHERMAL LOSSY MATERIAL Philippe Waro* and Yves Leroy lnstitut d’Electronique et de Microelectronique du Nord U.M.R. C.N.R.S. 9929 Departement Hyperfrequences et Semi-Conducteur Universit6 des Sciences et Technologie de Lille Avenue Poincarre BP 69 59655 Villeneuve d’Ascq Cedex, France

KEY TERMS Microwaves, radiometry, correlation temperature, lossy material ABSTRACT A microwave cornlator is connected to the inputs of an isothermal passive two port made of two antennas placed on both sides of an isothermal lossy material. We verifi in which way the output signal of the correlator depends on the scattering matrix of the two port, on the temperature and reference temperature of the radiometer, and on the delay time of the correlator. 0 I994 John Wiley & Sons, Inc. 1. INTRODUCTION

Radiofrequency and microwave correlation radiometers are generally used in radioastronomy and remote sensing 11, 21. Similar correlators work also as short-range sensors, such as in the Coherent detection of the thermal noise transmitted by two antennas coupled to a homogeneous lossy material [3-91 or by other types of passive symmetrical lossy two ports [9-121. Figure 4 Image interferograms of Figure 3 for nonzero tilt and (a) zero shear, (b) nonzero shear

*Present address: CSEE TRANSPORT, Z.I. de la vigne aux loups, rue Denis Papin, 91380 Chilly Mazarin, France.

MICROWAVE AND OPTICAL TECHNOLOGY LETERS / Vol. 7, No. 15, October 20 1994

723