Real-time polarization phase shifting technique for dynamic deformation measurement

Optics and Lasers in Engineering 31 (1999) 289}295

Real-time polarization phase shifting technique for dynamic deformation measurement Qian Kemao*, Miao Hong, Wu Xiaoping Department of Mechanics and Mechanical Engineering, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China Received 7 April 1999; received in revised form 21 May 1999; accepted 2 June 1999

Abstract By using a special Ronchi phase grating and polarization phase shifting method, four phase shifted patterns can be captured simultaneously, and then the dynamic deformation can be measured. The analysis and experiments are given.  1999 Elsevier Science Ltd. All rights reserved.

1. Introduction The typical high-sensitive optical-electronic methods for deformation measurement are TV-holography, electronic speckle pattern interferometry (ESPI) and moireH interferometry. These methods can be used to measure the static and dynamic behaviors such as deformation, vibration, #uid "eld and temperature "eld, etc. The systems with the phase shifting methods have higher sensitivity than those without the phase shifting methods. Based on the phase shifting theory at least three di!erent interference patterns are needed for measuring the `phasea which is connected to the measured physical situation. Suppose the intensities I of these interference patterns are G I(x, y)"a(x, y)#b(x, y)!2a(x, y)b(x, y)sin[u(x, y)!2h ] (1) G where i"1, 2, 3,2, a(x, y) and b(x, y) are amplitudes of two interference beams respectively and u(x, y) is the phase distribution which has relation with displacement,

* Corresponding author. E-mail address: [email protected] (Qian Kemao) 0143-8166/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 9 9 ) 0 0 0 2 2 - 6

290

Qian Kemao et al. / Optics and Lasers in Engineering 31 (1999) 289}295

vibration amplitude or refractive index. Generally the known phase changes 2h G should be limited from 0 to 2p. Though there are many techniques to introduce the phase changes h , there is G a di$culty for dynamic problems especially for transient ones because the acquisition of those interference patterns needs enough time. The piezo-electric crystal is widely used for phase shifting. Compared with PZT, the polarization phase shifting method has some bene"ts. Refs. [1,2] show two set-ups which use the polarization phase shifting method for real-time measurement, but they are complex. In this paper a new set-up is introduced. Four phase shifted interference patterns can be captured simultaneously by combining the polarization phase shifting method and a special Ronchi phase grating. This set-up is cheap and simple without any moving elements. So it can easily be used under various environments. Any transient phenomena can be recorded by using this phase shifting technique as long as the recorder is fast enough. We "rst reported this new system in a national conference on experimental mechanics [3], Ref. [4] shows a very similar system. This paper will introduce the special Ronchi phase grating, polarization phase shifting method and the new real-time polarization phase shifting system.

2. Optical system of real-time polarization phase shifting 2.1. Ronchi phase grating For simplicity without losing generality, parallel Ronchi phase grating (Fig. 1) will be analyzed. Its amplitude transmittance is



jk(n h#n h ) np!(p/4)(x(np#(p/4),    t(x, y)"t(x)" e jk(h#nh) np#(p/4)(x(np#(3p/4), e

(2)

where n is the refractive index of grating medium, p is the grating constant, k is the  wave number and n is the integer. Suppose that a monochromatic wave traveling in the Z direction impinges on the grating; let the complex amplitude of waves just

Fig. 1. Ronchi phase grating.

Qian Kemao et al. / Optics and Lasers in Engineering 31 (1999) 289}295

291

entering the grating and just leaving the grating be ; (x, y) and ; (x, y), respectively, G M then ; (x, y)"; (x, y)t(x, y). M G

(3)

Since t(x) is a periodic function, we can express it as the sum of Fourier series, hence > ; (x, y)" a ; (x, y)eHpKVN, M K G K\ where a are Fourier coe$cients of t(x), K



[eHIL\F#1];eHIF>LF, m"0,  a " sin(mn/2) K [eHIL\F!1];eHIF>LF, mO0. mn

(4)

(5)

Eq. (5) indicates that the Ronchi grating divides the incident wave into many separate waves. These waves are same except that they have di!erent intensities and travel in di!erent directions. Select h satisfying eHIL\F"!1, then



0,

m"2q,

a " !2;eHIF>LF/mp m"4q#1, K 2;eHIF>LF/mp m"4q!1,

(6)

where q is the integer. From Eq. (6) we have (1) For this special Ronchi grating, zero order and all even orders di!raction are vanished. (2) The energy ratio of all odd orders di!raction are 1 1 I : I : I :2"1 : : :2 .    9 25

(7)

(3) The energy of $1 orders di!raction is 80% of the total energy. As for a cross Ronchi grating four "rst-order di!raction waves are obtained. Its total energy is 64% of the incident energy. 2.2. Polarization phase shifting As an example Fig. 2 is a system for in-plane displacement measurement by moireH interferometry. A laser beam is split into two linearly polarized beams, one horizontal and the other vertical. They impinge symmetrically onto specimen grating. The "nal interference pattern falls on a CCD chip which is the image plane of the specimen. The beam expander is omitted in Fig. 2. The normal polarization phase shifting system without Ronchi grating is illustrated on the right-hand side of Fig. 2. The angle between the quarter wave plate's fast axis and x-axis is 453 (Fig. 2a). By this quarter wave plate, two linearly polarized beams are transformed into two circularly polarized

292

Qian Kemao et al. / Optics and Lasers in Engineering 31 (1999) 289}295

Fig. 2. Optical schematic of moireH interferometry with polarization phase shifting. Right: a normal polarization phase shifting method with one linear polarizer; left: a real-time polarization phase shifting method with a special Ronchi phase grating and four linear polarizer; L: imaging lens, Q: quarter wave plate, R: Ronchi phase grating, P: analyzer.

beams, one right-handed and one left-handed, respectively. The analyzer is a linear polarizer and has a polarization axis of angle h (Fig. 2b); the interference pattern can be seen in Fig. 2c. Let the Jones vectors of waves before the quarter wave plate and after the analyzer be

 E V E W

and



E V , E W

Qian Kemao et al. / Optics and Lasers in Engineering 31 (1999) 289}295

293

respectively, then E cos h  sin 2h E 1 1 i V "  ; ; V ,  sin 2h E sin h E (2 i 1 W  W

 



  

(8)

where  sin 2h   sin 2h sin h (2 i 1  are the Jones matrices of the quarter wave plate and the analyzer, respectively. Let E and E be aeHP and beHP, respectively, with u !u "u, Eq. (8) will result in  W   Eq. (1). If we change the angle h step by step, a series of interference patterns can be captured and the phase distribution u(x, y) will be calculated. 1

   1 i

and

cos h



2.3. Real-time polarization phase shifting system The left part of Fig. 2 is the schematic of a real-time polarization phase shifting system. The di!erences between normal polarization phase shifting system and the new real-time one are: (1) A special cross Ronchi grating R is inserted (Fig. 2d). It can split the wave into four, but no fringes is formed (Fig. 2e). The Ronchi phase grating is made by ion beam etching [5] and supplied by National Laboratory of Synchrotron Radiation (Hefei, China). (2) The analyzer is composed of four polarizers as Fig. 2f, in which the polarization axes are h"0, p/4, p/2, 3p/4 Four interference patterns can be seen on monitor simultaneously (Fig. 2g). The recorder plane (i.e. image plane) has a suitable distance from Ronchi phase grating in order that the split images will not overlap each other. Their intensities are I "(a#b)!2ab sin u (h "0),   I "(a#b)#2ab cos u (h "p/4),   I "(a#b)#2ab sin u (h "p/2),   I "(a#b)!2ab cos u (h "3p/4).   Then the expected phase u will be I !I . u"arctan  (9) I !I   At "rst the coordinates of each point on the specimen must be recognized at the four images. A program is needed to calculate the expected phase u(x, y).

3. Experiments The in-plane displacement is measured by moireH interferometry with this new realtime polarization phase shifting technique. Three sets of fringe patterns have been

294

Qian Kemao et al. / Optics and Lasers in Engineering 31 (1999) 289}295

Fig. 3. Sampled images and their corresponding phases.

sampled in sequence as shown in Fig. 3(a)}(c), and their corresponding calculated phases are shown in Fig. 3(d)}(f ).

4. Conclusion A new real-time polarization phase shifting method for dynamic deformation measurement has been realized. The set-up is simple and cheap and can be applied widely and easily. The perfect pixel matching of the four phase shifted images, the calibration of the system and the accuracy estimation of this method will be studied further.

References [1] Arjan JP, van Haasteren, Frankena HJ. Real-time displacement measurement using a multicamera phase-stepping speckle interferometer. Appl Opt 1994;33(19):4137}42. [2] Weijers AL, van Brug H, Frankena HJ. Real time deformation measurement using a transportable shearoghaphy system. SPIE 1996;2921:76}81.

Qian Kemao et al. / Optics and Lasers in Engineering 31 (1999) 289}295

295

[3] Qian Kemao, Miao Hong, Wu Xiaoping. Study on real-time measurement for dynamic deformation by polarization phase shifting Moire interferometry. In: Proceedings of the Ninth National Conference on Experimental Mechanics, 1998. p. 445}8 (in Chinese). [4] Kranz J, Lamprecht J, Hettwer A, Schwider J. Fiber optical single frame speckle interferometer for measuring industrial surface. SPIE 3407, 328}31. [5] Shaojun Fu, et al. Fabrication of Ronchi phase grating by ion beam etching. Chinese J Quantum Electro 1995;12(2):146}9.