In general, there are several methods of measuringarea [1–6]. They can roughly be categorized into twotypes, contact mode and noncontact mode. In contactmode, a lot of research has been done. The maindisadvantage is that the surface of the object maybecome corrosive or damaged after measurement.Lately, measurement methods based on laser reflec-tion and ultrasonic reflection [1–6] were applied innoncontact mode. The laser techniques have the ad-vantage of high speed. However, the object reflectiv-ity always plays an important role. If the objectreflectivity is bad, the system will work poorly oreven fail entirely. The ultrasonic technique has theadditional advantage of a wide angle range, but withthe same problem caused by object reflectivity.Another disadvantage is that these techniques al-ways require huge storage of image data. Thus, itis difficult to make a real-time area measuringsystem.
In this paper, a novel high-speed measuring setupusing circular Dammann grating (CDG) via a CCDcamera is developed. This measurement techniquecalculates the diameter of the spectrum directlywhile adjusting the location of CDG. The spec-trum-based measuring system can therefore obtainthe area between a CCD camera and an objectquickly. The experimental results also show that thisis a robust and real time system for measuringprojected area. Even in some area measurements, re-searchers need some optical devices that can supporta long distance range. Hence, CDG is one of the bet-ter candidates as it can generate the spectrum atlong distance with acceptable spectrum quality.
2. Background Study
A Dammann grating is a binary phase grating for al-lowing a high output of light energy. These gratingsare binary, and we can therefore employ planar fab-rication techniques for cheaper mass production pro-cesses. The diffracted spots could be one dimensionalor two dimensional depending on the applications[7]. The special feature of a high uniformity beamsplitter is to be found in its periodic nature togetherwith binary phase. This was proposed by Dammann
in the early 1970s [8,9]. The only free parameters arethe locations of transition points. At each of thesetransition points, the phase changes from 0 to πand vice versa, as shown in Fig. 1. Such a gratinghas many advantages, i.e., low cost, high efficiency,and most importantly, supporting N ×M array illu-mination [10–12]. Later, in 2003 Zhou et al. [13] pro-posed the concept of CDG based on the modulation ofthe Bessel function using a binary phase annulusmask. The phase and radius of each annulus canbe modified so that the intensity at the far fieldcan be manipulated. However the CDG does not havethe periodic nature most of the gratings require, andtherefore, it is only a diffractive optical element andequal separation cannot be achieved. Recently, Zhao[14–16] and colleagues proposed certain new designmethods for the periodic CDG and suggested someoptical systems employing CDG, e.g., measuring an-gular rotation of mirrors, fast focal length measure-ment, and distance measurement [17–19].The amplitude of the CDG is shown in Fig. 1,
where frig are the set of normalized phase transitionpoints along the grating, while r0 ¼ 0, rN ¼ 1 are theboundary values. Using the Hankel transform [20],the equation for the amplitude can be expressed by
AY�
x2ri
�→
riAq
J1ð2πqriÞ: ð1Þ
The grating has a binary structure, and hence themagnitude of each diffraction order, q, is given by
MðqÞ ¼ 1q
����2XN−1
k¼1
ð−1ÞkrkJ1ð2πqrkÞ þ rNJ1ð2πqrNÞ����:ð2Þ
The final intensity equations are given as follows:
nthorder:Iq ¼ jMðqÞj2 ¼ 1
q2
����2XN−1
k¼1
ð−1ÞkrkJ1
× ð2πqrkÞ þ rNJ1ð2πqrNÞ����2; ð3Þ
0thorder:I0 ¼�2XNk¼1
ð−1Þkrk þ 1
�2: ð4Þ
We follow the even-numbered design rule, by whichthe grating periodicity will be set symmetric with re-spect to the origin [11],
rk ¼ rk−N=2 þ 0:5; ð5Þ
as the central spot effect will theoretically be can-celed. The overall efficiency is therefore given as
η ¼Xnq¼1
qIq; ð6Þ
and the uniformity is defined as
uni ¼ maxfIg −minfIgmaxfIg þminfIg ; ð7Þ
where fIg is the set of normalized efficiency alongdiffracted orders
We have also utilized a fractional Fourier trans-form, which is the extension of the traditional Four-ier transform in the fractional order domain. It notonly has a relationship with the Fourier transformand Fresnel transform but also has its own specialfeatures that are widely applied in many domains[21–25]. In this paper, we have employed these fea-tures in our experiment. Figure 2 implements thefractional Fourier transform from the real conver-gent beam. We analyzed this characteristic in theparaxial condition, where f is the focal length of lens,D is measured distance and d is optical source dis-tance. From Ref. [26], we know that the distance Dat the viewing plane is determined by the positiond of the illuminated light source. Hence, we have
D ¼ frd: ð8Þ
The field distribution in the spectrum plane will thenbe
uðxFÞ¼c0exp�jk
ðf −dþLÞ ·x2F2½D ·ðf −dþLÞþf ·ðd−LÞ�
�Zþ∞
−∞
tðxDOEÞ
×exp�−jk
f ·xDOE ·xFD ·ðf −dþLÞþf ·ðd−LÞ
�dxDOE: ð9Þ
Fig. 1. Transition points in the cross section of the CDG. Fig. 2. Schematic diagram of a fractional Fourier transform.
Using classical optics theory [26], we can simplifyEq. (9) into
H ¼ 2Lf
d − fλT; ð10Þ
where λ is the input wavelength and T is the period ofthe grating.In our prototype, we have modified the setup of
Fig. 2 into Fig. 3. A spectrum from this optical grat-ing can be generated and projected onto the screen.By using a laser and a focusing lens, this configura-tion can be easily set up. The position of the CDG isexactly parallel to the screen. The parameter L is theCDG movement distance. By adjusting L, the spec-trum pattern of the first-order CDG can then be ob-served on the screen with one circle. The dimensionof the device under test (DUT) with a regular patterncan then be measured by the diameter of the pro-jected spectrum, which is manipulated by movingL as shown in Fig. 4. Hence, we know the area bygetting the product of H1 and H2, as H is the exactdiameter of the CDG spectrum.
3. Experimental Results
We employed a dry etching lithography process [16]for fabricating the 1st-order spectrum of CDG. Theetching depth is the distance for the light to have aphase shift of π [8,9]. The designed grating will haveanetchingdepth of617:4nm,as the refractive index ofglass is 1.513 at 633nmwavelength. In a half-period,its phase transition points are 0, 10.5, and 21 μm. Thegrating is then patterned on a 2 cm glass substrate. Inour CDG experiment, we used a He–Ne laser sourcewith an input power of 152 μW. The output spectrum
is shown inFig. 5. Themeasuredefficiencywas76.4%,and the zero-order powerwas found tobe0:53 μW.Thefocal lengths of the lens we have chosen were 200 and250mm with �1% variation. The spectrum width is2mm. The image size coming out from the lens isaround a quarter of the size of that chosen lens.The image distortion would not then be taken into ac-count. The thickness of the lens is around 5mm, andthe curvature of the lens is around 200mm. Hence,the thin lens formula could therefore be employed.Figures 6 and 7 show the image diameter versus Lwith different focal length. Comparing these findingswith theoretical results, we have found that the erroris better than 3%,, which is a small deviation.
We have also tried two different types of objects,DUT 1 and DUT 2, at longer focal length. The actualsize of DUT 1 is 36mm× 54mm, while that of DUT 2is 132mm× 182mm. In the following experiment,the focal length of the lens we have chosen is300mm. The spectrum is 5mm. For DUT 1 at D ¼2:06m away, shown in Fig. 8, the measured d is
Fig. 3. Distance and area measurement using the CDG.
Fig. 4. Measuring area with the diameter of the CDG.
35:1 cm. We have measured H1 ¼ 20 cm for the DUTheight, while H2 ¼ 30 cm for the DUT width. Byusing Eq. (10), the measured dimension is
35:5mm × 53mm. Compared with the theoreticalresult, the error is around 3%. For DUT 2 at D ¼11:25m, shown in Fig. 9, the measured d is30:8 cm. We have measured H1 ¼ 11:8 cm for theDUT height, while H2 ¼ 16:2 cm for the DUT width.By using Eq. (10), the measured dimension is133:4mm× 183:1mm. Compared with the theoreti-cal result, the error is around 3%, which is also a verysmall deviation.
4. Conclusion
In this paper, a spectrum-based measuring system isproposed to measure the area from a CCD camera toan object through the use of CDG. The proposed mea-suring system does not use expensive high-speed sig-nal processing methods to identify the diameter ofthe CDG spectrum but instead uses a simple lens de-sign and fractional Fourier transform. This methodnot only has more compact configurations than mostreported techniques but also offers comparable orbetter measurement precision. It should be empha-sized that the size of grating should be chosen tobe as large as possible for better resolution and accu-racy. The theoretical analysis and experimentalresults both demonstrate that an accuracy betterthan 3% can easily be achieved.
This project is supported by the General ResearchFund (No. 112507) of the Research Grant Council inHong Kong.
References1. A. Caarullo and M. Parvis, “An ultrasonic sensor for distance
measurement in automotive applications,” IEEE Sensors J. 1,143–147 (2001).
2. P. G. Gulden, D. Becker, and M. Vossiek, “Novel opticaldistance sensor based on MSM technology,” IEEE SensorsJ. 4, 612–618 (2004).
3. B. Culshaw, G. Pierce, and J. Pan, “Non-contact measurementof the mechanical properties of materials using an all-opticaltechnique,” IEEE Sensors J. 3, 62–70 (2003).
4. A. Yasuda, S. Kuwashima, and Y. Kanai, “Ashipborne-typewave-height mater for oceangoing vessels using micro-wave Doppler radar,” IEEE J. Ocean. Eng. 10, 138–143 (1985).
5. H. Yan, “Image analysis for digital media applications,” IEEEComput. Graph. Appl. 21, 18–26 (2001).
6. R. Cucchiara, M. Piccardi, and P. Mello, “Image analysis andrule-based reasoning for a traffic monitoring system,” IEEETrans. Intell. Transp. Syst. 1, 119–130 (2000).
7. J. F. Wen and P. S. Chung, “2D optical splitters with polymeroptical fiber arrays,” J. Opt. A: Pure Appl. Opt. 9, 723–727(2007).
8. H. Dammann and K. Gortler, “High-efficiency in line multipleimaging by means of multiple phase holograms,” Opt.Commun. 3, 312–315 (1971).
9. H. Dammann and E. Klotz, “Coherent optical generation andinspection of two-dimensional periodic structures,” Opt. Acta4, 505–515 (1977).
10. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoid-ally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
11. R. L. Morrison, “Symmetries that simply the design of spotarray phase gratings,” J. Opt. Soc. Am. A 9, 464–471 (1992).
12. J. N. Mait, “Design of binary-phase and multiphase Fouriergratings for array generation,” J. Opt Soc. Am. A 7, 1514–1528(1990).
13. C. Zhou, J. Jia, and L. Liu, “Circular Dammann grating,” Opt.Lett. 28, 2174–2176 (2003).
14. S. Zhao and P. S. Chung, “Design of circular Dammanngrating,” Opt. Lett. 31, 2387–2389 (2006).
15. J. F. Wen, S. Y. Law, and P. S. Chung, “Design of circularDammann grating (CDG) by employing circular spot rotationmethod,” Appl. Opt. 46, 5452–5455 (2007).
16. J. F. Wen and P. S. Chung, “A new circular Dammann gratingusing Hankel transform,” J. Opt. A: Pure Appl. Opt. 10,075206 (2008).
17. S. Zhao, J. F. Wen, and P. S. Chung, “Simple focal-lengthmeasurement technique with a circular Dammann grating,”Appl. Opt. 46, 44–49 (2007).
18. J. F. Wen and P. S. Chung, “The use of circular Dammanngrating for angle measurement,” Appl. Opt. 47, 5197–5200(2008).
19. J. F. Wen, Z. Y. Chen, and P. S. Chung, “A novel distancemeasurement technique based on optical fractional Fouriertransform OECC,” in Opto-Electronics and CommunicationsConference, 2008, and the 2008 Australian Conference onOptical Fibre Technology, OECC/ACOFT 2008 (IEEE, 2008),pp. 1–2.
20. R. N. Bracewell, The Fourier Transform and its Applications(McGraw-Hill, 1978).
21. H. M. Ozaktas and D. Mendlovic, “Fourier transforms offractional order and their optical interpretation,” Opt.Commun. 101, 163–169 (1993).
22. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, “Fractionalcorrelation,” Appl. Opt. 34, 303–309 (1995).
23. R. G. Dorsch and A. W. Lohmann, “Fractional Fouriertransform used for a lens-design problem,” Appl. Opt. 34,4111–4112 (1995).
24. Z. Zalevsky and D. Mendlovic, “FractionalWiener filter,”Appl.Opt. 35, 3930–3936 (1996).
25. B.-Z. Dong, Y. Zhang, B. Y. Gu, and G. Z. Yang, “Numericalinvestigation of phase retrieval in a fractional Fourier trans-form,” J. Opt. Soc. Am. A 14, 2709–2714 (1997).
26. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
