Improvements of Traditional Laser Fraunhofer Diffraction Experiment Using Digital Image Processing Method

JIN YuanweiWANG YabingZHAO BinYI ZhaoguangXU Shenghui

1 Department of Mathematics and Physics，Nanjing Institute of Technology，Nanjing 211167，China 2 School of Mathematics and Statistics,Institute of Space Weather，Nanjing University of Information Science and Technology，Nanjing 210044，China

Abstract：We present an improved digital image processing(DIP) method to calculate the widths of single slits. Different from the traditional laser Fraunhofer diffraction experiment in college physical experiments，by performing fast Fourier transform，inverse fast Fourier transform and the nonlinear leastsquare fitting on the diffraction pattern taken by a camera，the DIP method can quickly return an analytic expression，whose period is used to calculate widths of single slits. By comparing the measured results by the DIP method and the successional difference(SD) method，we find that for a single slit whose width is 60-372 μm，the DIP method is more accurate. Experimental results show that for single slits with widths between 40 μm and 160 μm，the relative error of the DIP method is less than 2.78%. Also，the DIP method can be used to measure the diameter of filament and fibres online in real time.

Key words:digital image processing method；traditional laser Fraunhofer diffraction experiment；width of single slit

Introduction

Fraunhofer diffraction is a common phenomenon in our daily life，which can be easily achieved by experiments and simulations[1].In college physical experiments and engineering applications，F(xiàn)raunhofer diffraction is commonly used to measure micro distances，such as the width of single slits，the diameter of filament and fibres，Yang’s modulus and linear expansion coefficient of metals[2-6].

Manually moving the photoelectric sensor and visually recording the positions of different fringes make the traditional laser Fraunhofer diffraction experiment a tedious and time-consuming method with large measurement errors[7-8].In recent years，the automatic collection and the processing of light intensity have been achieved by the adoption of the charge-coupled device(CCD)[9-11].However，the dynamic range of the ordinary CCD is so small that most of the optical energy falls on the central bright fringe.Other bright fringes are also greatly affected by noises.Even after a series of noise reduction processing，the result cannot be improved significantly[10，12].

In general，the traditional method for measuring the width of a single slit is to find the locations of dark fringes and then get the average width of the bright fringes[7].This method assumes that the diffraction patterns are equidistant.Strictly speaking，the width of each bright fringe is not exactly the same.In addition，the speckle noises of laser will affect the judgement of the positions of dark fringes.Therefore，the filtering process is necessary to improve the signals of the diffraction pattern.In this study，the diffraction pattern is processed by the digital image processing(DIP) method[13].By using fast Fourier transform(FFT)[14]，inverse fast Fourier transform(IFFT)[15]and the nonlinear least square methods，we find out the analytic expression of the diffraction light intensity，and then calculate the widths of single slits，which is precise and convenient.

1 Equipment and Principle of Laser Single Slit Fraunhofer Diffraction

The diagram of light propagation of single slit Fraunhofer diffraction is shown in Fig.1(a)，which is mainly composed of a laser whose wavelengthλ=532 nm，a single slit whose widthd=80 μm，a white screen with two vertical fluorescent strips (shown in Fig.1(b))，a camera with a resolution of 640×480，and a terminal computer.The camera is placed vertically below the optical axis with a tilt angle of -10°，so that the horizontal fringes will not be deformed.The entire experiment has to be carried out in a darkroom to reduce the effects of stray light[16].

According to the Fraunhofer diffraction theory，the diffraction pattern will be generated on the screen when the laser passes through a single slit whose width is comparable to the laser wavelength.There are two reasons that convex lens are not included along the optical path：(a) since the emission angle of the laser beam is about 10-3rad，and the width of the single slit is small，the laser can be considered as parallel light;(b) the distanceLfrom the single slit to the screen is 80.00 cm，which meets the condition ofd2/(8Lλ)<<1.[7]

Fig.1 Sketch of experimental equipment：(a) diagram of light propagation of single slit Fraunhofer diffraction；(b) a white screen with two vertical fluorescent strips

The equation for the appearance of dark fringes is：

dsinθj=jλ，j=±1，±2，±3，…,

(1)

(2)

wherejis the order of dark fringes，θjis the diffraction angle of thej-order dark fringe，andcjis the distance from thej-order dark fringe to the center of the diffraction pattern.From Eq.(1)，we can see that for both sides of the diffraction pattern，sinθjincreases in equal increments with the increase ofj，and the period of sinθjalso equalsλ/d.

2 Diffraction Pattern Analysis Using Digital Image Processing Methods

2.1 Calibration and center determination of diffraction pattern

Figure 2(a) shows one of the patterns taken by the camera，in which the upper part is the pattern of the two fluorescent strips，and the lower part is the pattern of laser single slit Fraunhofer diffraction.The two vertical fluorescent strips with a distance of 2.00 cm between their inner boundaries will be used for the calibration of the diffraction pattern.After the image binarization，the image in Fig.2(a) is converted to anM×Narray which consists only of 1 and 0，and it depends on the grayscale threshold selected during the image processing，where 1 represents white pixel and 0 represents black pixel.The binary pattern with a threshold of 0.80 is shown in Fig.2(b)，in which the boundaries of fluorescent strips can be seen clearly.

The actual distance between adjacent pixels is

Δ=h/(b+1),

(3)

wherebandhrepresent the number of black pixels and the actual distance between inner boundaries of two fluorescent strips，respectively.In order to reduce the influence of laser speckle noise on processing，we find the center of the diffraction pattern and only analyze one side of it.By the threshold of 0.95，F(xiàn)ig.2(c) shows the binary diffraction pattern which only consists of the center bright fringe.By performing row-by-row summation on the corresponding 2-dimensional array，we can obtain the rowPwith the largest sum value，and the columnQ，which is the center of all the white pixels in the rowP.The intersection point of the white cross is the center position (P，Q) of the diffraction pattern.

Fig.2 Image binarization：(a) one of patterns taken by camera；(b)binary pattern with a threshold of 0.80；(c) binary pattern with a threshold of 0.95

2.2 FFT and IFFT of diffraction pattern

Next,we will only analyze one side of the diffraction，i.e.，the pixels from the columnQto 1 in the rowP.Pixels are numbered from the center to the edge.The actual distance between the pixelnand the central pixel is (n-1)h/(b+1),wheren=1，2，…，Q.Substitute it into Eq.(2) and get：

(4)

So far，we can get the relationship betweensand sinθof each pixel.In Fig.3，crosses，triangles and circles represent measured data points，minimum points and maximum points，respectively.It can be clearly seen that the pixels are not evenly distributed，and the multiple points around the extreme points make it difficult to determine the location of minimum points.Even if we find the position of minimum point，it can be found that the difference between the abscissas of adjacent minimum points is not a constant value.Therefore，the method of calculating the width of the single slit by determining the position of minimum points is not precise[17].

Fig.3 Diagram of relationship between intensity s and sin θ of each pixel

We then perform FFT to the intensitysby

(5)

wheres(n) is the intensity of pixeln，Qis the total number of pixels，kis the data number，andS(k) is a complex，whose modulus |S(k)| represents the amplitude.Figure 4 shows the |S(k)| as a function ofk.

It can be seen from Fig.4 that there are two significant spikes，which correspond to a periodic signal.By filtering，we can find the minimum points on both sides of each spike and their corresponding data number，which are marked byk1，k2，k3andk4.We get theS′(k) by keeping all the values ofS(k) fromk1tok2，k3tok4，and set all the other values to 0.Then a new periodic intensity signals′ can be achieved by IFFT of theS′(k).Figure 5 shows the intensitys(marked by circle) and the new intensitys′ (marked by cross) as a function of sinθ.

Fig.4 Amplitude as a function of k

Fig.5 s and s′ as a function of sin θ

In Fig.5，the minimum points ofs-sinθands′-sinθare marked byA1toA10，andE1toE10，respectively.The coordinates of these points are shown in Table 1.From Table 1，we can find that the abscissas of the corresponding points in these two signals are very close，especially for the points fromA2toA8，andE2toE8.It shows that the oscillation period of data points ofs′-sinθis the same as the difference of sinθof adjacent dark fringes，i.e.，equalsλ/d.

2.3 Nonlinear least square fitting of new periodic intensity signal s′

Table 1 Minimum points of s-sin θ and s′-sin θ

(6)

The relative error is

Er=|dD-d|/d×100%=0.84%.

(7)

Fig.6 Diagram of fitting data points and fitting curve

3 Discussion

By processing the eight minimum points fromA1toA8with the successional difference(SD) method，the average difference of sinθof adjacent dark fringes can be obtained as

(8)

The width of the single slit calculated by the SD methoddS(μm) is：

(9)

The relative error is

Er=|dS-d|/d×100%=1.48%.

(10)

It is found that for a single slit with a width of 80 μm，the DIP method is better than the SD method.Compared with the method of the traditional photoelectric sensor，in which the location of the dark fringes is determined by using a photoelectric sensor，the SD method is more intelligent and accurate.

In order to verify the accuracy of the DIP method，we also measure the widths of other single slits with different standard values (20，40，80，120，and 160 μm)，and compare them with the measurement results using the SD method.Table 2 gives the comparison of the measurement results by these two methods.It can be seen from Table 2 that when the slit width is equal to 20 μm or 40 μm，the SD method is more accurate.The reason for this is that when the slit width is small，the more concentrated light makes it easy to determine the location of the dark fringes.However，it is not the case for the DIP method，in which the less bright fringe of the diffraction pattern is，the worse the nonlinear fitting is.When the slit width is 80，120 or 160 μm，the DIP method is significantly better than the SD method.For a single slit with a width of 160 μm，the relative error of the SD method even reaches 24.29%.That is because as the width of the slit increases，the gradually increasing number of fringes makes it difficult for the SD method to determine the location of the dark fringes.

Table 2 Comparison of measured single slit width and relative error by DIP and SD methods

It is necessary to determine the measurement range of the single slit width.When the width of a single slit is large，the distance between fringes becomes narrow.The actual distance between adjacent pixels in this study is 0.028 57 cm.In the diffraction pattern，a complete fringe should be determined by five pixels at least.That is to say，the fringe width should not be less than 0.114 3 cm，or the single slit width should not be more than 372 μm.When the width of a single slit is small，the distance between fringes becomes wide.In order to ensure the accuracy of the measurement，the order of the dark fringe should not be less than four.In this experiment，there are 640 pixels in the horizontal direction and the corresponding distance is 18.284 8 cm.Then we will know that the fringe width should not be wider than 2.285 6 cm.Therefore we cannot measure the single slit whose width is less than 19 μm.Theoretically，the single slit width that can be measured by the DIP method is from 19 μm to 372 μm.However，in actual experiments，due to the limitation of laser power，for a single slit with a width of 20 μm，only the first order dark fringe can be observed on the optical screen，which makes it impossible to use the DIP method.Experimental results show that for single slits with widths between 60 μm and 372 μm，the DIP method is more accurate and faster than the SD method and the traditional photoelectric sensor method.

The main factors affecting the measurement precision are the laser power，the camera resolution，the precision of the pattern calibration，the image distortion，and the stray light.Therefore a laser with appropriate power，a high-quality camera with wide-angle lens and a more accurate pattern calibration method can improve the measurement precision significantly.

4 Conclusions

In this study，we have presented that the width of a single slit can be measured easily and accurately by using the DIP method.Comparing with the traditional methods，it is faster and easier with a higher precision.This method shows a good improvement when measuring the width of a single slit which is between 60 μm and 372 μm，and hopefully，sheds a light on the digitization of the traditional laser Fraunhofer diffraction experiment and accurate measurement in engineering applications.