Application of DBPIP to Phase Errors Detection in Coherent Beam Combination
Wang Xiaohua*a,b,c Shen Fenga,b aThe Laboratory on Adaptive Optics, Institute of Optics and Electronics,
Chinese Academy of Sciences, Chengdu 610209，China bThe Key Laboratory on Adaptive Optics, Chinese Academy of Sciences,
Chengdu 610209, China cGraduate School of Chinese Academy of Sciences, Beijing 100049, China
ABSTRACT
A new phase sensing technique is used to beam combining, which is called DBPIP (the Displacement of the Brightest Point of the Interference Pattern). DBPIP is a method of detecting piston errors between two beams based on the far-field interference pattern. It finds the position of the brightest point of the interference pattern, and then calculates the related piston errors. The advantage of DBPIP is very convenient using to detect the piston aberration in a coherent beam combination (CBC) system. Theoretically, only a few iterations can compensate the piston error completely if we get the diffractive pattern image of enough resolution. The simulation results represent that, one iteration is enough to correct the piston error to less than λ/20; if higher accuracy is required, such as less than λ/50, two iterations is enough. Moreover, this technique can also compensate the tilt aberrations of beams in a few iterations. Keywords: DBPIP, Piston Error, Phase Sensor, Coherent Combination, Adaptive Close Loop Control.
1. INTRODUCTION
In recent years, the coherent beam combining (CBC) technologies, which can combine several lower-power single frequency lasers to a single higher-power laser, have been paid much attention and research. The key technology of CBC is the phase locking. Now the active phase locking technology is a research hotspot. There are several techniques of active phase locking, such as heterodyne method [1, 2], LOCSET (Locking of Optical Coherence by Single-Detector Electronic-Frequency Tagging) [3-5], AO (Adaptive Optics) [6~8], and some passive phasing methods [9-13], etc. Both heterodyne method and the LOCSET method are based on the signal of a single intensity detector such as photo diode. There are some methods [14, 15], which are based on the far-field diffraction images, to phase multiple apertures of astronomical telescopes with segmented mirrors. These methods can also be used for beam combining. A new phase sensing technique, which is like the broadband phasing algorithm in [16], is represented in this paper, called DBPIP. DBPIP is a method of acquiring phase error between two beams from the far-field interference pattern, which converts the phase sensing to displacement measuring. It finds the brightest point of the interference pattern, and then calculates the piston error between two beams. There is a fact that the brightest point displaces linearly with the piston error, so it is easy to get the piston error with 2π periods ignored. The advantage of DBPIP is very convenient using to detect the piston errors in a CBC system. To validate its efficiency, we designed a series of simulation of combining two beams coherently. The results represent that, theoretically if we get the accurate linearly dependent coefficient and efficient resolution diffractive pattern image, just one iteration can compensate the piston aberration, with the residual error is less than λ/20; if higher accuracy is required, such as less than λ/50, two iterations is enough to compensate the piston aberration. Moreover, this technique can also compensate the tilt aberrations of beams simultaneously.
2. PRINCIPLE OF DBPIP
DBPIP comes from the methods of phasing the mirror segments of telescopes. Its principle is introduced in this section. We used rectangular shaped apertures at first to simplify the formulas calculation. Then in following sections we change the aperture shape to circle and other irregular shape to testify the conclusion.
2.1 Displacement of maximum intensity for rectangular apertures’ interference pattern
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In order to calculate easily, two rectangle shaped apertures are chosen (Figure 1). It is assumed that the two source beams have their initial phase respectively, with φ1 and φ2. The difference between φ1 and φ2 is called piston error Δφ, which should be phase-locked, and also assumed that the beams are parallel plane waves of equal intensity, polarization and frequency. So the complex amplitude of the Wavefront behind the apertures is shown as:
1
2
1 1
1 1 1 1
/ 2 ( ) / 2
0
i
i
e a x D b y bU x , y e D x a b y b
other
ϕ
ϕ
−
−
⎧ − ≤ ≤ −Δ − ≤ ≤⎪= Δ ≤ ≤ − ≤ ≤⎨⎪⎩
，， (1)
Figure 1: Rectangle shaped apertures.
From Fourier Optics, the interference pattern is the two apertures’ far-field diffractive pattern [17]. So the intensity distribution along the x axis is:
22
20
64 1 ( ) ( )( ,0) lim ( , ) sin cos2 2y
b B A x B A xI x I x yx
ϕπ→
− + + Δ⎡ ⎤= = ⎢ ⎥⎣ ⎦， ,
2k D kaA B
f fΔ
= = (2)
Where, Δφ = φ2 – φ1, I(x, 0) is the intensity distribution along the x axis, which is shown in Figure 2 .
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40
50
100
150
200
250
300
350
400
x : mm
I(x,0
) (*6
4*b2 / π
2 )
Figure 2: Intensity distribution along the x axis
where a = 7mm, λ = 1.064μm, f = 1000mm, ΔD = 0, Δφ = π/2 All the maximum and minimum intensity for the far-field pattern will be displaced due to the piston error. Their positions are given by the differential equation below,
2
2 3
( ,0) 64 1 ( ) ( )sin cos2 2
[ cos( ) cos( ) sin( ) sin( )]2 2 2 2
0
dI x b B A x B A xdx x
Bx Bx Ax Ax Bx Ax
ϕπ
ϕ ϕ ϕ ϕ
− + + Δ⎡ ⎤= ⎢ ⎥⎣ ⎦Δ Δ Δ Δ
+ − + − + + +
=
(3)
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Then the positions of the zeros for the intensity are given by the following linear relationship. (2 1) , a
nx nB A
ϕ π−Δ ± += = ± ±
+0, 1, 2. . . (4)
where, xa represent the positions of the zeros for the intensity. The positions of the bright fringe are not easy to get like above, but they have following nonlinear relationship with the piston error Δφ:
1 cos cos sin sin2 tansin sin cos cos
Bx Bx Ax Ax Bx AxBx Bx Ax Ax Bx Ax
ϕ − − − +⎛ ⎞Δ = ⎜ ⎟− + −⎝ ⎠ (5)
The relationship above is a periodic function, and approximately is equivalent to 2 , , 0, 1, 2,...
( )l
nx nK A Bϕ π π ϕ πΔ +
= − − ≤ Δ < = ± ±+
(6)
with Kl is a constant number. When assuming ΔD = 0 we get Kl = 1.34 by numerical analysis. The peak intensity should be the 0th order bright fringe, so the location of the peak is
, ( )l
xK A B
ϕ π ϕ πΔ= − − ≤ Δ <
+ (7)
That is to say, the location of the peak intensity shifts linearly with the piston error when the shape of the beam is rectangle.
2.2 Displacement of maximum intensity for circular apertures’ interference pattern
This section we change the shape of the apertures to circle. The two circular apertures are separated from each other by 16mm, shown in Figure 3.
Figure 3: Shape apertures used in analysis.
According to Fourier Optics, the intensity distribution of far-field diffraction of the apertures above is
2 2 24 1 4(2 )( , ) 2 1 cos , ba
bb
r x yrJI x y r xr fππρ ϕ ρ
ρ λ⎡ ⎤⎛ ⎞ +⎛ ⎞
= + + Δ =⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦
(8)
Where, ra (= 8mm) is half of the two apertures’ separation; rb (= 5mm) is the radius of the apertures; Δφ is the piston error between the apertures; J1(x) represents Bessel function of first kind for the 1th order. As the symmetry in the direction of y axis, the brightest point locates on the x axis. And we can get the intensity distribution along x axis by set y = 0 easily. By numerical salvation of the above equation by set y = 0, the position of the maximum point has a relation with the piston error Δφ shown in Figure 4, with f = 1000mm, λ= 1.064μm. Exactly, the relationship is nonlinear, but we can express the relationship in line approximately as
= -0.0096x ϕΔ (9)
Where the linearly dependent coefficient 0.0096 (unit: mm/rad) depends on ra, rb, f and λ. From Figure 4, we can see that the position of the peak intensity also shifts linearly with piston error when the shape of the apertures is changed to circle.
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Figure 4: Positions of the maximum point in different piston errors.
2.3 Displacement of maximum intensity for irregular apertures’ interference pattern
This section we use two irregular shaped apertures to testify if the shape of apertures has effect on the linear relationship. The shape of the apertures is showed in Figure 5, these apertures are symmetric just in y axial orientation. The diameter of inscribed circle of the hexagon in center is 1 mm. The overlaps of the hexagon and the two circles are the two irregular apertures.
Figure 5: Irregular apertures (color filled). unit: mm
Some simulations are done to get the relationship between the displacement of the brightest point x and the piston error Δφ. Assuming the wavelength λ = 1.064μm, lens’s focus f = 300mm. Let Δφ step 0.1π from -0.9π to 0.9π. For each Δφ, we simulate the two apertures’ far-field diffractive pattern, find the brightest point, and calculate its displacement from its original position which is the position when Δφ= 0. The simulation results are shown in the following figure Figure 6. The results represent that, when there is a piston error Δφ between the two apertures the brightest point shifts almost linearly to
0.0759 , ( )x ϕ π ϕ π= − Δ − < Δ < (10)
Compare the above equation (10) with the equations (7) and (9), the location of the brightest point shifts with piston error Δφ as:
, ( )Px K ϕ π ϕ π= − Δ − < Δ < (11)
Where, in an exact condition Kp is a constant number, with unit of mm/rad, depends on the shape and location of the apertures, the lens’ focus length f and the wavelength λ. The conclusion is the displacement of the brightest point has a linear relationship with the piston error between the two apertures. But this linear relationship only exists in [-π π) period, and beyond this interval their relationship is a periodic function with 2π period. If all the 2π periods were ignored, the piston error Δφ could be got by detecting the displacement x as
/ Px KϕΔ = − (12)
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If found the proper coefficient Kp and detected the displacement x, we can get the piston error accurately.
Figure 6: Results of the above simulation. Displacement of the brightest point
shifts linearly with the piston error between the two apertures.
3. SIMULATION AND RESULTS
A simulation is designed to combine two laser beams using DBPIP detecting their piston error. In this simulation the two circular beams have a diameter of 2mm and a wavelength of 1.064μm. Three hexagon shaped lenslets arranged side-by-side were used as Wavefront sensor. All the lenslets have the same focus length of 300mm. The shapes of beams and lenslets are shown in Figure 7. The simulated CCD camera used to record the far-field diffraction pattern image has 256×256 pixels with 8 bits, each pixel has an area of 10.6μm×10.6μm.
Figure 7: Shapes of the 2 beams (circles) and 3 lenslets (hexagons). unit: mm
In coherent beam combination, piston errors between beams are the leading phase error, tilt errors are secondary. The two lenslets on sides are used to detect the beams’ tilt, and the central one is used to detect the piston error.
3.1 Simulation in case with piston only
First we consider the beams have just piston error. The flow chart of the simulation is shown in Figure 8.
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Figure 8: Flow chart of the simulation.
The details of simulation steps are as follow: Step 1: Acquire the coefficient Kp in equation (12). From equation (10), Kp = 0.0759 mm/rad. Step 2: Generate a group of random phases, calculate the two beams’ Wavefront and their far-field interference pattern. Then calculate the Wavefront behind the lenslets and the diffraction patterns on the focus plane of the lenslets. The symbol phase=[φ1 φ2] represents the two beams’ phases are φ1 and φ2 respectively. In this simulation, the generated phases are phase = 2π*[0.9572 0.4854] rad, the Strehl ratio of the far-field interference pattern is Strehl = 0.5179, and the figures are shown as Figure 9:
Figure 9: Initial figures before compensation.
Wavefront of the two beams (upper-left); Interference pattern of the two beams (left lower); Wavefront behind the lenslets (upper-right); Diffraction patterns on the lenslets’ focus plane (right lower).
Step 3: Find the maximum intensity point of the central lenslet’s diffraction pattern. Calculate its displacement, d, from the original position where the location of the maximum point when Δφ= 0. Here, d= 15 pixel(0.159mm), then Δφ= 0.5209*2π rad from equation (10) which is almost the worst case. Step 4: If Δφ< 2π/20, saying piston error is small enough, then go to the next step. Otherwise, compensate the phase by
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replacing the phases as phase = phase – [Δφ 0], then back to step 3. Here, the piston error in step 3 is bigger than 2π/20, so the compensated phases are phase = 2π*[0.4362 0.4854], after calculation in step 3, we get d =1 pixel (0.0106mm), Δφ= -0.0347*2π, Strehl = 0.9763, and the figures are shown in Figure 10. As Δφ< 2π/20, then go to the next step.
Figure 10: Figures after once iteration.
Step 5: Exit the close loop compensation. The simulation represents just one iteration can compensate the initial piston error Δφ= 0.9436π which almost is the worst case in coherent beam combination (CBC). So the DBPIP is very effective to be used in CBC.
3.2 Simulation in case with both piston and tilts
With tilts in x and y axial orientation, the complex amplitude of one beam is:
0 0( , ) exp{ [( ) ( ) ]}x yU x y i i x x t y y tϕ= + − + − (13)
Where, the (x0, y0) is the position of the optical axis; tx, ty are tilts along the x and y axial orientations; φ is the phase. We use (φ, tx, ty) as one beam’s parameters. As the intensity distribution of far-field diffraction pattern is the Fourier transform of the complex amplitude, the pattern of the beam shown in equation (13) will displace from the original position, where the pattern is when the beam has no tilts. From the Shift theorem of Fourier transform, the displacement along the x, y axial orientations are dx, dy:
,2 2x x y y
f fd t d tλ λπ π
= − = − (14)
To be contrasted with the above simulation all the other parameters except the tilts are the same in this simulation. And the initial Wavefront of the two beams are (0.9572 -0.3730 0.1324)*2π and (0.4854 0.4134 -0.4025) *2π. The compensated parameters after every iteration are shown in Tabel 1, where Δφ= φ1 - φ2, and the figures are shown in Figure 11~Figure 13.
Because a piston error of 2π has the same effect as the piston error of 0 in far-field intensity distribution, piston errors of 2π can be ignored. After once iteration the Strehl ratio, which equals 0.9208, is big enough for CBC experiments. And after the second iteration the piston error with 2π ignored is less than 2π/1000 which is a perfect state for CBC.
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4. CONCLUSIONS
From the formula calculation and numerical analysis, the brightest point of two apertures’ far-field interference pattern shifts almost linearly with their piston error, and the shape of the apertures does not change the character of linearity. And the linearity dependent coefficient was affected by the shape and scale of the apertures, the focus length of the lens and the wavelength of the wave. But the accurate effects of all these four factors can not be got obviously now. For regular shaped apertures, the linearity dependent coefficient can be got by solving the differential equations like equation (3) using numerical analysis. But for irregular shaped apertures like Figure 7 the coefficient can be got just by numerical simulations. Two factors are needed to get the right piston error, one is the right linearity dependent coefficient, and the other is the exact displacement of the brightest point of the interference pattern. And the later factor is must be considered in system designing. From the static simulation results, if got the right coefficient and the accurate displacements, less than 3 iterations are enough to compensate both the static piston error and the static tilt error. And more than being used in CBC, the DBPIP can also be used in some other piston or tilt errors detections such as phasing the mirror segments of large telescopes.
Figure 11: Figures of initial statement.
Figure 12: Figures after one iteration.
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Figure 13: Figures after two iterations.
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