Focal spot enhanced by artificial iterative phase conjugated twin-line planar array

Ting Li 1 Department of underwater weaponry and chemical defense, Dalian Naval Academy

Dalian China Zhu kou Department of underwater weaponry and chemical defense, Dalian Naval Academy

Dalian China JiajingWang Department of underwater weaponry and chemical defense, Dalian Naval Academy

Dalian China

ABSTRACT It has been proven that the artificial iterative phase conjugated linear array can enhance the focus. In this paper, twin-line planar array based on artificial iterative phase conjugated processing focuses a point-like source in a homogenous medium. Theory analysis and numeri- cal simulations give conclusions. Contrasting with phase conjugation, artificial iterative phase conjugated twin-line planar array focuses a small focal spot and low sidelobes. Meanwhile, focal size and sidelobe amplitude decrease as iterate number increases. Furthermore, the focal spot size becomes sharp if the distance between the two linear array is larger than half a wavelength.

1. INTRODUCTION

Getting an exact location of the source is always a popular topic for noise imaging and location domain [1-9], especially for low-frequency case[10].Phase conjugation is also referred to as time reversal in time domain. It can refocus the incident wave back and further achieving self-adaptive focusing. It is a popular method in acoustic focusing and imaging. However, because of the diffrac- tion limitation, focal spot size focused by phase conjugation is larger than half a wavelength. Artifi- cial iterative phase conjugated processing has been proven to enhance the focus and the diffraction limitation is overcome using uniform linear array[11]. Since the twin-line arrays can distinguishing the port/starboard problem, it is widely used in source location. Using a twin-line arrays to focus a point-like source by phase conjugation and artificial iterative phase conjugated processing may be an interesting research. He has treated the twin-line array as the planar array to focus a source and points out that the optimum space between two single lines is the one fourth of the wavelength[12]. In this paper, we employ the twin-line planar array to focus a point-like source in the free space us- ing phase conjugation and artificial iterative phase conjugated processing. Conclusions come out that focal spot calculated by artificial iterative phase conjugated processing is smaller than that cal- culated by phase conjugation. Increasing the iteration number, focal spot becomes sharp when the iteration number is a digit. If the space of the twin-line is larger than half a wavelength, the focus calculated by artificial iterative phase conjugated processing performs better.

1 litingyouxiang@sina.com

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2. ARTICIAL ITERATIVE PHASE CONJUGATED PROCESSING USING TWIN-LINE PLANAR ARRAY

s r In an infinite, homogeneous and isotropic medium, the point-like source is located at in Car- tesian coordinates in figure 1. The received sound is calculated by artificial iterative phase conju- gated processing (AIPCP) by computer[11].

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Figure 1: Sketch map of the linear array, source, and field points In the Cartesian coordinates as shown in figure 1, source and field points are in the same plane

s r s R (XOZ plane). The distance between the point-like source and the origin point is . The angle

s R s  s  between and Z axis is denoted as in the YOZ plane and the azimuth angle is . The dis- r R tance between field point and transducer located at the origin of the coordinate is and the azi-  y d 0 N muth angle is . The space of two linear array is . Each linear array has elements spaced in

x d . The received sound are then phase conjugated and send back to the field. There two receiving and sending forms: the monopole one and the dipole one described as follows. If it is sent by monopole transducers, then this array form of PCM is called PC/M. For transducer j

        , , , , , , * / r r r r r r j j s j G G I M jPC  (1)

The normal derivate of pressure is conjugated and emitted by dipole sources. This array form of PCM is called PC/D. For transducer j

     



     r r r r r r

*

, ,

, , , ,

G G I jPC (2)

j j s j

D / 



n

n

    , , j s r r G s r j r where denotes the angular frequency, is the Green function from to ,

s P and denotes the sound pressure radiated by the source. There are two forms of AIPCP, the monopole one and the dipole one described as follows after N iteration loops:

      

     r r r r r r j j s j G G I PC (3)

1 N * M / j N , , , , , ,

     



1 *

N







 

G G I    r r r r r r

, ,

, , , ,

j j s j (4)

N jPC/D

  

  





n

n

 where denotes the normal derivative operator of the array.

n 

Taking the two element array located at Y axis for example, the sound pressure received by the mth element is described as follows.

In the far field, according to Fraunhofer far-field approximation

       

d m R jk P      r r s

cos sin 1 exp ,

1 1 π 4

s s y s m R

1

s

(5)

       

     r r (6)

cos sin 1 exp , R

d m R jk G

1 1 π 4

y m

1

Taking the linear array located on the X-axis for example, the sound pressure received by the nth element is described as follows:

       

d n R jk P      r r s

sin sin 1 exp ,

1 2 π 4

s s x s n R

1

s

(7)

       

     r r

sin sin 1 R exp , R

d n jk G x n

1 2 π 4

1

The normalized beam function of the twin-line planar array using AIPCP in the far-field ：

     

  



, , ,

D D D

1 2

   

    

                                 

N 1 π sin cos sin cos sin 2

d

y s s

   

    

N 1 π sin cos sin cos 2sin

d

y s s

   

         

N 1 π sin sin sin sin sin

d N

x s s

0

   

N 1 π sin sin sin sin sin

d N

x s s

0

(8) The focal spot size is described by the full width at half maximum (FWHM) when it falls to half of the peak height, that is, the half maximum width or half peak width.

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R r R R N d        (9)

3 sin sin 2π 1

1 1 s1

s h s

 

y

According to the equation 9, increasing the number of iterations N can reduce the focal spot as other facts remain unchanged. When the iteration number N remain the same, increasing the space of the twin line or element number can also reduce the focal spot size.

3. FOCUSING ANALYSIS BY NUMERICAL SIMULATION

In this section, a point-like source locates at the origin of the Cartesian coordinate system in the free space. Focal spot size by PCM and AIPCP is discussed in quite near distance, near field and far field of the source. Effects of the array parameter and iteration number are also discussed for AIPCP.

3.1. Criterion of near field and far field

The twin-line planar array has two linear arrays that parallel to the x-axis and has Z-axis sym- metry as shown in figure 2. The length of linear array is 12 meters. A point-like source located at the origin of the Cartesian coordinates sends a single frequency wave at 344Hz and the wavelength is 1 meter. According to the criterion of near field and far field shown as follows:

1.356 Z L  Quite near field:

2 1.356 L L Z 

   Near field:

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1.356 L Z L 

2

   Far field:

3.2. Comparisons between PCM and AIPCP

The distance between the two linear array is 0.25m and the element distance of each linear array is also 0.25m as shown in figure 2. The twin-line array locates at 8m, 40m and 1000m away from the source. They are corresponding to a quite near distance, a near field and a far field away from source according to the criterion in section 3.1.

Figure 2: Elements location

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Figure 3: Comparisons between PCM and AIPCP recorded at 8m away from the source In Figure 3, the planar array locate at 8m away from the source, AIPCP performs better than PCM both in focal spot size and sidelobe distance. AIPC/M and AIPC/D have the same perfor- mance at the same iteration number. If iteration number becomes large, the focal spot size becomes sharp and amplitude of sidelobe becomes big.

Figure 4: Comparisons between PCM and AIPCP recorded at 40m away from the source

Figure 5: Comparisons between PCM and AIPCP recorded at 1000m away from the source. In the near and far field, AIPCP performs better than PCM both in focal spot size and sidelobe distance. AIPC/M and AIPC/D have the same performance at the same iteration number. The focus unchanged when the iteration number increases from 6 to13.

40m

3.3. Comparisons between AIPCP with different array parameters and different iteration number

1) Different iteration number at different distance away from the source Since the focus unchanged when the iteration number increases from 6 to13 in the near and far field, iteration number is discussed from 2 to 6.

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Figure 6: Comparisons with different iteration number for AIPC/M recorded at 8m away from

the source.

Figure 7: Comparisons with different iteration number for AIPC/M recorded at 40m away

from the source.

Figure 8: Comparisons with different iteration number for AIPC/M recorded at 1000m away

from the source. When iteration number increases, the focal spot size decreases as shown in Figure 6 to 8. The fo- cal spot is enhanced by increasing the iteration number.

2) Different element parameters in the same iteration number In the legends of this part,”0.25AIPC/M3” means the focus calculated by AIPC/M with iteration number 3 when the space between two linear arrays is 0.25m.

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Figure 9: Comparisons with different element space for AIPC/M and AIPC/D at 8m away

from the source.

Figure 10: Comparisons with different element space for AIPC/M and AIPC/D at 40m

away from the source.

Fm. A 40m

Figure 11: Comparisons with different element space for AIPC/M and AIPC/D at 1000m

away from the source. When the space between two linear arrays is larger than 0.5m, the focal spot size is small. In the near and far field, focus by AIPC/M and AIPC/D are similar in the same iteration number.

4. CONCLUSIONS

To check the ability of artificial iterative phase conjugated twin-line array, we analyzed it by theory analysis and numerical simulation. conclusions come out that contrasting with phase conju-

Fm. A

gation, artificial iterative phase conjugated twin-line planar array focuses a small focal spot and low sidelobes. Increasing the iteration number can enhance the focus. For a small iteration number, the focal spot size deduces quickly as iteration number increases. Meanwhile, when the space between the two linear arrays is larger than half a wavelength, AIPCP performs better. 5. ACKNOWLEDGEMENTS

We would like to gratefully acknowledge the support of the College Development Found of Da- lian Naval Academy (Grant No.DJYKYKT2021-024) and the Key discipline construction project.

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