Method and program for predicting bending in optical products
In the optical product bending prediction method according to the present invention, bending δ of a substrate having a film is predicted by a computer performing calculation (steps S 11 to S 13 ) of the following Expressions (A1), (B1), etc., in which a and T d are set as fitting parameters and have been optimized in each coating condition using a plurality of samples. Here, σ int is intrinsic stress, E f is a Young's modulus of a multilayer film, v f is a Poisson's ratio of the multilayer film, α f is a linear coefficient expansion of the multilayer film, α s is a linear coefficient expansion of the substrate, T is a room temperature, d is a film thickness, b is a substrate thickness, l is a radius of the substrate, E s is a Young's modulus of the substrate, and v s is a Poisson's ratio of the substrate. [ Mathematical 1 ] δ = 3 l 2 b 2 ( 1 - v s E s ) d σ ( A1 ) σ = a σ i n t + ( α s - α f ) ( E f 1 - v f ) ( T - T d ) ( B1 )
1. A computer implemented optical product bending prediction program configured to predict a bending δ of a substrate having a film by execution thereof by a computer,
wherein the computer includes a non-transitory, computer-readable storage medium storing the optical product bending prediction program and an integrated film designing program, and control means capable of referring to the non-transitory computer-readable storage medium to execute the optical product bending prediction program and the integrated film designing program stored therein,
wherein the optical product bending prediction program stored in the non-transitory, computer-readable medium includes the following Expressions (A1) to (I), in which a is set as a first fitting parameter and T d is set as a second fitting parameter, the fitting parameters having been optimized for each coating condition using a plurality of samples,
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having the film using the Expressions (A1) to (I),
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
(
A1
)
σ
=
a
σ
int
+
(
α
s
-
a
f
)
(
E
f
1
-
v
f
)
(
T
-
T
d
)
(
B1
)
d
=
d
H
+
d
L
(
C
)
σ
i
n
t
=
d
~
H
σ
H
+
d
~
L
σ
L
(
D
)
d
~
H
=
d
H
/
d
(
E
)
d
~
L
=
d
L
/
d
(
F
)
E
f
=
d
~
H
E
H
+
d
~
L
E
L
(
G
)
v
f
=
d
~
H
v
H
+
d
~
L
v
L
(
H
)
α
f
=
d
~
H
α
H
+
d
~
L
α
L
.
(
I
)
wherein σ int is intrinsic stress, E f is a Young's modulus of a multilayer film, E H is a Young's modulus of a high refractive index layer, E L is a Young's modulus of a low refractive index layer, v f is a Poisson's ratio of the multilayer film, v H is a Poisson's ratio of the high refractive index layer, v L is a Poisson's ratio of the low refractive index layer, α f is a coefficient of linear expansion of the multilayer film, α H is a coefficient of linear expansion of the high refractive index layer, α L is a coefficient of linear expansion of the low refractive index layer, α s is a coefficient of linear expansion of the substrate, T is a room temperature, d is a film thickness, b is a thickness of the substrate, 1 is a radius of the substrate, E s is a Young's modulus of the substrate, v s is a Poisson's ratio of the substrate, d H is a total film thickness of the high refractive index layer, d L is a total film thickness of the low refractive index layer, σ H is intrinsic stress of the high refractive index layer, and σ L is intrinsic stress of the low refractive index layer, and
wherein the film designing program stored in the non-transitory, computer-readable medium, automatically passes values determined by the film designing program to the bending prediction program, whereby an accurate bending δ of the substrate can be determined during or at the completion of executing the film designing program.
2. A computer implemented optical product bending prediction program configured to predict a bending δ of a substrate having a film by execution thereof by a computer,
wherein the computer includes a non-transitory, computer-readable storage medium storing the optical product bending prediction program and an integrated film designing program, and control means capable of referring to the non-transitory, computer-readable storage medium to execute the optical product bending prediction program and the integrated film designing program stored therein,
wherein the optical product bending prediction program stored in the non-transitory, computer-readable storage medium includes the following Expressions (A1) to (I), in which a H is set as a first primary fitting parameter, a L is set as a first secondary fitting parameter, and T d is set as a second fitting parameter, the fitting parameters having been optimized for each coating condition using a plurality of samples,
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having a film using the Expressions (A1) to (I),
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
(
A1
)
σ
=
a
H
σ
H
d
~
H
+
a
L
σ
L
d
~
L
+
(
α
s
-
a
f
)
(
E
f
1
-
v
f
)
(
T
-
T
d
)
(
B2
)
d
=
d
H
+
d
L
(
C
)
d
˜
H
=
d
H
/
d
(
E
)
d
~
L
=
d
L
/
d
(
F
)
E
f
=
d
~
H
E
H
+
d
~
L
E
L
(
G
)
v
f
=
d
~
H
v
H
+
d
~
L
v
L
(
H
)
α
f
=
d
~
H
α
H
+
d
~
L
α
L
.
(
I
)
wherein σ int is intrinsic stress, E f is a Young's modulus of a multilayer film, E H is a Young's modulus of a high refractive index layer, E L is a Young's modulus of a low refractive index layer, v f is a Poisson's ratio of the multilayer film, v H is a Poisson's ratio of the high refractive index layer, v L is a Poisson's ratio of the low refractive index layer, α f is a coefficient of linear expansion of the multilayer film, α H is a coefficient of linear expansion of the high refractive index layer, α L is a coefficient of linear expansion of the low refractive index layer, α s is a coefficient of linear expansion of the substrate, T is a room temperature, d is a film thickness, b is a thickness of the substrate, 1 is a radius of the substrate, E s is a Young's modulus of the substrate, v s is a Poisson's ratio of the substrate, d H is a total film thickness of the high refractive index layer, d L is a total film thickness of the low refractive index layer, σ H is intrinsic stress of the high refractive index layer, and σ L is intrinsic stress of the low refractive index layer, and
wherein the film designing program stored in the non-transitory, computer-readable medium, automatically passes values determined by the film designing program to the bending prediction program, whereby an accurate bending δ of the substrate is determined during or at the completion of the execution of the film designing program.
3. The computer implemented optical product bending prediction program according to claim 1 , wherein the Expression (A1) is replaced with the following Expression (A2), in which δ s is set as a third fitting parameter, and
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having the film using the Expressions (A2) to (I) for which optimization has been performed in each of the coating conditions using the plurality of samples
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
+
δ
s
.
(
A2
)
4. The computer implemented optical product bending prediction program according to claim 2 , wherein the Expression (A1) is replaced with the following Expression (A2), in which δ s is set as a third fitting parameter, and
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having the film using the Expressions (A2) to (I) for which optimization has been performed in each of the coating conditions using the plurality of samples
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
+
δ
s
.
(
A2
)
5. The computer implemented optical product bending prediction program according to claim 1 , wherein the Expression (A1) is replaced with the following Expression (A3), including bending δ 0 of the substrate on which the film has not yet been formed, and
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having the film using the Expressions (A3) to (I) for which optimization has been performed in each of the coating conditions using the plurality of samples
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
+
δ
0
.
(
A3
)
6. The computer implemented optical product bending prediction program according to claim 2 , wherein the Expression (A1) is replaced with the following Expression (A3), including bending δ 0 of the substrate on which the film has not yet been formed, and
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having the film using the Expressions (A3) to (I) for which optimization has been performed in each of the coating conditions using the plurality of samples
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
+
δ
0
.
(
A3
)
7. The computer implemented optical product bending prediction program according to claim 1 , wherein the Expression (A1) is replaced with the following Expression (A4), including a third fitting parameter δ s and bending δ 0 of the substrate on which the film has not yet been formed, and
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having the film using the Expressions (A4) to (I) for which optimization has been performed in each of the coating conditions using the plurality of samples
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
+
δ
0
+
δ
s
.
(
A4
)
8. The computer implemented optical product bending prediction program according to claim 2 , wherein the Expression (A1) is replaced with the following Expression (A4) including a third fitting parameter δ s and bending δ 0 of the substrate on which the film has not yet been formed, and
wherein the control means is configured to execute the optical product bending prediction program to perform calculations to predict the bending δ of the substrate having the film using the Expressions (A4) to (I) for which optimization has been performed in each of the coating conditions using the plurality of samples
δ
=
3
l
2
b
2
(
1
-
v
s
E
s
)
d
σ
+
δ
0
+
δ
s
.
(
A4
)