Ilkka Kallioniemi, Jyrki Saarinen, and Erkki Oja, "Characterization of diffraction gratings in a rigorous domain with optical scatterometry: hierarchical neural-network model," Appl. Opt. 38, 5920-5930 (1999)

Characterization of microstructures with features from
submicrometers to hundreds of micrometers requires versatile
methods. Profilometry and optical microscopy cannot cope with
submicrometer features, and atomic-force microscopy, scanning-electron
microscopy, and near-field microscopy are inherently slow, off-line
methods. In optical scatterometry, the laser light scattered from a
sample is measured and the sample profile is subsequently
characterized. We propose the use of a two-stage model based on
neural networks: rough categorization followed by refinement, thus
reducing the need for prior information on the sample. We simulate
the method for a submicrometer diffraction grating characterized by
five parameters. It is shown that intensity measurements of few
diffraction orders by use only of one wavelength are enough to yield
rms errors of less than 2 nm for the parameters (approximately 2–3%
of the optimal values of the parameters).

Bryan D. Maione, David Luo, Matthew Miskiewicz, Michael Escuti, and Michael W. Kudenov Appl. Opt. 55(31) 8667-8675 (2016)

References

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Angles of Reflection θ_{-1} and
θ_{1} for Diffraction Orders R_{-1}
and R_{1} with Three Angles of Incidence
θ_{i} = 10°; 15°,
30°a

Diffraction Order

θ_{
i
} (°)

Angle of Reflection θ_{-1
}/θ_{
1
} (°)

R_{-1
}

10

-32.2

R_{
1
}

10

61.7

R_{-1
}

15

-26.6

R_{
1
}

15

75.0

R_{-1
}

30

-11.9

R_{
1
}

30

>90

Angle of reflection >90° means that
the corresponding diffraction order is not propagating. The
corresponding values for diffraction order R_{0}
are θ_{0} = θ_{
i
}, i.e.,
specular reflection. The sign of θ_{0} is chosen
positive.

Table 2

Classification Performance of the Level-A Network for Test
Sample Sets T_{B1
} and
${T}_{B}^{c}$, Representing the Range of
Network B_{1} and the Complement of the Range of Networks
B_{1}, B_{2}, and B_{3},
Respectivelya

Classification Measure

Percent Classified Correctly At

σ = -∞ dB

σ = -20 dB

σ = -13 dB

p_{
1
}(V_{
B
1
}, T_{
B
1
})

94.8

57.5

59.0

p_{
1
}(V_{
B
}, T_{
B
1
})

100.0

97.2

97.3

p_{
1
}(V_{
B
}, ${T}_{B}^{c}$)

12.0

16.5

18.2

The number of samples is 1000 in both
cases. The classification is performed with respect to both groove
depths h_{1} and h_{2}, and the
classification measure is defined in Eq. (1). Input data
consist of diffraction intensities R_{0} and
R_{-1} calculated with three angles of incidence,
θ_{
i
} = 10°, 15°, 30°, and two states
of polarization, TE and TM, i.e., twelve elements in
total. Gaussian noise with zero mean and a standard deviation of
σ = N dB is added to the input data. In test one
(the first column), N = -∞. In tests two
(the second column) and three (the third column), N
= -20 and N = -13, respectively.

Table 3

Root-Mean-Square Errors of Prediction between Output
${\overrightarrow{y}}_{{B}_{1}}$ from Network B_{1} and
Target Parameters $\overrightarrow{O}$, Averaged over 200 Test
Samplesa

Noise Level (dB)

rms_{
h
1
} (nm)

rms_{
h
2
} (nm)

rms_{
t
1
} (nm)

rms_{
t
2
} (nm)

rms_{
t
3
} (nm)

-∞

0.39

0.71

2.33

2.98

2.28

-30

1.04

1.80

3.53

5.78

3.04

-20

1.58

3.46

4.78

6.39

3.70

-13

2.69

5.79

6.98

11.19

4.82

The errors are presented for groove
depths h_{1} and h_{2} as well
as for transition points t_{1}, t_{2}, and
t_{3}. Input data consist of diffraction
intensities R_{0} and R_{-1}
calculated with three angles of incidence, θ_{
i
}= 10°, 15°, 30°, and two states of polarization, TE and TM,
i.e., 12 elements in total. Gaussian noise of zero mean and values
of σ -∞ dB (no noise), -30, -20, or -13 dB relative to
the intensity is added to test input vectors
${\overrightarrow{I}}^{\left(m\right)}$. Different networks are
taught for each amount of noise.

Table 4

Same as Table 3 Except That Input Data Consist of
Diffraction Intensities R_{0} and
R_{-1} Calculated with Three Angles of Incidence,
θ_{i} = 10°, 15°, 30°, and One State of
Polarization, TE, i.e., Six Elements in Total

Noise Level (dB)

rms_{
h
1
} (nm)

rms_{
h
2
} (nm)

rms_{
t
1
} (nm)

rms_{
t
2
} (nm)

rms_{
t
3
} (nm)

-∞

2.27

1.54

9.62

11.92

11.94

-30

2.46

2.47

11.13

16.06

12.41

-20

3.63

4.35

14.92

19.92

15.20

-13

4.97

6.47

17.66

19.88

17.00

Table 5

Target Values and Predictions Given by Networks
B_{1}, B_{2}, and B_{3} for Four Sample
Gratings Chosen from within the Combined Range of the Three-Level B
networksa

The rightmost column shows the sum of
absolute deviations between the predictions and the target values as
the sums are taken over h_{1}, h_{2},
t_{1}, t_{2}, and
t_{3}. Input data consist of diffraction
intensities R_{0} and R_{-1}
calculated with three angles of incidence, θ_{
i
}= 10°, 15°, and 30°, and two states of polarization, TE and
TM, i.e., 12 elements in total. If the correct sample parameters
are within the range of a specific network, the corresponding line is
typed boldface.

Table 6

Same as Table 5 Except That Here Gaussian Noise with
Standard Deviation σ = -13 dB Relative to the Intensity Is
Added To the Test Input Data

Angles of Reflection θ_{-1} and
θ_{1} for Diffraction Orders R_{-1}
and R_{1} with Three Angles of Incidence
θ_{i} = 10°; 15°,
30°a

Diffraction Order

θ_{
i
} (°)

Angle of Reflection θ_{-1
}/θ_{
1
} (°)

R_{-1
}

10

-32.2

R_{
1
}

10

61.7

R_{-1
}

15

-26.6

R_{
1
}

15

75.0

R_{-1
}

30

-11.9

R_{
1
}

30

>90

Angle of reflection >90° means that
the corresponding diffraction order is not propagating. The
corresponding values for diffraction order R_{0}
are θ_{0} = θ_{
i
}, i.e.,
specular reflection. The sign of θ_{0} is chosen
positive.

Table 2

Classification Performance of the Level-A Network for Test
Sample Sets T_{B1
} and
${T}_{B}^{c}$, Representing the Range of
Network B_{1} and the Complement of the Range of Networks
B_{1}, B_{2}, and B_{3},
Respectivelya

Classification Measure

Percent Classified Correctly At

σ = -∞ dB

σ = -20 dB

σ = -13 dB

p_{
1
}(V_{
B
1
}, T_{
B
1
})

94.8

57.5

59.0

p_{
1
}(V_{
B
}, T_{
B
1
})

100.0

97.2

97.3

p_{
1
}(V_{
B
}, ${T}_{B}^{c}$)

12.0

16.5

18.2

The number of samples is 1000 in both
cases. The classification is performed with respect to both groove
depths h_{1} and h_{2}, and the
classification measure is defined in Eq. (1). Input data
consist of diffraction intensities R_{0} and
R_{-1} calculated with three angles of incidence,
θ_{
i
} = 10°, 15°, 30°, and two states
of polarization, TE and TM, i.e., twelve elements in
total. Gaussian noise with zero mean and a standard deviation of
σ = N dB is added to the input data. In test one
(the first column), N = -∞. In tests two
(the second column) and three (the third column), N
= -20 and N = -13, respectively.

Table 3

Root-Mean-Square Errors of Prediction between Output
${\overrightarrow{y}}_{{B}_{1}}$ from Network B_{1} and
Target Parameters $\overrightarrow{O}$, Averaged over 200 Test
Samplesa

Noise Level (dB)

rms_{
h
1
} (nm)

rms_{
h
2
} (nm)

rms_{
t
1
} (nm)

rms_{
t
2
} (nm)

rms_{
t
3
} (nm)

-∞

0.39

0.71

2.33

2.98

2.28

-30

1.04

1.80

3.53

5.78

3.04

-20

1.58

3.46

4.78

6.39

3.70

-13

2.69

5.79

6.98

11.19

4.82

The errors are presented for groove
depths h_{1} and h_{2} as well
as for transition points t_{1}, t_{2}, and
t_{3}. Input data consist of diffraction
intensities R_{0} and R_{-1}
calculated with three angles of incidence, θ_{
i
}= 10°, 15°, 30°, and two states of polarization, TE and TM,
i.e., 12 elements in total. Gaussian noise of zero mean and values
of σ -∞ dB (no noise), -30, -20, or -13 dB relative to
the intensity is added to test input vectors
${\overrightarrow{I}}^{\left(m\right)}$. Different networks are
taught for each amount of noise.

Table 4

Same as Table 3 Except That Input Data Consist of
Diffraction Intensities R_{0} and
R_{-1} Calculated with Three Angles of Incidence,
θ_{i} = 10°, 15°, 30°, and One State of
Polarization, TE, i.e., Six Elements in Total

Noise Level (dB)

rms_{
h
1
} (nm)

rms_{
h
2
} (nm)

rms_{
t
1
} (nm)

rms_{
t
2
} (nm)

rms_{
t
3
} (nm)

-∞

2.27

1.54

9.62

11.92

11.94

-30

2.46

2.47

11.13

16.06

12.41

-20

3.63

4.35

14.92

19.92

15.20

-13

4.97

6.47

17.66

19.88

17.00

Table 5

Target Values and Predictions Given by Networks
B_{1}, B_{2}, and B_{3} for Four Sample
Gratings Chosen from within the Combined Range of the Three-Level B
networksa

The rightmost column shows the sum of
absolute deviations between the predictions and the target values as
the sums are taken over h_{1}, h_{2},
t_{1}, t_{2}, and
t_{3}. Input data consist of diffraction
intensities R_{0} and R_{-1}
calculated with three angles of incidence, θ_{
i
}= 10°, 15°, and 30°, and two states of polarization, TE and
TM, i.e., 12 elements in total. If the correct sample parameters
are within the range of a specific network, the corresponding line is
typed boldface.

Table 6

Same as Table 5 Except That Here Gaussian Noise with
Standard Deviation σ = -13 dB Relative to the Intensity Is
Added To the Test Input Data