## Comparison of iterative angular spectrum and optimal rotation angle methods in designing beam-fanners

Optics Express, Vol. 17, Issue 17, pp. 14825-14831 (2009)

http://dx.doi.org/10.1364/OE.17.014825

Acrobat PDF (185 KB)

### Abstract

We compare the iterative angular spectrum (IAS) and the optimal rotation angle (ORA) methods in designing two-dimensional finite aperture diffractive optical elements (FADOEs) used as beamfanners. The transfer functions of both methods are compared analytically in the spatial frequency domain. We have designed several structures of 1-to-4 and 1-to-6 beamfanners to investigate the differences in the performance of the beamfanners designed by ORA method for near field operation. Using the three-dimensional finite difference time-domain (3-D FDTD) method with perfect matched layer (PML) absorbing boundary condition (ABC), the diffraction efficiency is calculated for each designed FADOE and the corresponding values are compared.

© 2009 OSA

## 1. Introduction

1. J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express **7**(6), 237–242 (2000). [CrossRef] [PubMed]

2. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A **14**(1), 34–43 (1997). [CrossRef]

3. D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A **15**(6), 1599–1607 (1998). [CrossRef]

1. J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express **7**(6), 237–242 (2000). [CrossRef] [PubMed]

3. D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A **15**(6), 1599–1607 (1998). [CrossRef]

5. J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. **36**(32), 8435–8444 (1997). [CrossRef]

6. J. Stigwall and J. Bengtsson, “Design of array of diffractive optical elements with inter-element coherent fan-outs,” Opt. Express **12**(23), 5675–5683 (2004). [CrossRef] [PubMed]

8. F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A **5**(7), 1058–1065 (1988). [CrossRef]

9. F. Di, Y. Yingbai, J. Guofan, T. Qiaofeng, and H. Liu, “Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm,” J. Opt. Soc. Am. A **20**(9), 1739–1746 (2003). [CrossRef]

10. T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express **16**(10), 7203–7213 (2008). [CrossRef] [PubMed]

11. S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express **8**, 706–722 (2001). [CrossRef]

## 2. Iterative angular spectrum method

*z*=

*z*

_{obs}. It is assumed that a plane wave propagating along the z-axis interacts with the DOE at the input plane and it is expected to produce the desired pattern at the observation plane.

_{0}) and includes DOE, while the AS region extends from the output plane located at

*z*=0 to the observation plane (P

_{1}). The thickness of DOE region is equal to the maximum etch depth (

*d*

_{max}). The refraction index of the dielectric material used is

*n*

_{2}, while the surrounding medium is assumed to be air (

*n*

_{1}=1).

*τ*=2

*n*

_{1}/(

*n*+

_{1}*n*

_{2}) is Fresnel transmission coefficient, Π(

*x*/L

*,*

_{x}*y*/L

*) is equal to 1 for any point inside a rectangular aperture of L*

_{y}*×L*

_{x}*, and equal to zero for other points. The phase function according to phase only approximation is φ(*

_{y}*x*,

*y*)=

*k*

_{0}Δ

*nd*(

*x*,

*y*), where Δ

*n*is the difference between the two refractive indices,

*k*

_{0}=2π/λ, λ is the wavelength in free space and

*d*(

*x*,

*y*) is the etch depth. We assume an incident plane wave,

**u**

_{inc}, that is propagating normal to the aperture. The wave at the output plane of the first region is

**u**

_{0}=

*t*(

*x*,

*y*)

**u**

_{inc}. The angular spectrum of

**u**

_{0},

*i.e.*,

**U**

_{0}, is defined as [4]Using the fast Fourier transform (FFT) technique,

**U**

_{0}can be computed. Assuming a square aperture, L

*=L*

_{x}*=L, the distance between two adjacent points in spatial frequency domain is*

_{y}*δf*=

*δf*=

_{x}*δf*=1/L. To improve the resolution in the frequency domain, we can compute the FFT on an area larger than the DOE area. This is particularly useful if the desired intensity profile in the observation plane extends beyond the DOE area.

_{y}**u**

_{1}, can be obtained bywhere

*k*

_{2}=

*n*

_{2}

*k*

_{0}and λ

_{2}=λ/

*n*

_{2}are the wave-number and wavelength in the AS region, respectively [4]. Using the inverse fast Fourier transform (IFFT),

**u**

_{1}can be calculated as

**u**

_{1}=IFFT(H

**U**

_{0}), whereis the angular spectrum transfer function. The maximum spatial frequency is

*f*

_{max}=1/(2

*δ*), where

*δ*is the minimum feature size of the DOE. To consider all propagating waves, the condition

*f*

_{max}>1/λ

_{2}, which is equivalent to δ

*<*λ

_{2}/2, should be satisfied.

**u**

_{1}, is modified such that the difference between the obtained amplitude and the desired amplitude is reduced. The modified field distribution is then propagated backward to find the corresponding field distribution at the DOE output plane. Also, using the phase of the back going filed, the DOE structure is modified to satisfy the new field distribution. This modified DOE is quantized and used to repeat the above procedure.

## 3. Optimal rotation angle method

5. J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. **36**(32), 8435–8444 (1997). [CrossRef]

6. J. Stigwall and J. Bengtsson, “Design of array of diffractive optical elements with inter-element coherent fan-outs,” Opt. Express **12**(23), 5675–5683 (2004). [CrossRef] [PubMed]

*k*-th cell iswhere

*k*-th pixel, (

*x*,

_{k}*y*), respectively, and φ

_{k}*is the amount of phase modulation produced by the DOE. Considering the*

_{k}*m*-th spot in the observation plane, located at (

*x*,

_{m}*y*,

_{m}*z*

_{obs}), the field at this spot can be represented aswhere h

*is a complex value quantity that relates the field at the*

_{k,m}*k*-th cell of the DOE to the field at the

*m*-th spot of the observation plane. Using the Helmholtz-Kirchhoff integral, it is obtained as [5

5. J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. **36**(32), 8435–8444 (1997). [CrossRef]

*x*-

_{k}*x*)

_{m}^{2}+(

*y*-

_{k}*y*)

_{m}^{2}+

*z*

_{obs}

^{2}]

^{½},

*a*and

*b*are the

*x*and

*y*dimensions of each DOE cell, respectively.

*φ*and this procedure is repeated until the phases of the cells are converged. For the formula of Δ

_{k}*φ*, the reader is reffered to [5

_{k}**36**(32), 8435–8444 (1997). [CrossRef]

## 4. Transfer functions

*z*

_{obs}is changed. We use Eq. (4) as the transfer function of IAS method. For the ORA method, we start with the Helmholtz-Kirchhoff integral [4] and obtain the spatial impulse response of the propagation medium between DOE and observation plane aswhere

*r*=(

*x*

^{2}+

*y*

^{2}+

*z*

_{obs}

^{2})

^{1/2}. The above spatial impulse response is a continuous function of space, while

*h*in Eq. (7) is a discrete quantity. Equation (7) can be obtained from Eq. (8) using sampling theorem [4]. Next,

_{k,m}**u**

_{1}can be represented by using the 2-D convolution asWe use the 2-D fast Fourier transform (FFT) to calculate the ORA transfer function in the spatial frequency domain. The phase of both transfer functions are the same, but their amplitudes are different.

_{obs}smaller than 10λ the frequencies near edge are amplified. Second, for

*z*

_{obs}greater than 10λ, the spatial frequency bandwidth of the ORA transfer function becomes smaller (See Fig. 3 ). The first difference produces high spatial frequency components in the DOE structures designed by ORA method, which can destructively affect the field distribution. Figure 4 shows the effect of these high spatial frequency components in two typical DOE profiles. The second difference eliminates high spatial frequencies that are important specially in cases that the spots are located at a large distance from each other compared with the DOE dimensions.

## 5. Design and analysis

_{0}made of silicon with a refractive index of

*n*

_{2}=3.4 and λ

_{0}is set to 5μm. The spatial quantization steps along the three directions (

*x*,

*y*and

*z*),

*i.e.*,

*a*,

*b*, and δ, and the etch depth quantization step are chosen equal to λ

_{0}/25=.0.2μm. As mentioned in section 2, to improve the frequency resolution, we put additional zeroes around the above matrix to increase its dimension by a factor of 10. As a result, the spatial frequency resolution will be δ

*=1/(100λ*

_{f}_{0}) and therefore, the transmission function is represented by a 2500×2500 complex-valued matrix.

*x*-

*x*)

_{m}^{2}+(

*y*-

*y*)

_{m}^{2}]/(2

*w*

_{0}

^{2})]}, where (

*x*,

_{m}*y*) corresponds to the center of the

_{m}*m*-th spot. In our simulations, we have used

*w*

_{0}= 0.14λ. This width is very small compared to the distance between the spot points (3λ to 8λ). Therefore, the two distribution profiles are approximately the same in numerical calculations.

12. D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave and Guided Wave Letters **7**(7), 184–186 (1997). [CrossRef]

*i.e.*, δ

*=δ*

_{x}*=δ*

_{y}*=δ. Also, δ*

_{z}*t*=δ/(2

*c*), where

*c*is the velocity of light in free space, was chosen as the time step. The depths of PML and scattered field were set to 7

*δ*. The incident field was assumed to be a TEM wave with a sharp Gaussian time varying amplitude,

*E*

_{xinc}=exp{-(

*t*-

*t*

_{d})

^{2}/(2

*w*

_{t}

^{2})}, where

*t*

_{d}and

*w*

_{t}are the time delay and the spread parameter of the Gaussian pulse, respectively. Using the FDTD method, the distributions of the electric and magnetic fields were obtained for 200 time steps.

*P*

_{0}plane by the FDTD method, the angular spectrum technique was used to obtain the fields at the observation plane,

*i.e.*,

*P*

_{1}. The distribution of light intensity,

*I*, at

*P*

_{1}is then calculated. The diffraction efficiency defined asis then computed, where

*W*is the detectors window which consists of four (six) circular areas of radius equal to λ

_{0}, centered at the four (six) spots positions.

_{4}and S

_{6}are separation distance of adjacent spots in the 1-to-4 and 1-to-6 beamfanners, respectively. It can be seen, that the IAS method provides better designs than the ORA.

## 6. Conclusions

*z*

_{obs}) as an important variable, we found, that if

*z*

_{obs}is less than or equal to 10λ, the high spatial frequency components appear stronger in the ORA method. This results in increasing the number of discontinuities in the DOE phase profile. Vice versa, by increasing

*z*

_{obs}, the ORA frequency bandwidth becomes smaller. This puts a limit in the separating distance between the spots.

## Acknowledgments

## References and links

1. | J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express |

2. | D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A |

3. | D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A |

4. | J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005). |

5. | J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. |

6. | J. Stigwall and J. Bengtsson, “Design of array of diffractive optical elements with inter-element coherent fan-outs,” Opt. Express |

7. | R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) |

8. | F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A |

9. | F. Di, Y. Yingbai, J. Guofan, T. Qiaofeng, and H. Liu, “Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm,” J. Opt. Soc. Am. A |

10. | T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express |

11. | S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express |

12. | D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave and Guided Wave Letters |

13. | D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press series on RF and microwave technology, 2000. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1220) Diffraction and gratings : Apertures

(050.1970) Diffraction and gratings : Diffractive optics

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: May 4, 2009

Revised Manuscript: July 1, 2009

Manuscript Accepted: July 24, 2009

Published: August 6, 2009

**Citation**

Seyyed H. Kazemi, Mir M. Mirsalehi, and Amir R. Attari, "Comparison of iterative angular spectrum and optimal rotation angle methods in designing beam-fanners," Opt. Express **17**, 14825-14831 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14825

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### References

- J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express 7(6), 237–242 (2000). [CrossRef] [PubMed]
- D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14(1), 34–43 (1997). [CrossRef]
- D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15(6), 1599–1607 (1998). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005).
- J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. 36(32), 8435–8444 (1997). [CrossRef]
- J. Stigwall and J. Bengtsson, “Design of array of diffractive optical elements with inter-element coherent fan-outs,” Opt. Express 12(23), 5675–5683 (2004). [CrossRef] [PubMed]
- R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–250 (1972).
- F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5(7), 1058–1065 (1988). [CrossRef]
- F. Di, Y. Yingbai, J. Guofan, T. Qiaofeng, and H. Liu, “Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm,” J. Opt. Soc. Am. A 20(9), 1739–1746 (2003). [CrossRef]
- T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008). [CrossRef] [PubMed]
- S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express 8, 706–722 (2001). [CrossRef]
- D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave and Guided Wave Letters 7(7), 184–186 (1997). [CrossRef]
- D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press series on RF and microwave technology, 2000.

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