Computation of the cascaded optical fractional Fourier transform under different variable scales

Optics Communications 285 (2012) 997–1000

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Computation of the cascaded optical fractional Fourier transform under different variable scales Jiong Wang ⁎, Yudong Zhang, Guojun Li, Xuecheng Xin, Xiangang Luo State Key Laboratory of Optical Technologies for Microfabrication, Institute of Optics and Electronics, Chinese Academy of Sciences, P. O. Box 350, Chengdu 610209, China

a r t i c l e

i n f o

Article history: Received 20 May 2011 Received in revised form 24 September 2011 Accepted 29 September 2011 Available online 24 October 2011 Keywords: Fractional Fourier transform Optical fractional Fourier transform Computation

a b s t r a c t A simple but effective method for calculating the cascading of the multiple stages of optical fractional Fourier transform was proposed. Unlike the previous methods for solving the problems of cascading the optical fractional Fourier transform, we did not change the structures of the optical setups proposed by Lohmann but calculate the cascaded optical fractional Fourier transform with modified transform orders obtained by our method. Computer-simulation and experimental results were presented and a good agreement between them was obtained. © 2011 Elsevier B.V. All rights reserved.

1. Introduction

2. Theory analysis

The fractional Fourier transform (FRT) has many applications in the fields of quantum mechanics, signal processing and optics. After being introduced into optics by Ozaktas,Mendlovic [1,2] and Lohmann [3], it has been used successfully in image encryption [4] and spatially variant filtering [5,6]. FRT is regarded as a general case of Fourier transform. It is related to Fresnel diffraction in the same way as the standard Fourier transform is related to Fraunhofer diffraction [7]. So, it is also a useful tool of researching on diffraction optics. To implement the optical FRT, two ways have been proposed. Ozaktas and Mendlovic suggested a technique for optical implementation of FRT by utilizing the propagation of light beams through quadratic graded index media. Lohmann, almost at the same time, pointed out two configurations by making use of thin lenses to realize optical FRT. Implementing a single stage of the FRT may be valueless in the optical processing system such as optical correlator and spatially variant filtering systems. However, computation of the cascaded optical Fractional Fourier Transform (FRT’s) is never an easy task since the additivity properties of FRT is not satisfied unconditionally in optical implementations [8–10]. Then the question is how to calculating the optical FRT’s with multiple stages. In this paper, the cascading systems consisting of Lohmann-single-lens setup are demonstrated to discuss this problem in detail and a solution is presented.

FRT was invented by mathematicians perhaps approximately 80 years ago in relation to quantum mechanics. The original mathematical definition can be expressed as:

⁎ Corresponding author. E-mail address: [email protected] (J. Wang). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.09.070

F

p

  ϕ π n h io
0  exp i 4 þ i 2
0 02 2 0 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∫f x exp iπ x cotðϕÞ þ x cotðϕÞ−2xx cscðϕÞ dx ; f x j sinðϕÞj

ð1Þ where ϕ = pπ/2 is the transform angle, and p is called transform order. For this definition the additivity properties holds naturally,   i.e., F p1 F p2 ¼ F p1 þp2 . As Lohmann proposed two setups illustrated in Fig. 1 to perform the optical FRT’s, the definition is rewritten as: h i

iπ 2 p 2 F ½f ðx0 Þ ¼ c∫f ðx0 Þ exp dx0 ; 0 x0 cotðϕÞ þ x cotðϕÞ−2xx0 cscðϕÞ λf ð2Þ

Fig. 1. The optical implementations of fractional Fourier transform (FRT) proposed by Lohamann (a) single lens, (b) double lenses.

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J. Wang et al. / Optics Communications 285 (2012) 997–1000

where f′ is the standard focal length. It acts as a scaling parameter that scales the transform. For the single-lens setup, the equations f ′ = f sin(pπ/2) and z1 = f ′ tan(pπ/4) are satisfied. For the double-lenses setup, we have f ′ = f tan (pπ/4) andz2 = f ′ sin(pπ/2), where f is the focal length. Let us assume qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi ′ ′ that X 0 ¼ x0 = λf and X ¼ x= λf . By inserting them into Eq. (2), we

(a)

can obtain:

p

F ½f ðx0 Þ ¼ C∫f

qffiffiffiffiffiffiffi n h io 2 2 λf ′ X 0 exp iπ X 0 cotðϕÞ þ X cotðϕÞ−2XX 0 cscðϕÞ dX 0 ;

ð3Þ From a comparison of Eq. (3) with Eq. (1), it is obvious that Eq. (3) qffiffiffiffiffiffiffi ′ represents the FRT of the function f λf X 0 . Therefore, the optical qffiffiffiffiffiffiffi ′ FRT defined in Eq. (2) is a scaled FRT indeed. Let S ¼ λf be the scale factor. By analyzing the equations above, we may deduce that S is determined not only by the parameters of the optical setup such as wavelength λ and the focal length of the lens f, but also by the transform order p. Eq. (2) also implies the cascading condition for optical implementations of FRT, which requires all the stages of transform must have the same scale factors. We consider cascading two stages of optical FRT’s performed by two Lohamann-single-lens setups, as shown in Fig. 2. The transform orders are p1 and p2 respectively. The focal lengths of lens 1 and lens 2 are f1 and f2 respectively. The distances between the input plane and lens 1, lens 1 and lens 2, lens 2 and the out plane are z1, z1 + z2 and z2, respectively. According to the cascading condition, S1 = S2, we have p π. p π sin 1 : f 1 =f 2 ¼ sin 2 2 2

ð4Þ

From Eq. (4) we can see the cascading condition is only satisfied in a few special cases. To perform two stages of optical FRT with arbitrary p1 and p2 while keeping the additivity properties holding, f1 and f2 must be chosen to satisfy Eq. (4) according to the variations of p1 and p2. Maybe we can use zoom lenses to fulfill the requirement, however, it is inconvenient. In order to explain the problem more clearly, the optical setup of Fig. 2 with parameters z1 = 21.80mm, z2 = 109.20mm and f1 = f2 = 200mm are used to carry out an experiment. Fig. 3(a) is the output image when a square aperture with the size of 3mm × 3mm is set at the input plane. By contrast with the experimental result, a numerical simulation is performed by using the calculated values of p1 = 0.3 and p2 = 0.7, which are calculated according to the values of z1 and z2 respectively. Fig. 3(b) is the simulation result obtained by implementing the FRT's of the square aperture with orders 0.3 and 0.7 successively. Comparing the experimental result shown in Fig. 3(a) with the simulation result shown in Fig. 3(b), it is found

Fig. 2. Schematic illustration of cascading of two stages of optical fractional Fourier transform with two Lohmann-single-lens configurations.

(b)

Fig. 3. (a) Experimental result obtained in the optical setup shown in Fig. 2 with parameters z1 = 21.80mm, z2 = 109.20mm, i.e., p1 = 0.3, p2 = 0.7; (b) Simulation result obtained by implementing the FRT’s of the input object with orders p1 = 0.3, p2 = 0.7 one after the other.

that the computer simulation result do not coincide with the experimental one because the cascading condition is not satisfied. Now the question is how to compute the cascaded optical FRT's while the cascading condition is not satisfied. The solution will be given in the following section. 3. The method for computing the cascaded optical FRT's We only analyze the one-dimensional case for simplicity. The result acquired is also valid for two-dimensional case. Inserting the scale factor S into Eq. (2), the equation is rewritten as h i

iπ 2 p 2 F ½f ðx0 Þ ¼ C∫f ðx0 Þ exp 2 x0 cotðϕÞ þ x cotðϕÞ−2x0 x cscðϕÞ dx0 : S ð5Þ

Fig. 4. Values for p2′ versus p2 in three different cases.

J. Wang et al. / Optics Communications 285 (2012) 997–1000 Table 1 Parameters used for performing the numerical simulations and the experiments. Case

Aperture size (mm × mm)

p1

z1(mm)

p2

z2(mm)

ϕ′2

p2′

1 2 3

3×3 2×2 2×2

0.3 0.6 0.8

21.80 82.44 138.20

0.7 0.8 0.6

109.20 138.20 82.44

1.3167 1.3011 0.8640

0.8383 0.8283 0.5500

After implementing the first stage of optical FRT in the optical setup shown in Fig. 2, we obtain i

iπ h 2 p 2 F 1 ½f ðx0 ÞðkÞ ¼ C 1 ∫f ðx0 Þ exp x0 cotðϕ1 Þ þ k cotðϕ1 Þ−2x0 k cscðϕ1 Þ dx0 : 2 S1

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the input plane, it will be impossible to get the correct FRT output simply by scaling the output plane. Thus, we must use the right scaling factor at the input plane. For this reason, Shutian Liu et al. [11] proposed a method to solve the scale problem. They suggested the exact optical FRT can be obtained by using a scaled function f(sx) instead of the original input function f(x) as a new input function, where s is the reciprocal of S. Then we can implement the first step of the optical FRT in the optical setup of Fig. 2 exactly by substituting f(x0) with f(s1x0), and the process can be expressed as

p

p

F 1 ½f ðx0 ÞðkÞ ¼ F 1 ½f ðs1 x0 ÞðK 1 Þ;

ð7Þ

ð6Þ As is mentioned above, the optical FRT of f(x) actually indicates a scaled FRT, which is the FRT of the function f(Sx). Unfortunately, the FRT operation is scale variant. If one uses the wrong scaling factor at

where K1 = k/S1, s1 = 1/S1. The approach to calculate the second step of the optical FRT is what we really concern about.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 5. Simulation results (left column) obtained by implementing the FRT’s of the input image with orders p1 and p2 one after the other. Experimental results (middle column) obtained in the setup of Fig. 2 with parameters z1 and z2. Simulation results (right column) obtained by implementing the FRT’s of the input image with orders p1 and p2′ one after the other.

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J. Wang et al. / Optics Communications 285 (2012) 997–1000

Letting GðK 1 Þ ¼ F p1 ½f ðs1 x0 ÞðK 1 Þ, K2 = k/S2, the result of the second stage of the optical FRT may be written as p

F 2 ½GðK 1 Þ ¼ F

p2

 

  k K S p ¼F 2 G 2 2 : G S1 S1

ð8Þ

According to similarity theorem of the FRT, we have  39 8 2

  < X2 cos ϕ2′ = S2 4 5 cot 1− K ¼ C ðϕ2 Þ exp i F G : 2 S1 2 cosðϕ2 Þ ;  1 0 ′ sin ϕ2 p2′ A;  F fGðK 2 Þg@X sinðϕ2 Þ

the computer-simulation and experimental results obtained in different cases. Obviously, the simulation images (shown in Fig. 5(a, d, g)) obtained by implementing the FRT’s of the input image with orders p1 and p2 successively differ from the experimental images (shown in Fig. 5(b, e, h)). On the contrary, the simulation results (shown in Fig. 5 (c, f, i)) obtained by using our method are in good agreement with the experimental results.

p2

5. Conclusion ð9Þ

where X = x/S2, ϕ2 = p2π/2, ϕ′2 = p′2π/2 and ϕ′2 = arctan[(S2/S1) 2 tan (ϕ2)]. Thus it can be seen that ϕ′2 is a new transform angle. It is related not only to the transform angle ϕ2 but also to the scale factors determined by the optical setups which constitute the cascaded system. Therefore, p2′ should be used as the new transform order of the second step of optical FRT instead of using p2. Fig. 4 shows the values of p2′ as a function of p2 when f1 = f2 = 200mm and the wavelength is 532nm. Obviously p2′ increases with p2 and reaches a maximum p2′ = 1 when p2 is equal to 1. The three curves indicate the different increase behaviors of p2′ when the values of p1 are 0.2, 0.5 and 0.8 respectively. We mark the curves with dots to show the cases in which the cascading conditions are satisfied. To obtain an exact result, a multiplication with a constant complex multiplier should be implemented. If we only intend to obtain an intensity pattern at the output plane, the complex constant related to ϕ2 can be ignored. Then the intensity of the output image is  12  0   sin ϕ′2  p′2  A : I ¼ F fGðK 2 Þg@X sinðϕ2 Þ  

To conclude, we have analyzed the problem of computing the cascading of the optical FRT’s with different variable scales and provided a solution by taking the optical configuration constituted of two Lohamann-single-lens setups as an example. Comparisons of numerical simulation and experimental results are presented to show the feasibility of our method. For calculating the cascading of two stages of the optical FRT’s with arbitrary orders p1 and p2, the first stage of optical FRT should be implemented by substituting the original function f(x) with a scaled function f(sx) to obtain an exact optical FRT with order p1 of the input object. A second step of FRT with order p2′ instead of order p2 should be followed. The value of p2′ can be calculated according to the equation p′2 = 2 arctan[(S2/S1) 2 tan(ϕ2)]/π. The output results obtained by using our method may have an additional quadratic phase distribution. However, it can be ignored if we only intend to obtain the intensity patterns at the output plane. Because the amplitude and phase results obtained by implementing the first stage of the optical FRT are accurate, the method is capable of performing numerical simulations of spatial-variant filtering implemented by thin lenses. It will be helpful for design and evaluation the FRT filtering systems.

References 4. Numerical simulations and experimental verification As the FRT plays an important role in the area of optics and signal processing, the discretization of the FRT has attracted a considerable amount of attention. To date, various methods have been proposed. We use the method proposed by Soo-Chang Pei and Min-Hung Yeh [12] and improved by Candan [13] to perform the numerical simulations since this method preserves many properties of continuous transform and approximates the transform quite closely. The optical system shown in Fig. 2 is taken into account again. Assuming the wavelength λ is 532 nm, and the focal lengths of the first and the second lens are both 200 mm. Table 1 shows the parameters used in numerical simulations. The optical systems are also aligned according to these parameters to verify our method. Fig. 5 shows

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