Application of the Talbot effect for three-dimensional ...

Application of the Talbot effect for three-dimensional and step-height measurement using an LCD Elizabeth Vela-Esparza*a, Gustavo Ramírez-Zavaletab, Marco Antonio Rosales-Medinaa, Ponciano Rodríguez-Monterob, Eduardo Tepichín-Rodríguezb a Universidad de las Américas, Puebla (UDLAP), Ex-hacienda Santa Catarina Mártir s/n, San Andrés Cholula, Pue., México 72820; b Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Apdo. Postal #51, Puebla, Pue., México 72000 ABSTRACT The self-imaging phenomenon under coherent illumination or Talbot effect has been widely used in different fields including optical metrology. According to the Talbot effect, when a periodic grating is illuminated with spatially and temporary collimated coherent light, the grating is self- imaged at a certain distance, ZT, which depends on the inverse of the wavelength λ of the incident light and the square of the period, d, of the grating. This means that they can be seen at different positions ZT by changing either d or λ. Using the wavelength dependence of the self-images, and a fixed period, d, an application of the Talbot effect for three-dimensional and step-height measurement using a two-wavelength laser, appeared recently in the literature. We propose in this work to use an LCD to display a tunable grating. In our setup, we used a fixed wavelength and a dynamic 1-D grating to adjust the step-height measurement capabilities. We also analyze the possibility of measuring continuously varying surfaces with this technique. We include the preliminary results of our proposal. Keywords: Talbot effect, tunable grating, LCD, dynamic grating.

1. INTRODUCTION The Talbot effect, also known as self-imaging phenomenon, was first discovered by H. F. Talbot1 in 1836. This phenomenon consists of the replication of the periodic object at certain distances along the optical axis, when illuminated with spatially and temporary coherent light. The distance between consecutive self-images is constant, and depends only on the wavelength and the period of the grating. Lord Rayleigh2 observed this phenomenon in 1881; he demonstrated the longitudinal periodicity of the self-images and gave the analytical expression for the separation between the self-images. J. M. Cowley and A. F. Moodie3 observed, in 1957, the behavior of the self-images phenomenon using a spherical wave front and the influence of the change on the size of the source of illumination. In 1961, A. W. Lohmann4 presented the first application of the Talbot effect which consisted of a device capable of characterizing the magnitude of the wavelength of a source. Then, in 1965, J. T. Winthrop and C. R. Worthington5 studied the Talbot effect using coherent and incoherent incident light on periodic objects placed at finite distances. Two years later, in 1967, W. D. Montgomery6 established the necessary and sufficient condition for the periodicity of the diffracted field along the longitudinal axis of a periodic object. The self-imaging phenomenon has attracted many researchers who had developed several applications in different areas like optical metrology and spectrometry7-9. Also, applications using the Talbot interferometer have been presented on the existing literature9-14. For this research work, even though two-wavelength, multiple-wavelength or tunable lasers are easily available, we fix it *[email protected].

Optics and Photonics for Information Processing III, edited by Khan M. Iftekharuddin, Abdul Ahad Sami Awwal, Proc. of SPIE Vol. 7442, 74420F · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.826396

Proc. of SPIE Vol. 7442 74420F-1

to one wavelength. A change in the period of the grating can be achieved by displaying the grating in a liquid crystal display (LCD) controlled by software. The software allows us to show many different periods of the gratings without mechanical modification of any of the components needed. This phenomenon is applied to measurement of step-height objects by making incident the self-images on the discontinuous object. The LCD can display different period grating until the self-image shows a high contrast of the patterns and, the Talbot distance can be calculated by an analytical expression presented in the next section. 2. PRINCIPLE OF THE TWO-WAVELENGTH TALBOT EFFECT According to the Talbot effect, when illuminated with spatially and temporary coherent light a grating will be reproduced or self-imaged at a certain distance, ZT. This distance is constant and depends only on the wavelength of the light λ and the period of the grating, d. We have that for a plane wave front the distance is given by:

ZT =

2nd 2

λ

,

(1)

where n is an integer or half-integer, d is the period of the grating and λ is the wavelength of the light. The integer corresponds to the positive self-images and the half-integer to the negative self-images. In this research work we will only use the positive Talbot images, i.e. the integer self-images. From Eq. (1) it can be seen that the self-image plane distances from the object are directly proportional to the period of the grating and inversely proportional to the wavelength of light. This phenomenon can also been reproduced illuminating the periodic grating with a spherical wave front; but in our research we will focus only on plane wave front15,16. In 2006, Takeda17 et al. published an application of the Talbot effect in three-dimensional step-height measurement using two different wavelengths. This method consists in using two laser beams with different wavelengths λ1 and λ2, make them incident on the periodic grating and implementing this for depth measurement in three-dimensional step-height objects. Fig. 1, shows a scheme of the technique using two different wavelengths.

Fig.1. Diagram of the principle of two-wavelength Talbot self-imaging system.

Fig. 1 shows the schematic diagram of the common-path two-wavelength Talbot self-imaging system. A red He–Ne laser with a wavelength of λ1 = 632.8 nm and a diode-pumped frequency doubled Nd:YAG green laser with a wavelength of λ2 = 532 nm, are used in this method. Both of the lasers have 5 mW of power. They were first mixed at a beam splitter;


then passed through a beam expander and a spatial filter. They were collimated and then made incident on the Ronchitype grating. The first Talbot self-image distance from the grating for the wavelength λ1 can be given by

Z1 =

2d 2

λ1

,

(2)

,

(3)

and for the wavelength λ2,

Z2 =

2d 2

λ2

Subtracting Eq.(2) from Eq. (3) we have that the distance between the first Talbot self-images plane for the wavelength

λ1 and λ2 is given by the following expression:

ΔZ = ZT 2 − ZT 1 =

2 ⎛ 1 1⎞ ⎜ − ⎟, d 2 ⎝ λ2 λ1 ⎠

(4)

where λ1 corresponds to the red laser wavelength (632.8 nm) and λ2 to the green laser wavelength (532 nm). Using the two-wavelength Talbot effect is possible to shift the self-image plane without any mechanical movement of the periodic grating. This was implemented for measuring depth of step-height objects. Fig. 2 shows a schematic of a step-height object. The size of the object was chosen to our convenience for demonstration purposes of our suggested method.

Fig.2. Diagram of the step-like object used for the experimental demonstration of the method for measuring depth using the Talbot effect. It is a discontinuous object having the first plane, A, a second plane, called B; they are placed at a fixed distance from the periodic object. The method suggested in this work measures the distance between plane A and B, i.e. the depth of the three-dimensional step-like object.

First, we will explain the technique demonstrated by Takeda17 and collaborators. The experimental setup of the twowavelength Talbot effect used by these authors is shown in Fig. 3. The self images were recorded by a CCD detector and then analyzed in a computer. The distance between the first plane of the object, A, and the plane B was fixed to 9.36 cm. The procedure followed to implement the plane shift by changing the wavelength of the laser light without mechanical movement was to switch on the red laser and record the images projected on the top and the bottom of the object. Then the red laser was switched off and the green laser was switched on, and the patterns that were projected on the first and second plane were recorded without mechanically moving any component in the setup. We observed that for the red laser wavelength, the grating was self-imaged on the first plane (plane A) of the step-like object, while for the second plane (plane B) the image does not correspond to the self-image. For the green laser, the grating was self-imaged on the plane B of the object and we observed that on the first plane of the object (plane A) the image did not correspond to a self-image for the green wavelength. Using the Eq. (1) we calculate the Talbot distance for the red and the green wavelengths. The first self-image for λ1 is seen at 49.3 cm and for λ2 the first self-image is projected at 58.66 cm. The total measured distance between the two


self-images, ΔZT, was calculated using the Eq. (4), it is equal to 9.36 cm. These results demonstrate that the plane of the self-images of the Talbot effect can be shifted without any mechanical movement of the grating or any other component of the setup. It was demonstrated experimentally that the measurement of depth on step-height three-dimensional objects is possible using the dependence of the Talbot effect on the wavelength of the light. In the next section we proceed to explain the essence of our experimental technique.

Fig. 3. Schematic of the experimental setup for the measurement of the step height of a discontinuous object using the twowavelength Talbot effect. Two different wavelengths are made incident on the periodic object, Ronchi-type grating and the projected images are recorded by a CCD Camera and then displayed on a PC.

3. PRINCIPLE OF THE TALBOT EFFECT USING A LCD The principle of the Talbot effect using an LCD for measuring depth in step-height three-dimensional objects consists in exploiting the dependence of the Talbot distances from the period of the object (See Eq. (1)). For corroborating this method it was used a liquid crystal display (LCD) with an active area of 26.6 cm x 20.0 cm and a resolution of 800 x 600 pixels of 32 µm of side each one. The grating is displayed on the LCD controlled by software. The software was developed as part of this research work and consisted in a loop that allowed us to display different pitches for the grating in real time without any mechanical modification of the setup. Fig. 4 shows a schematic diagram of the experimental setup of our technique for measuring depth in step-height treedimensional objects. We used a He-Ne red laser with a wavelength λ = 632.8 nm that was made incident on the LCD that displayed the periodic grating controlled by software, we observed the images projected on the planes of the object and captured them using a CCD camera. The same type of object was used for the corroboration of this technique; in the lab it was simulated with two planes as shown in Fig. 5, the plane A and the plane B, defined in Fig. 2, were placed at 34.95 cm between them. The red laser was made incident on the LCD that displayed a grating with a period d1 and, its projections on both of the planes were recorded. Then, using the software to control de LCD the period of the grating was changed to d2, the red laser was made incident on the LCD and the projected images were recorded.


Fig. 4. Schematic diagram of the experimental set up using one fixed wavelength (632.8 nm) and a dynamic grating displayed on a LCD. On the computer, software controls the period of the grating displayed on the LCD, changing the distances of Talbot.

It was observed that for the period d1 the grating was replicated on the plane A and for the period d2 the first self-image was projected on the plane B. For this technique we have that the depth can be calculated using the following Eq.:

ΔZ T =

1

λ

(d 2 2 − d12 ),

(5)

where d1 is the period which projected the first self-image on the plane A of the object, and d2 is the period which projected the first self image on the plane B and λ is the wavelength of the light, in this case we used λ = 632.8 nm.

Fig. 5. Diagram showing how the object was simulated for confirming the technique of depth measurement using a LCD. The depth of the object is given by the distance between both planes, ΔZT.

A brief analysis of both methods will be presented before continuing explaining our suggested technique. From Eq. (1) is clearly seen that the Talbot distances have a direct dependence on the period of the object and that they are inversely proportional to the wavelength of the light. Theoretical plots are presented in Fig. 6. The pitch of the grating used for reproducing the graphs is 0.2 mm (5 cycles/mm). It is seen on the same figure (Fig. 6) that using greater wavelengths


allows us to have smaller Talbot distances. Also, it is important to mention that there are not available in the market lasers for all of the wavelengths shown on the plots; wavelengths in the range of the visible spectrum were the only we considered (400- 700 nm). Fig. 6(a) shows a point which represents the wavelength of a Krypton laser at 416 nm and another one representing a Ruby laser with wavelength of 694 nm, they correspond to the smallest wavelength and the greatest one available, respectively. We can notice in the same figure (Fig. 6) that the maximum distance between self-images of Talbot will be given by the distance of the first self-image projected by the laser with wavelength of 416 nm at 19.23 cm. Also, the minimum distance of Talbot will be given by the laser with wavelength of 694 nm at 11.53 cm. The plot presented in Fig. 6(b) shows a point which represents the wavelength λ1= 632.8 nm and the other point denotes a wavelength λ2 = 532 nm, both were used by Takeda17 for the experimental corroboration of their technique. It is important to mention that in Figure 6(b) we have that the range of measurement of the technique of Takeda17 goes from 19.23 cm to 11.53 cm. That evidences that for the method for depth measurement using two different wavelengths the rage of measurement is in the order of centimeters and, for this range there are not lasers that let us have all the possible self-images.

(a)

(b)

Fig. 6. Plots of the distances of the first self-image of Talbot depending on the wavelength. On (a) the points show the maximum and the minimum wavelength for lasers in the range of the visible spectrum. Plot (b) shows the distances for the first self-image for the wavelengths 532 nm and 632.8 nm, which were used for the experimental corroboration of Takeda17.

We have mentioned that one limitation of the technique for depth measurement using the Talbot effect with two different wavelengths is the difficulty to find lasers for all the electromagnetic spectra and the restricted range of the distances of Talbot. In contrast, the range and mathematical behavior for our proposal is analyzed. The directly proportional behavior of the period of the grating with respect to the Talbot distances is illustrated on Fig. 7. The plots show the position of the first image of Talbot versus the period of the grating. The black curve supposes a continuously display of periods on the LCD, yet, due to the pixelization of the liquid crystal display, we can only present a discrete number of periods, they are illustrated in Fig. 7 by the circles. We suppose that the pixels have the shape of squares and a period corresponds to one pixel switched on and one switched off. For this experiment we chose the minimum lines of the grating to display a self-image as ten; corresponding to this, the maximum possible period to be displayed is 2.62 mm. We observe on Fig. 7(b) that the minimum possible period to display on the LCD is 64 µm which gives us the position of the first self-image of Talbot at 12.95 mm and the first self-image at 21.76 m for the greatest period possible to display, which is 2.64 mm. The minimum difference between self-images, ΔZTmin is 38.83 mm, which corresponds to the separation between two blue circles on the discrete plot of Fig. 7. Next section describes in a more extensive manner the methodology of our experiment and the results obtained.


(a)

(b)

Fig. 7. Plots of the Talbot distances depending on the period of the grating. (a) Shows the theoretical (straight line) and the discrete (circles) plot. The blue circles represent the possible periods to be displayed with the LCD used in the experiment. (b) The minimum period possible to display on the liquid crystal display is 64 µm which self-images at a distance of 12.9 mm. The maximum period displayed on the LCD would self-image at 21.76 m.

4. RESULTS The former section explains the fundamentals of the method for measuring depth of step-like objects using the Talbot effect and a dynamic grating displayed on an LCD. The step-like object was simulated with two planes, plane A and plane B, placed at arbitrary distances. In the images below, the ones placed at left correspond to the projection recorded by the CCD camera on the plane A; and the projection on plane B correspond to the image on the right. The next series of images show the experimental results obtained for our suggested method of depth measurement. First, the minimum period was displayed on the LCD, d1 = 64 µm, its pattern does not self-images on neither of the planes A or B, as shown and Figure 8(a) and 8(b). The next step consisted of display the consecutive periods, d2 and d3, and capture their images projected on the planes (Figure 8(c) and 8(d)). For the period d2, there is no self-image observed on the planes, on the other hand, we observe that for the period d3 it corresponds to a replica of the periodic object on plane A. Having this information, we can calculate the distance at which the fixed plane A is, given by ZT = 11.65 cm. Thus, more periods continue to be displayed until one self-images on plane B. Figures 9 and 10 show the projected images for periods d4 = 256 µm, d5 = 0.32 mm and d6 = 0.384. The same way we obtained a self-image for period d3 on plane A; we can distinguish the replica of the periodic grating on plane B for period d6. We present also the images for the projections of periods d7 = 0.448 mm, d8 = 0.512 mm, d9 = 0.576 mm and d10 = 0.64 mm. All of images presented in this section demonstrate that, using the dependence of the Talbot effect from the period of the grating, we can measure depth in step-like three-dimensional object without making any mechanical modification of the setup.


(a)

(b)

(c)

(d)

(e)

(f)

Fig. 8 (a) and (b) Images captured by the CCD camera in both planes for the period d1 = 64 µm. In (c) and (d) we observe the projection of the period d2 = 128 µm. In (e) y (f), for the period d3 = 192 µm. It is seen that on the planes (a), (b), (c), (d) and (f) the periodic grating is not replicated; but it is perfectly self-imaged in (e).


(a)

(b)

(c)

(d)

(e)

(f)

Fig. 9. (a) and (b) Images captured for period d4 = 256 µm, we can see that this period does not self-image on neither of the planes. (c) and (d) Images captured for period d5 = 0.32 mm; no Talbot image is observed. In (e) and (f) it is seen the projections for period d6 = 0.384 mm, a self image is observed on the plane B.


(a)

(b)

(c)

(d)

(e)

(f)

Fig. 10 (a) and (b) Images captured by the CCD camera for both planes, A and B, displaying the period d7 = 0.448 mm. In (c) and (d) it is seen the projection for the period d8 = 0.512 mm. And on (e) and (f), for d9 = 0.576 mm. None of the periods projected self-image on neither of the planes simulating the object.


(a)

(b)

Fig. 11 (a) and (b) Images recorded using a period d10 = 0.64 mm. (a) corresponds to the image on the first plane and (b) to the image projected on the second plane. It is observed that for neither of the planes the period replicates the grating.

The figures above presented in this section demonstrate that the technique for depth measurement on three-dimensional step-like objects using the dependence on the period of the Talbot effect is effective. Also, it has been pointed out the possibility of changing in real time the period of the grating without any modification of the setup, which reduces the sources of error.

ACKNOWLEDGEMENTS One of the authors (E.V.E.) thanks CONACyT for its support through project PY-103638.

CONCLUSIONS The principle of depth measurement in step-like three-dimensional objects using an LCD was described and corroborated experimentally. This technique was compared to the method suggested by Takeda17 and his collaborators whose range of measurements is in the order of millimeters. When the dependence on the period of the grating of the Talbot self-images is used, a greater range of measurement was validated: 12.9 mm to 21.76 m, this is the main advantage of the technique discussed in this work. Further, the pitch of the grating can be changed in real time without any mechanical modification of the setup, diminishing the source of error.

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