A method of changing wavelength to simulate large scale problems

Fig 3.4 Numerical results of using original parameters and changed parameters up to 40°

For any kind of material, its relative permittivity can be expressed as

e

_r:

1 2

r

i

e = + e e (102)

Where

2 0

e s

=

we (103)

s

is the electrical conductivity ,

w

is the laser frequency and

e

₀is the permittivity in free space and

e

1^,

e

₂ are the real part and imaginary part of the relative permittivity.

And for any material, its relative permittivity (relative permeability assumed to be 1 here since it applies to most materials ) can be derived from the formula below, since refractive index

n

and extinction number

k

can be easily found from literatures:

2 2

1

2

2

n n

e k

e k

ì = - í =

î (104)

For dieletrics,

e

₂is very small (usually assumed to be 0) since

k

is very small. In this case, n>

k

,

e

1is a positive number (and over 1) , so the standard FDTD can be applied.

For dispersive materials,

k

is not 0 anymore, but if n>

k

,

e

₁ is still a positive number from (103), which means the standard FDTD algorithm still stands. The only difference here is that the electrical conductivity is not 0 anymore.

From (103), re-express laser frequency as

2 p l c /

₀^{, yields:}

0

=2

0 2 2

c 60 e

sl p e e = (105)

Since for a certain material at certain laser radiant frequency,

n k

are constants, so do

e

₁

e

₂. In the regard, (105) is a constant, which means for a certain problem in standard FDTD algorithm, the product of wavelength and electrical conductivity is a constant. This is a very important conclusion here for laser simulations, because we can simply increase the wavelength and decrease the electrical conductivity to keep the same material properties, and remain the same reflectance. So it’s possible for us to simulate large scale problems with standard FDTD algorithm. Take the previously discussed case as an example, the keyhole welding simulation, if the original wavelength is adopted, 10¹² grids

are needed. But if the changed wavelength is adopted, for instance,

l

₀can be increased to 100um, so for the same size problem, only 100 grids in each direction, the total number of grids is reduced to only 10⁶.

But for some metals, their extinction number

k

are larger than refractive index

n

, which leads to a negative

e

₁, and this can be simulated only with a dispersive code in general. But the previous work which has been discussed in (3.3) about changing the value of

n

and

k

, has shed a light upon this matter. Now discuss the possibility of this method, changing

n

and

k

while keeping the same reflectance pattern, and using the standard FDTD code to simulate a dispersive material (especially when

n

is smaller than

k

).

1) When

n

>

k

:

a. If

k

=0, then

e

₂=0,

s

=0, then it’s a lossless dielectric, which can be simulated by using the standard FDTD code very easily.

b. If 0<

k

<

n

. Equation (105) is derived, a standard FDTD algorithm still works.

2) When

n

<

k

This will cause

e

₁<0. But since reflectance is governed by (99) , (100). We can choose another set of

n

,

k

which meets the requirement of

n

>

k

while keep the same reflectance pattern. After chosen a proper set of

n

,

k

values, we can use the standard FDTD code to simulate dispersive materials too. Now the problem is : can different set of

n

,

k

give us the same reflectance pattern. Consider iron Fe at wavelength of 1.06

m

m，which processes the values of

n

=3.81,

k

=4.44. For simple problems, for example, there is only one incident angle in the whole domain. In this case ,we can control the

n

,

k

value to maintain the exact the same reflectance according to (99),(100). But for some complicated problems, in which there are more than one incident angles in a problem at the same time, thus , there will be different

n

,

k

values at different incident angle, which is rather difficult to achieve in the code. So a global set of

n

,

k

value for one simulation is required , even for complex problems. As the incident angle covers the range from 0 to 90 degrees, it is possible that we choose a certain angle to get our desired

n

,

k

. It’s totally arbitrary , but our goal is to minimize the errors. For a pre-chosen angle

q

, make

k

=0, then we can get the minimum and maximum values of

n

, which is the possible range for

n

. Choose a serial values of

n

in between, then plot the results for both P and S polarization.

n

-

k

relations and analytic reflectance solutions for changing

n

,

k

at different angles (0°,30°,45°,75.2°(Brewster's angle),82°) are shown as below:

Fig 3.4

n

-

k

relations at 0°. For both P and S polarization, the

n

-

k

value relations are the same, so we only have the same set of

n

-

k

for both S and P polarization.

n

and

k

are listed in Table 1.:

Table 1.

n

0.11 0.111 1.01 1.91 2.82 3.72 4.62 5.53 6.43 7.33 8.23 9.14

k

^0.0 0.111 2.71 3.61 4.14 4.42 4.51 4.42 4.14 3.61 2.71 0.0

Values of chosen

n

,

k

at 0°for Fe at 1.06μm(

n

=3.81,

k

=4.44)

Fig 3.5 Analytic results of different

n

,

k

by choosing reference angle at 0°. Red lines are original values (

n

=3.81,

k

=4.44), blue and green lines are results by changing

n

,

k

while keeping the same reflectance at 0°. Green lines are P –polarization, blue lines are S-polarization.

Brown lines are two special cases which

n

=

k

=0.111 or 4.512. The S-polarization result of 0.111overlaps the result of n being 4.512 P-polarization, in the lower part of the graph; also the 0.111 P-polarization also overlaps 4.512 S-polarization in the upper part of the graph, which is the green lines zone. The same overlapping problem happens when

k

=0, which are the two extreme cases (

n

_min= 0.109457 and

n

_max= 9.13597). For the other cases of

n

,

k

in the figure, P lines(green) go down with

n

increasing while S lines go up with n increasing .

Fig 3.6

n

-

k

relations at 30°. For P and S polarization, the

n

-

k

value relations are different, so there are 2 different sets of

n

-

k

for S and P polarization.

n

and

k

are listed in Table 2

Table 2.

P

n 0.15 1.28 2.41 3.54 4.67 5.80 6.93 8.06 9.19

k 0.0 3.00 3.92 4.38 4.52 4.38 3.92 3.00 0.0

S

n 0.08 1.21 2.34 3.46 4.60 5.72 6.84 7.97 9.09858

k 0.0 2.98 3.90 4.36 4.51 4.37 3.90 2.98 0.0

Values of chosen

n

,

k

at 30°for Fe at 1.06μm(

n

=3.81,

k

=4.44)

Fig 3.7 Analytic results of different

n

,

k

by choosing reference angle at 30°. Red lines are original values (

n

=3.81,

k

=4.44). Green lines are P –polarization, blue lines are S-polarization.

Fig 3.8

n

-

k

relations at 45°. For P and S polarization, the

n

-

k

value relations are different, so there are 2 different sets of

n

-

k

for S and P polarization.

n

and

k

are listed in Table 3.

Table 3.

P

n 0.22 1.35 2.48 3.62 4.75 5.89 7.02 8.16 9.29

k 0.0 3.00 3.93 4.40 4.54 4.40 3.93 3.00 0.0

S

n 0.06 1.12 2.18 3.24 4.3 5.4 6.4 7.5 9.1

k 0.0 2.91 3.82 4.31 4.50 4.43 4.10 3.43 0.0

Values of chosen

n

,

k

at 45°for Fe at 1.06μm(

n

=3.81,

k

=4.44)

Fig 3.9 Analytic results of different

n

,

k

by choosing reference angle at 45°. Red lines are original values (

n

=3.81,

k

=4.44). Green lines are P –polarization, blue lines are S-polarization.

Fig 3.10

n

-

k

relations at 75.2°. For P and S polarization, the

n

-

k

value relations are different, there are 2 different sets of

n

-

k

for S and P polarization.

n

and

k

are listed in Table 4.

Table 4

P

n 1.31 2.61 3.90 5.21 6.47 7.77 9.07 10.37 11.70

k 0.0 3.43 4.49 5.03 5.20 5.04 4.51 3.47 0.0

S

n 0.007 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

k 0.0 3.73 4.23 4.47 4.47 4.24 3.74 2.83 0.0

Values of chosen

n

,

k

at 75.2°for Fe at 1.06μm(

n

=3.81,

k

=4.44)

Fig 3.11 Analytic results of different

n

,

k

by choosing reference angle at 75.2°. Red lines are original values (

n

=3.81,

k

=4.44). Green lines are P –polarization, blue lines are S-polarization.

Fig 3.12

n

-

k

relations at 82°. For P and S polarization, the

n

-

k

value relations are different, there are 2 different sets of

n

-

k

for S and P polarization.

n

and

k

are listed in Table 5.

Table 5.

P

n 2.59 4.76 6.93 9.10 11.26 13.42 15.58 17.78 19.95

k 0.0 5.74 7.52 8.40 8.68 8.41 7.53 5.74 0.0

S

n 0.002 2.00 3.00 4.00 5.00 6.00 7.00 7.98 8.99

k 0.0 3.73 4.24 4.46 4.47 4.24 3.73 2.83 0.0

Values of chosen

n

,

k

at 82°for Fe at 1.06μm(

n

=3.81,

k

=4.44)

Fig 3.13 Analytic results of different

n

_,

k

by choosing reference angle at 82°. Red lines are original values (

n

_=3.81,

k

=4.44). Green lines are S –polarization, blue lines are P-polarization.

As the above figures illustrated, when

n

is over a certain value ,for example 1, all the lines follow the same pattern: the larger

n

is, the lower P polarization lines and the higher S polarization lines after the reference angles(0°,30°,45°,75.2°(Brewster's angle),82°); the larger

n

is, the higher P polarization and the lower S polarization before the reference angles.

And another notable point here is, the relative errors of reflectance between the new sets of

n

,

k

and the original one. The larger the reference angle is, the larger the error is:

Fig 3.14 incident- angle- dependent relative error of P and S-polarization at 0°.

Fig 3.15 incident- angle- dependent relative error of P and S-polarization at 30°

Fig 3.16 incident- angle- dependent relative error of P and S-polarization at 45°

Fig 3.17 incident- angle- dependent relative error of P and S-polarization at 75.2°

Fig 3.18 Incident- angle- dependent relative error of P and S-polarization at 82°

As we can see from the above figures, the incident- angle- dependent relative errors become larger as the reference angles become larger. For angle at 0°,30°and 45°, the errors are quite close because the analytic reflectance curves coincide with the original

n

,

k

values very well, in which case, at small incident angles (0°- 60°). And this is obvious from Fig 3.5, while huge errors are generated at larger angles (> 60°).

One more important thing is, by looking at all the figures, some very import conclusions can be made:

1).S-Polarization has the best simulation results, the error is extremely small in the whole range while the P-polarization has small errors when the angles are small, but huge when the angles become larger.

2).Reference angle at small angles (0°,30°,45°)give us the best results and they are almost the same. But for angle at 0°, only one set of

n

,

k

is required since the P and S polarization share the same

n

,

k

. But for other angles(30°or 45°)，there are 2 sets of

n

,

k

.So in the regard, we can choose angle at 0°for simplicity.

As is for choosing proper

n

,

k

, the overall relative errors for different sets of

n

,

k

with respect to refractive index

n

can be calculated.

Fig 3.19 Overall error versus refractive index

In the above figure, square marks lines are P-polarization and triangle mark lines are S polarization. Different colors represent different reference angles. The S-error becomes extremely small when refractive index is larger than 1, and the P errors are strongly related to the reference angles. When reference angles are set at 0°，30°or 45°，the P errors are quite close. But when the angle increases further (75.2°，82°)， the errors are huge alone with the change of

n

range. The minimum errors are around

n

=3.81(original material property), and this is understandable. So the guide line for choosing

n

,

k

is to choose the smallest

n

which is larger than

k

and most close to the original values. One subtle requirement is that we have to make sure that

e

₁is larger than one, and this is not difficult as long as we have a small difference between

n

and

k

according to (104).

All the discuss above based on Fe at 1.06um wavelength, but the method can be applied to all other materials， too.

The method can be described in the scheme as below:

In another word, every material can be simulated with a standard FDTD algorithm, and reflectance can be calculated with a enlarged wavelength.

In order to verify this method, now try Fe at 1.06um wavelength (

n

=3.81,

k

=4.44).

For dispersive material, use Drude Model. Retrieve plasma frequency

w

pand damping constant

g

_p(listed in Table 6) from (106) and the following formula (consider

e

_¥=2.0 as a free parameter ):

2 2 2

1 2

-

2 ^p

(

2 ^p 2

)

3 ^{p p} 2

r

p p p

i j

j

w w w g

e e e e e

w wg w g w wg

¥ ¥

æ ö

ç ÷

= + = - = - + + - ç è + ÷ ø (106)

Table 6.

n k e

1

e

₂

e

_¥

w

p

(rad/s)

g

p

(rad/s)

3.81 4.44 -5.19 33.83 2.0 2.29274x10¹⁶ 8.35895x10¹⁵

Then for changing

n

,

k

by using our method we can get all the required values listed in Table 6:

Table 7.

n

(original)

k

(original)

n

new

knew

e

¹

e

₂

l

₀

s

(S/m)

Dispersive material n

,

k

n

>

k

Standard FDTD Choose proper n

,k

( n

>

k )

yes no

(new) (μm)

3.81 4.44 4.6227 4.512 1.00011 41.7268 1.06 656081.761

Note that in Table 7 , we can easily change

l

_{0 and}

s

since we have an important conclusion as

0

=2

0 2 2

c 60 e sl p e e =

Verify the results by running both standard and dispersive codes using the corresponding parameters in Table 6 (dispersive) and Table 7 (standard).

Fig 3.20 simulation results by using standard FDTD and dispersive FDTD codes. Dashed lines are S- polarization and solid lines are P-polarization. Red and green lines are analytic solutions for original values and changed values, respectively. Blue lines are standard FDTD numerical results for changed values while Brown lines are dispersive FDTD numerical results for original values.

The S and P polarization beam propagation of Fe at 1.06um with original parameters at steady state for different incident angles (from 0°- 60°) are listed in Fig 3.21 (S-polarization) and Fig 3.22 (P

polarization) as a demonstration, scheme is just like Fig.3.1, only the source location is different and the vacuum-material interface tilts.

a.0° b.10°

c.20° d.30°

e.40° f.50°

f.60°

Fig 3.21(a.-f.) beam propagations ,S polarization for Fe at 1.06um with original parameters for different incident angles

a.0° b.10°

c.20° d.30°

e.40° f.50°

f.60°

Fig 3.22(a.-f.) beam propagations, P polarization for Fe at 1.06um with original parameters for different incident angles

As we can see that the results of two different codes agree quite well. Even for circular-polarization

Fig 3.24 simulation results by using standard FDTD and dispersive FDTD codes. Dashed lines are S- polarization ,solid lines are P-polarization and the lines with cross-mark are circular-polarization. Red and green lines are analytic solutions for original values and changed values, respectively. Blue lines are standard FDTD numerical results for changed values while Brown lines are dispersive FDTD numerical results for original values.

So the method is valid and we can use this method to calculate dispersive materials by changing

n

,

k

with standard FDTD algorithm.