Verification methods of the shaping effect by simulation in design process of a diffractive optical element

Verification methods of the shaping effect by simulation in design process of a diffractive optical element

Optik - International Journal for Light and Electron Optics 178 (2019) 310–315 Contents lists available at ScienceDirect Optik journal homepage: www...

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Optik - International Journal for Light and Electron Optics 178 (2019) 310–315

Contents lists available at ScienceDirect

Optik journal homepage: www.elsevier.com/locate/ijleo

Original research article

Verification methods of the shaping effect by simulation in design process of a diffractive optical element ⁎

Shuzhen Nie , Jin Yu, Zhongwei Fan

T



Department of Optoelectronic Engineering, Academy of Opto-Electronics, Chinese Academy of Sciences, Beijing 100094, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Diffractive optics Laser beam shaping DOE

In design process of a diffractive optical element (DOE), the shaping effect of calculated phase data is the key to determine whether the DOE can meet the requirement. A spatial light modulator (SLM) usually is chosen as the tool to detect real output beam and the phase data can be accepted if the detected results meet the designed requirements. In this paper, we propose verification methods by simulation in design process of a DOE which do not need to build an experimental system and are easy to carry out. Fresnel, Kirchhoff, Rayleigh-Sommerfeld and the angular spectrum diffraction theories are deduced briefly by Nyquist sampling theorem and applied to simulate the shaped effects. The results of converting a Gaussian beam to a micrometer-scale uniform flattop beam are presented, and the results show that the shaping effects of the DOE calculated by the above diffraction theories are similar which can all meet the designed requirements. The verification methods are feasible and will provide valid and rapid ways to estimate the phase data.

1. Introduction Diffractive optics refers to the technology of producing surface-relief, and computer-generated diffractive patterns directly on the optical material by means of fabrication technology in the microelectronics industry, offering a very flexible way of controlling and shaping of phase fronts or bending of rays [1–3]. It is a developed branch in optics based on the principle of optical diffraction. A diffractive optical element (DOE) is consist of computer-generated diffractive patterns and modulates the phase of an incident wave front. Therefore, several optical functions can be simultaneously achieved by only one DOE. The DOEs have many advantages, such as light weight, small volume, and cheap fabrication. Beam shaping, as part of the field of diffractive optics, is used widely for optical conversions [4,5]. DOEs could easily convert the laser beams into needed distributions, so they are widely used in many applications, such as laser welding, laser-simulated etching, laser scanning and branding [6–9]. The design process of a DOE can be considered as a phase-retrieval problem [10]. Based on the input and output laser distributions, how to solve the optimal phase date of the DOE is the final goal. Various algorithms have been proposed to get the optimal date, such as the direct binary search algorithm, the Gerchberg-Saxton (GS) algorithm, and some modified GS algorithms [11,12]. Nevertheless, the real shaping effect at the output plane by a DOE is the key to determine whether the calculated phase data is feasible. The shaping effect can be tested by a SLM [13] which needs to build an experimental system. The phase date is loaded into the SLM and the real distribution of the shaped beam can be tested at the output plane. The experimental results can verify the correctness of the calculated phase data of the DOE. In this paper, we present some verification methods of the shaping effect by ⁎ Corresponding authors at: Department of Optoelectronic Engineering, Academy of Opto-Electronics, Chinese Academy of Sciences, Beijing 100094, China. E-mail addresses: [email protected] (S. Nie), [email protected] (Z. Fan).

https://doi.org/10.1016/j.ijleo.2018.10.010 Received 9 August 2018; Accepted 2 October 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 178 (2019) 310–315

S. Nie et al.

Fig. 1. Reference coordinate of the diffractive calculation.

simulation in design process of a DOE, which do not need to build an experimental system and are easy to carry out. These methods are based on simulation and make use of Fresnel, Kirchhoff, Rayleigh-Sommerfeld and the angular spectrum diffraction theories [14], which are chosen in different input conditions according to Nyquist sampling theorem. 2. Principle The reference coordinate of the diffractive calculation is shown in Fig. 1. The input beam is irradiated on the DOE and the shaped beam can be achieved at the output plane. We assume that the coordinate at the DOE and output plane are denoted as (x 0 , y0 ) and (x , y ) , respectively. Generally, the phase-only DOE is considered to maximize the diffraction efficiency. Therefore, the transmission function to the DOE can be written as

t (x 0 , y0 ) = exp [iφ (x 0 , y0 )]

(1)

where φ (x 0 , y0 ) ranged from 0 to 2π represents the phase distribution of the DOE. Then, the light field immediately behind the DOE is expressed as follows:

U (x 0 , y0 ) = Uin (x 0 , y0 ) exp [iφ (x 0 , y0 )]

(2)

Finally, the complex amplitude distribution U (x , y ) at the output plane can be given by the Fresnel, Kirchhoff, RayleighSommerfeld and the angular spectrum diffraction theories which are discussed below. The shaping effect of the DOE can be verified by the simulation results according to Nyquist sampling theorem. 2.1. Fresnel diffraction integral Fresnel diffraction integral can be expressed as

U (x , y ) =

exp(ikd ) iλd



∫ ∫ U0 (x 0 , y0 ) exp { 2ikd [(x 0 − x )2 + (y0 − y)2] } dx 0 dy0 −∞

(3)

where λ is the wavelength of the incident light, k = 2π / λ is the wave number in the free space, d is the propagation distance. The Eq. (3) can be deduced by Fourier transform and the laws of convolution as

Fig. 2. The simulated output beam distribution: (a) 3D distribution (b) 2D profile. 311

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Fig. 3. The calculated phase data of the DOE.

Fig. 4. The simulated output beam calculated by Fresnel diffraction transfer function: (a) 3D distribution (b) 2D profile.

Fig. 5. The simulated output beam calculated by analytical solution of Fresnel diffraction transfer function: (a) 3D distribution (b) 2D profile.

F {U (x , y )} = F {U0 (x 0 , y0 )} F ⎧ ⎨ ⎩

exp(ikd ) ik exp ⎡ (x 2 + y 2 )⎤ ⎫ iλd 2 ⎣ d ⎦⎬ ⎭

(4)

where ‘F{}’represents the two-dimension FFT algorithm operation. The frequency coordinate is denoted as (fx , f y ) and Fresnel diffraction transfer function is shown as

HF (fx , f y ) = F ⎧ ⎨ ⎩

exp(ikd ) ik exp ⎡ (x 2 + y 2 )⎤ ⎫ iλd ⎣ 2d ⎦⎬ ⎭

(5) 312

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Fig. 6. The simulated output beam calculated by Kirchhoff diffraction transfer function: (a) 3D distribution (b) 2D profile.

Fig. 7. The simulated output beam calculated by Rayleigh-Sommerfeld diffraction transfer function: (a) 3D distribution (b) 2D profile.

Fig. 8. The simulated output beam calculated by the angular spectrum diffraction transfer function: (a) 3D distribution (b) 2D profile.

Eq. (5) has analytical solution:

λ2 2 HF (fx , f y ) = exp ⎧ikd ⎡1 − (f + f y2 ) ⎤ ⎫ ⎢ ⎥ ⎨ 2 x ⎦⎬ ⎩ ⎣ ⎭

(6)

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S. Nie et al.

Therefore, the complex amplitude distribution at output plane can be expressed as

U (x , y ) = F −1 {F {U0 (x 0 , y0 )} HF (fx , f y )}

(7)

According to Nyquist sampling theorem, Eq. (7) can be used when ΔL0 ≥ plane.

Nλd . ΔL0 is the calculated width of the diffractive

2.2. Kirchhoff diffraction integral Kirchhoff diffraction integral can be expressed as ∞

U (x , y ) =

∫ ∫ U0 (x 0 , y0 ) −∞

exp (ik d 2 + (x 0 − x )2 + (y0 − y )2 ) ⎡ 1 ⎤ d dx 0 dy0 ⎢2 + 2 + (x − x ) 2 + (y − y ) 2 ⎥ iλ d 2 + (x 0 − x )2 + (y0 − y )2 d 2 0 0 ⎣ ⎦

(8)

The Eq. (8) can be deduced by Fourier transform and the laws of convolution as

F {U (x , y )} = F {U0 (x 0 , y0 )} F

⎧ exp (ik d 2 + x 2 + y 2 ) ⎡ 1 d ⎤⎫ ⎢2 + 2 + x 2 + y2 ⎥ ⎬ ⎨ iλ d 2 + x 2 + y 2 d 2 ⎦⎭ ⎣ ⎩

(9)

Kirchhoff diffraction transfer function is shown as

HJ (fx , f y ) = F

2 2 2 ⎧ exp (ik d + x + y ) ⎫ [ d 2 + x 2 + y 2 + d] ⎨ 2iλ (d 2 + x 2 + y 2 ) ⎬ ⎩ ⎭

(10)

Therefore, the complex amplitude distribution at output plane can be expressed as

U (x , y ) = F −1 {F {U0 (x 0 , y0 )} HJ (fx , f y )}

(11)

Eq. (11) can be used when Δs ≤ λ d 2 + ΔL02 /2 / ΔL0 based on Nyquist sampling theorem. Δs is the sampling interval. 2.3. Rayleigh-Sommerfeld diffraction integral Rayleigh-Sommerfeld diffraction integral can be expressed as ∞

U (x , y ) =

∫ ∫ U0 (x 0 , y0 )

d⋅exp (ik d 2 + (x 0 − x )2 + (y0 − y )2 )

−∞

iλ (d 2 + (x 0 − x )2 + (y0 − y )2)

dx 0 dy0

(12)

The Eq. (12) can be deduced by Fourier transform and the laws of convolution as

F {U (x , y )} = F {U0 (x 0 , y0 )} F

2 2 2 ⎧ d⋅exp (ik d + x + y ) ⎫ 2 2 2 ⎨ ⎬ iλ (d + x + y ) ⎩ ⎭

(13)

Rayleigh-Sommerfeld diffraction transfer function is shown as

HR (fx , f y ) = F

2 2 2 ⎧ d⋅exp (ik d + x + y ) ⎫ 2 2 2 ⎨ ⎬ iλ (d + x + y ) ⎩ ⎭

(14)

Therefore, the complex amplitude distribution at output plane can be expressed as

U (x , y ) = F −1 {F {U0 (x 0 , y0 )} HR (fx , f y )} Eq. (15) can be used when Δs ≤ λ

d2

+

ΔL02 /2 / ΔL0

(15) based on Nyquist sampling theorem

2.4. The angular spectrum diffraction integral The angular spectrum diffraction integral can be solved by Fourier transform:

F {U (x , y )} = F {U0 (x 0 , y0 )} exp (ikd 1 − (λfx )2 − (λf y )2 )

(16)

The angular spectrum diffraction transfer function is shown as

HB (fx , f y ) = exp (ikd 1 − (λfx )2 − (λf y )2 )

(17)

Therefore, the complex amplitude distribution at output plane can be expressed as

U (x , y ) = F −1 {F {U0 (x 0 , y0 )} HB (fx , f y )}

(18) 314

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According to Nyquist sampling theorem, Eq. (18) can be used when ΔL0 ≥

Nλd .

3. Design and result We have designed a DOE to shape the incident Gaussian beam into a micrometer-scale uniform flattop beam. The phase date of the DOE has been calculated by modified GS arthrogram. The radius of the Gaussian beam at 1/e2 is 4.95 mm. The wavelength is 532 nm. The DOE is 10mm × 10mm and the propagation distance is 10 mm. The diameter of the shaped flattop beam is 2 μm. The simulated output beam is shown in Fig. 2. The calculated phase date of the DOE can be achieved and shown in Fig. 3. We now apply the proposed diffractive simulation methods to verify the shaping effects of the DOE. Before simulation, we have noticed that these simulation methods of the DOE satisfy the Nyquist sampling theorem. The simulation results are shown in Fig. 4–8. From the results, we can find that the shaping effects of the DOE calculated by different diffraction theories are similar which can verify the shaping effect. The verification methods of the shaping effect by simulation in design process of a DOE are proved to be feasible and easy to carry out. 4. Conclusion In conclusion, we have proposed verification methods of the shaping effect by simulation in design process of a DOE that can be used to test the output beam distribution without building an experimental setup. The verification methods are based on Fresnel, Kirchhoff, Rayleigh-Sommerfeld and the angular spectrum diffraction theories. A micrometer-scale uniform flattop beam is designed by a DOE and the correction of the phase data has been tested by the verification methods. The shaping effects simulated by these diffraction transfer functions are similar and can prove that verification methods by simulation in design process of a DOE are valid. These methods will provide easy ways to test the shaping effects of DOEs, improve the design efficiency and guarantee the correction of the calculated phase data of DOEs. Acknowledgments This work is supported by National key research and development program of China (Grant No. 2017YFB1104500), National key scientific and research equipment development project in China (Grant No. ZDYZ2013-2), National key foundation for exploring scientific instrument in China (Grant No. 2014YQ120351) and China innovative talent promotion plans for innovation team in priority fields (Grant No. 2014RA4051). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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