Numerical simulations

To show the feasibility of the proposed algorithm, I next describe simulations with realistic fibre parameters. A step-index fibre with core NA of 0.2 and core radius of 4 µm is considered. The wavelength of the monochromatic beam is 977 nm. These

Advanced waveguides for high power optical fibre sources D. B. S. Soh

parameters result in a normalised frequency V =5.14. By using the boundary condition in equation (7.7), the modal numbers l,p and the effective refractive index

eff

n of each mode was calculated. These are the five LP modes listed in Table 7.1. Note that some modes are split into cosine and sine modes as in equation (7.4) depending on the value of i. Figure 7.2 shows the mode intensity profile of each pair of mode representations ei

( )

x,y and e_Fi

( )

x,y , calculated using equation (7.26).

To evaluate the algorithm, three cases are simulated. In the first case, the exact mutual intensity profile at z= f was assumed to be known (simulation A). To show the practicability of the theory, more simulations were carried out that included retrieval of the mutual intensity profiles by use of the method described in the previous section (simulation B and C). Simulation B assumes the ideal CCD camera response, but the retrieval can be affected by the finite number of pixel points in the CCD array (the number of picture elements). Simulation C included the nonideal response of the CCD camera as well as the finite-grid effect. In simulation C, it was assumed that the image intensity was controlled to be well below the saturation level of the CCD camera by proper attenuation as in [100]. In this case, the non-ideality in CCD camera response is predominantly the intrinsic noise, such as dark current, blooming, shot noise, amplifier noise and quantization noise in the analogue-to-digital converter [104]. The signal-to-noise ratio in a conventional modern CCD cameras is

i

e

(a) i F

e

(b)

Figure 7.2. The calculated intensity distribution of each mode: (a) the modal solutions inside fibre, (b) the modal solutions after the spherical lens

Advanced waveguides for high power optical fibre sources D. B. S. Soh

more than 30 dB. For example, for one CCD camera (COHU 1100) the signal-to- noise ratio is specified at more than 50 dB. On the other hand, reference [104] explains that the overall noise sum is proportional to the averaged mean intensity of an image (which was taken as the sum of all the intensities at each pixel divided by the total number of pixels) and shows that the experimental value for the variance of noise is approximately 1% of the averaged mean intensity. Therefore simulation C includes a normally distributed random noise with variance of 1% of the averaged mean intensity for each image. In addition the linearity (or the photon-electron charge transfer efficiency) of conventional CCD cameras reaches 0.99999 [104]. Therefore the nonlinearity in the response curve was omitted in the simulation.

In the simulations an arbitrary set of test modal power weights c_i 2 as in table 7.1 were assigned to evaluate the accuracy of the algorithm. For this, a step index fibre with core NA of 0.2 and the core radius of 4 µm is used and the wavelength considered is 977 nm. This results in V number of 5.14. Figure 7.3 shows the intensity of the test multimode beam of concern. Assuming that the collimated free-space beam propagates in free-space according to the Fresnel integration in equations (7.22) and (7.37), one could obtain the calculated CCD images taken at the z=D plane for various values of

θ

, .

φ

Calculated Results

Modal Weights c_i 2( Error (%) )

Modes Effective Index Test Modal Weight 2 i

c Simulation A Simulation B Simulation C

LP01 cos 1.4686 64 64.03 (0.05) 60.39 (5.64) 60.01 (6.23) LP02 cos 1.4604 25 24.90 (0.40) 27.45 (9.80) 27.53 (10.1) LP11 cos 1.4655 4 3.988 (0.30) 4.391 (9.78) 4.434 (10.9) LP11 sin 1.4655 9 9.045 (0.50) 9.829 (9.21) 9.868 (9.64) LP21 cos 1.4615 1 1.003 (0.30) 1.100 (10.0) 1.093 (9.30) LP21 sin 1.4615 9 9.027 (0.30) 8.630 (4.11) 8.690 (3.44) LP31 cos 1.4570 4 3.981 (0.48) 4.241 (6.01) 4.350 (8.75) LP31 sin 1.4570 1 1.002 (0.20) 0.910 (9.00) 0.894 (10.6)

Table 1. The table lists the modes of the step-index fibre (0.2 NA and 4 µm core radius) considered in the simulations and their effective indices at 977 nm wavelength.

Advanced waveguides for high power optical fibre sources D. B. S. Soh

For the numerical verification of our method, each mode was excited with a power represented by the test modal weight. This was then retrieved with three different simulation procedures, A (without retrieval of mutual intensity), B (with retrieval of mutual intensity, but without CCD noise), and C (with retrieval of mutual intensity including CCD noise), with the results shown in the table.

To realise the inverse Radon transform for calculation of the modal power weights, 30 different uniformly distributed angles in the interval

[

−

π

/2,

π

/2

]

were used for each combination

θ

, . Assuming that the focal lengths of the cylindrical lenses were

φ

20 cm, d_i was varied from 0.87 m to 1.24 m (i=1,2) and D was varied from 1.2 m to 2.0 m to provide full coverage of the required angles

θ

, . This is similar to the

φ

procedure in reference [100]. Hence the distances d_m and d_M in the relation (7.23)

(x

y

z

f)

e

,

;

=−

(a) (b)

_e(x,y;z

=

_f)

(x

y

z

f)

e

,

;

=−

(a) (b)

(x

y

z

f)

e

,

;

=−

(a) (b)

_e(x,y;z

=

_f)

Figure 7.3. A test multimode beam with arbitrary assigned initial modal power: (a) the beam intensity inside fibre, (b) the free-space beam intensity at z= f .

Advanced waveguides for high power optical fibre sources D. B. S. Soh

were 0.87 m and 2 m, respectively, which gave the condition on the focal length f of the spherical lens as 1.4 cm << f << 87 cm. From this the focal length f of the spherical lens was chosen as 11 cm.

In simulation A the modal power weights c_i2 were calculated by using the relations (7.29) – (7.32) and by assuming that the exact mutual intensity at z= f was known. In simulation B the values of the c_i2s were calculated by retrieving a mutual intensity profile Γ_F from the intensity patterns after the two cylindrical lenses. Examples of the intensity patterns are shown in figure 7.4 for some values of

θ

, .

φ

From these intensity patterns the Wigner function at the z=0 plane with the inverse Radon transformation was calculated. From this Wigner function, the mutual intensity profile Γ_F was obtained through a two-dimensional inverse Fourier transform. Now by applying equations (7.29) – (7.32), the modal power weights were calculated as shown in Table 7.1. For simulation C, the same process was used except that a noise was added to the intensity patterns. Figure 7.5 shows the modal power distribution as

Figure 7.4. Selected CCD images for different

θ

, , i.e., with different positions

φ

of the lenses and the CCD array.

Advanced waveguides for high power optical fibre sources D. B. S. Soh

a percentage of the total power of the multimode beam. The test modal power distribution and the calculated results from simulations A, B and C are also shown.

Figure 7.6 shows a comparison between the intensity profiles of the test multimode beam inside the fibre and the calculated reconstructed beam. Note that it is impossible to reconstruct the multimode image by using the calculated c_i 2. For a full reconstruction, the phases must also be known. (Note that the retrieved modal power distribution is independent of the randomly assigned phases.) However, purely for demonstration purposes it was assumed that all the test modal weights are positive numbers with zero phase. The reconstruction through simulation A is almost exact, while simulations B and C resulted in some discrepancy. In the simulations, 50 grid points along each of the x and y axes were used, i.e., 2500 points for one image. The errors between the initially assigned modal weights and the calculated ones were within 0.5% for simulation A and within 10% and 11% for simulation B and C, respectively. This implies that the proposed method of calculating modal distribution is quite accurate provided the mutual intensity profile is known sufficiently well.

Since the difference in errors between simulation B (without noise) and simulation C (with noise) was approximately within 1%, it was concluded that the main cause of the simulation errors in simulation B and C was the limited number of position pixels

0 10 20 30 40 50 60 LP01 cos LP02 cos LP11 cos LP11 sin LP21 cos LP21 sin LP31 cos LP31 sin Modes P o w e r p o rt io n (

%) Given test modal weight

Simulation A Simulation B Simulation C

Figure 7.5. The modal power distribution among modes. The test modal power distribution and the results with simulations A, B and C are shown.

Advanced waveguides for high power optical fibre sources D. B. S. Soh

Figure 7.6. Comparison between the test multimode beam and reconstructed beam from calculated results. The figure shows the intensity pattern of the test multimode beam inside the fibre (solid line), the reconstructed image from simulation A (dashed line), the reconstructed image from simulation B (dotted line), and the reconstructed image from simulation C (dash-dotted line). Figure 7.6(b) is a magnified portion of figure 7.6(a) to clarify the small differences between results.

(a)

(b) (a)

Advanced waveguides for high power optical fibre sources D. B. S. Soh

and angles rather than the noise. Therefore the finer the grids and the more pixel points one uses, the more accurate the results one obtains, even in the presence of noise. For instance, the retrieval of the Wigner function from the CCD camera through the Radon transform depended on the number of angles, while the retrieval of mutual intensity profiles from the Wigner function by Fourier transform depended on the number of image pixels. Considering that CCD images normally provide more than 256×256 pixel data points, the accuracy might be quite satisfactory provided that limitations such as a nonlinear response are controlled. In addition, a larger number of angles can be used for more accurate results.

7.3. Modal power decomposition using wavelength

In document Advanced waveguides for high power optical fibre sources (Page 167-174)