Simulation verification and Discussions

In this section, we provide the support evidence of our approach. The simulation setting of the observation aperture A₁ was the same as that in section 4.2.3. Without loss of generality, the width variation form of the embedded-aperture

A2 is chosen as a sine form, i.e.,

d ( ) t

=

d

₀ +

α

sin

(

2

π ft )

. The half width of the embedded aperture is chosen as 70 m

μ

, and the amplitude of variation α is set to be 10 nm and the frequency of vibration is taken to be 100 Hz .

The retrieving results of the proposed three-detector configuration are shown in Fig. 4-5. The relation between the time delay, τ , to the inverse mean error of the retrieving embedded-signal variation Δ^d

( )

^t is shown in Fig. 4(a) and hence, the time

delay is chosen as 0.0294 sec, so as to have a higher performance. The errors,

( )

t a

( )

t

a Δ

Δ ' − and Δb'

( )

t −Δb

( )

t are shown in Figs. 4(b) and (c) in thick lines, while the

setting aperture width variations ^Δa

( )

^{t and Δb}

( )

^t are plotted in gray thin lines as a reference, and the error percentage, the ratio that the difference between exact setting and numerically retrieving value to the amplitude of the setting aperture width variations value, are plotted in the base portion. Referring to Fig. 4-5, one can see that with the three-detector embedded-aperture configuration, the subwavelength signatures can be retrieved in a good precision and the retrieving aperture width variations, Δa' and Δb' are with an error ratio below 1%.

Fig. 4-5. (a): The inverse mean error of the retrieving embedded-aperture variation versus the time

delay, τ . (b) the error and error percentage between setting and retrieving aperture width variations:

Δa and Δa', (c) the error and error percentage between setting and retrieving aperture width variations:Δb and Δb'.

There are two points that should be noted. First, the embedded variation of the

reference aperture, ^Δd

( )

^t , is unnecessary to be in the subwavelength scale, but could be in a larger scale to be more easily to carry out. A larger half aperture width of

mm

d

₀ =0.5 , that varies with a larger amplitude of

α

=1

μ m

can still be used to retrieve the two-dimension subwavelength with an acceptable error ratio below 3%

signal. Secondly, a better result could be obtained with more recorded data. For example, in our simulation, 2000 recording data was used to achieve the maximum error ratio bellow 1%. But with 600 recording data can only have a best maximum error ratio of 1.94%.

4.4 Summary

While retrieving 1-D subwavelength-scale variation from far-field measurement has been demonstrated to be possible, we want to find approaches to retrieve subwavelength-scale variation toward more realistic situation. Thus, in this chapter, a rectangular aperture varying in 2-D with in subwavelength-scale has been discussed.

A scheme to retrieve the coupled two-dimension subwavelength signatures by a three-detector embedded-aperture configuration accompanying time-delayed correlation method from far-field irradiance measurement was proposed. The precision of the proposed measurement was numerically verified could successfully characterize the two-dimensional dynamical signatures of subwavelength variations

information with error ratio below 1%.

Chapter 5

1-D Marginal Roughness Measurement

5.1 Introduction

In the chapter 2 and chapter 4, it has been shown that the subwavelength temporal variation of a simple case, i.e., a 1-D and 2-D subwavelength-scale temporal variation can be retrieved from a far-field irradiance measurement. It is of interests to express the possible implementation of subwavelength spatial variation in terms of far-field characteristics, and thus to retrieve the subwavelength-scale spatial variation from far-field measurement. In this chapter, a conceptual construction will be proposed as an optical ruler, which could be used to identify the spatial

subwavelength scale marginal-roughness variation from only far-field irradiance

measurement.

5.2 Constructed-aperture roughness measurement system

Referring to Fig. 5-1, a constructed-aperture measurement system behaving as an optical ruler was proposed to retrieve the marginal roughness of the test sample. The diffraction aperture Σ was constructed by a slit-like aperture and the margin of the

test sample, where the width of the slit-like aperture along the

η

direction was denoted as b. A monochromatic plane wave of amplitude 1 and wavelength λ was assumed to be orthogonally illuminated onto the constructed aperture Σ . The margin of the test sample was situated relative to the straight margin of the slit-like aperture in a base width

a

₀. The dimensions of

a

₀ and b were both above several wavelengths for satisfying the assumptions of the scalar diffraction theorem.

The front view of the constructed aperture was shown in Fig. 5-1 (b). A detector with the size WXW was positioned behind the aperture Σ at a distance

Z

₀ in the far-field region, or by introducing a focal lens just behind the aperture Σ , situated the detector at the focal plane of the lens. Thus the diffraction pattern on the detector plane could be evaluated by the Fraunhofer diffraction. The overall power collected by the detector was denoted as P_z.

Fig. 5-1. Schematic diagrams of (a) constructed-aperture marginal roughness measurement system, (b)

a front view of the constructed aperture of the system.

To illustrate the main idea of the proposed scheme, we recalled that in chapter 2, 1-D subwavelength temporal variation can be retrieved from a far-field irradiance

measurement in a precision better than 1nm by the following approximation, i.e.,

⎟⎠

⎜ ⎞

⎝

≅ ⎛ ₌

/ _{a a}0

z

da

P dPz

a

Δ

Δ . (5-1)

In the proposed constructed-aperture measurement system, while the width deviation from the base width

a

₀, Δa, of a rectangular aperture in a dimension

a

₀Xb, was in a scale of subwavelength, the aperture width deviation Δa could be retrieved

from a physical parameter, i.e., the derivative intensity

a0

da

a

dPz

= . ΔP_z is the deviation of the overall power comparing to that of the base width

a

₀,

P

_z

( ) a

0 , i.e.

( ) a P ( ) a

0

P

P

_z = _z − _z

Δ . The key idea of the proposed measurement is that, while the

width of the optical ruler b is small enough as compared to the spatial-variation scale of the marginal roughness

T

_d, the margin of the test sample could be estimated as a straight surface. The width of the constructed aperture (or say the averaging width) was denoted as

a , which could be retrieved by far-field irradiance

measurement. The procedures are stated as below.

First, we need to derive the derivative intensity of an aperture varying in one-dimension with only one side, in a correspondence to the constructed measurement system here. It is for the reason that while measuring the marginal roughness, the test sample will be moved in the

η

direction. Because the marginal

roughness is varied spatially from point to point, the width of the constructed aperture Σ , a, will be changed as the sample is moved. The constructed aperture will vary

only on one side and the derivative intensity of measured power, dP_z/da, could be deduced following the same approaches as in chapter 2.

The exact form of the diffraction pattern on the detector U can be derived from

Fraunhofer diffraction, and the intensity is

U

². The overall power collected by the detector is the integration of the intensity over all the complete detector area, which is also the function of the varying aperture width a,

( )

^{a Z f}

(

^{W a}

) (

^{f W b}

)

intensity is the derivation of overall power P_z over the varying aperture width a, which can be derived as

The numerical value of the derivative intensity

a0

da

a

dPz

= can be evaluated from Eq.

(5-3), and can be substituted into Eq. (5-1) to retrieve the margin position of the test sample as a=a₀+Δa.

5.3 Thickness effect of test sample

In section 5.2, the constructed aperture is estimated as an ideal planar aperture.

However, in a real situation, the test sample will have a thin thickness d inevitably.

As shown in the Fig. 5-2, the actual constructed aperture Σ will have an inclination '

angle,

(

0

)

1 /

tan⁻

d a

φ

= , to the incident plane wave.

Fig. 5-2. Schematic diagram of a side view of the constructed-aperture marginal roughness

measurement system.

The introducing power deviation from the thickness of the test sample cannot be neglected as comparing to the power deviation caused by the constructed-aperture width variation in the subwavelength scale. In other words, a solution to recover the

influence caused by the sample thickness has to be considered. Referring to Fig. 5-2,

considering that a plane wave passes through a ideal plane rectangular aperture Σ , the diffraction optical field on the detector D behind the aperture in the far-field distance

Z

₀ can be evaluated by Fraunhofer diffraction, i.e.,

( ) ( ) ( ) (

ξ η

)

ξ η

λ η π

λ U ξ j Z ^{x y d d}

Z y x y C

x

U_P ⎥

⎦

⎢ ⎤

⎣

⎡− +

=

∫∫

Σ 0

0 0

0 2

exp , ,

, , (5-4)

where for an aperture,

U

0

( ) ξ

^,

η

=¹ and

C

₀

( ) x

,

y

is the phase term. What we want to do is to find a suitable orientation of the detector, which could have an analytic solution of derivation intensity. Hence, the width variation of the

constructed-aperture could be retrieved by correlating with far-field irradiance variation from the value, derivation intensity.

Referring to Fig. 5-2, considering the situation that the constructed aperture Σ ' has an inclination angle

φ

to the incident plane wave, the diffraction optical field on the plane orthogonally behind the aperture Σ in the far-field distance '

Z

₀ can be evaluated as an oblique plane wave incident on a plane aperture Σ , i.e., '

( ) ( ) ( ) (

ξ η

)

ξ η

Referring to Fig. 5-2 and comparing Eqs. (5-4) and (5-7), this means that if we position the detector at a new position, the only difference in these two equation is the

diffraction aperture width of constructed aperture Σ that effective aperture width is ' Ψ

/

a , and the tilt factor Ψ=cos

φ

. The analytic derivative intensity of the detector at a new position could be solved as

( )

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ Ψ

= Ψ

0 0

, 2 2 /

8 Z

Si kaW b W k f

Z da dP

p

z , (5-8)

and thus can be substituted into Eq. (5-1) to retrieve the margin position of the test sample as a=a₀+Δa. This means that if we position the detector at a new position, which is:

(i) Rotating the original detector D relative to the aperture in an angle

φ

to be parallel to the constructed aperture Σ , and '

(ii) Shifting it in a distance Δ

x

'=

Z

₀sin

φ

along the direction of +x'-axis, the corresponding derivative intensity of the detector at a new position could be deduced, and to substituted into Eq. (5-1) to retrieve the margin roughness of the test sample. While the thickness of the test sample is small compare to the aperture width, tilt factor Ψ ≅1; the influence of Ψ is small and thus can be further neglected.

5.4 Simulation verifications and discussions

The feasibility of the proposed marginal roughness measurement will be numerically demonstrated as below. We first set the marginal roughness of test sample, and use a base width

a

₀= 50um to evaluate the base overall power

P

_z

( ) a

₀ . While moving the test sample, the corresponding overall power variation ΔP_z will be substituted into Eq. (5-1) to retrieve Δa, the deviation from marginal position to

the base width, and the exact marginal position of the test sample is simply the sum of the base width

a

₀ and the deviation Δa. The width of the optical ruler, b, was chosen as 6 m

μ

, which is about 10 times the size of the light source wavelength, 632.8nm that considered here. The base-width of the aperture

a

₀ was 50

μ m

. The width of the detector W was also 50 m

μ

, and the detector was placed behind the aperture at a distance of 30cm.

Without loss of generality, two marginal roughness profiles, i.e., sine variation

( ) a ( T

_d

)

a η

= ₀ +

α

×sin 2

πη

/ and quasi-periodic variation

( ) a ( T

d

) ( T

d

)

a

sin2 2 /

/ 2 2 2 sin

0

α πη α πη

η

= + × + × were used to simulate the marginal

roughness, and the thickness of the test sample was taken as one wavelength. The amplitude of the marginal roughness fluctuations, α , was set to 10 nm, and

T

_d was the spatial variation scale of the surface roughness.

The simulation processes are that: the marginal form of constructed-aperture is varying as the pre-setting function, a

( )

η . The corresponding collected power on the detector '

D

is evaluated numerically, and then substitute the overall power variation

Pz

Δ into Eq. (5-1) to get the width deviation Δa from bath width

a

₀, then we get

the estimated marginal roughness form: a=a₀ +Δa and compare it with the

pre-setting center roughness value of the constructed aperture. The retrieving results of the proposed marginal roughness measurement of different spatial scale

T

_d were shown in Fig. 5-3. Three different spatial variation scales,

T

_d= 2b, 5b and 10b, were used to explore the feasibility and the precision limitation of the proposed measurement. The deduced results of two marginal roughness forms are shown separately in Figs. 5-3 (a) and (b) by comparing the width deviation Δa to that of the pre-setting value, and the maximum error percentage of the retrieving results were shown in the figures. The pre-setting marginal roughness was plotted in thin lines as reference, spatial variation scale

T

_d= 2bwas plotted in gray thin lines,

T

_d= 5bwas plotted in thick lines, and

T

_d= 10bwas plotted in dot lines.

Fig. 5-3. The retrieving results of two different marginal roughness profiles: (a) sine variation. (b)

quasi-periodic variation.

It can be seen that the proposed method of roughness measurement is workable.

The relation between optical ruler width b and the spatial variation scale

T

_d

determines the measurement precision of the proposed scheme. If the width of the optical ruler is 1/5 of the spatial variation scale

T

_d, the precision of the proposed

marginal roughness measurement will exhibit a maximum error percentage below 10%. If the width of the optical ruler is 1/10 of the spatial variation scale

T

_d, the maximum error percentage will be further reduced to below 3%. From our simulation results, while the width of optical ruler is smaller than 1/5 of the roughness spatial variation scale

T

_d , the measuring precision of roughness varying in an amplitude 10nm is better than 1nm,

Besides, we should note that while using a shorter wavelength, the width of optical ruler could be reduced. It means that by using an optical ruler with a shorter wavelength: (1) a higher precision can always be achieved for measuring the same sample, and (2) the restriction of the spatial variation scale

T

_d of the test sample will

be released. Furthermore, it should be noted that the proposed measurement is still workable even when the fluctuation amplitude of the marginal roughness, α , is

increased to the value of one wavelength. And, the tunable embedded-aperture

interferometer configuration illustrated in the chapter 3 could be further implemented to increase the detection sensitivity.

5.5 Summary

In summary, a constructed-aperture measurement system behaving as an optical ruler was proposed to measure the marginal roughness of the test sample. The precision of the proposed method of roughness measurement is only depending on the relation between optical ruler width b and the roughness spatial variation scale

T

_d.

It has been numerically demonstrated that with the proposed method while the width of the optical ruler is 1/10 of the spatial variation scale

T

_d, the maximum error percentage or the retrieving subwavelength-scale marginal roughness could be below 3%. Better retrieving results can be further obtained by choosing an optical ruler with a shorter width.

Chapter 6

Conclusions and Future Works

6.1 Conclusions

Retrieving subwavelength information is an extensive and important topic and thus has been widely investigated. Several measurements have been proposed to retrieve the subwavelength feature detail of specimen while specimen size was in mesoscopic or nanoscopic region. While the retrieving of dynamic signature of subwavelength variation yields some more interesting information than the static features, particularly in determining physical origins and in identifying the generation mechanism, e.g., thermal characteristic, vibration, deformation. Thus, to retrieve subwavelength-scale dynamically variation is another important issue should be further investigated.

Owing to the experimental result that subwavelength feature variations of an object can affect the corresponding far-field diffraction pattern in a measurable way.

The far-field optical measurement was provided as a potential approach to have real-time high-precision measure of subwavelength-scale dynamical variation of

structure. Thus, in this thesis, we have investigated the approaches to retrieving dynamic signature of 1-D subwavelength-scale variation, to enhance detection sensitivity while measuring 1-D subwavelength variation, and to decouple 2-D subwavelength variation with measuring far-field optical characteristics. Besides, an extension application to identify the subwavelength-scale marginal roughness from only far-field irradiance measurement has also been proposed.

6.1.1 Retrieving of 1-D subwavelength variation information

We investigated approaches to retrieve 1-D subwavelength dynamic signatures of two simple diffraction structures, slit and rectangular aperture. Two correspondence far-field characteristics variation, irradiance and diffraction pattern was proposed as good feature quantities to retrieve 1-D subwavelength dynamic signatures.

First, with the method that retrieving subwavelength variation of the diffraction structure by measuring far-field irradiance variation. A physical quantity, derivative

intensity, of these two simple diffraction situations was deduced to retrieve the 1-D

dynamical subwavelength variation of these two diffraction structures. The dynamic subwavelength variation signature of both two diffraction structures are shown to be

determinable from its far-field irradiance with a precision of better than 1nm. Secondly, with the method that retrieving subwavelength variation of the diffraction structure by measuring far-field diffraction pattern variation. The analytical approximation relation between the dark line locations of the diffraction pattern and 1-D dynamical subwavelength variation was derived. The shifting of the diffraction pattern associated with subwavelength variation was verified holding a good linear correlation and was in an order about hundred times to the subwavelength-scale feature size variation, thus behaving as a good feature quantity to retrieve 1-D subwavelength feature variation.

6.1.2 Enhancement of detection sensitivity of 1-D subwavelength variation measurement

While measuring the structure that varying in subwavelength scale, a weak signal is usually retrieved and hence, it is a critical issue to enhance the detection sensitivity of the measurement. Therefore, effective measurement methodologies must be developed to retrieve subwavelength variation from far-field measurement with higher detection sensitivity. For avoiding damage the sample, an approach to enhance detection sensitivity without increasing the light power transmitting through the test sample was investigated.

A tunable asymmetrically-embedded-aperture interferometer configuration was proposed could efficiently enhance detection sensitivity of 1-D subwavelength variation measurement. The interferometer configuration is similar to the Mach-Zehnder interferometer structure but with an embedded aperture posited asymmetrically relative to the observing aperture with the subwavelength variation to be identified. With this configuration, the detection sensitivity of 1-D subwavelength variation measurement could be successfully increased to a desired order by increasing the light power passing through the embedded aperture. Besides, by simply shifting the relative position of the embedded aperture, the detection sensitivity could be enhanced at any specific detection width.

6.1.3 Deconvolution of 2-D subwavelength variation information

The investigation on far-field measurement schemes and detection sensitivity enhancement scheme of retrieving subwavelength dynamics signatures was started from that the diffraction feature was only varying only in 1-D. However, in more realistic situations, the structure may vary in two or three dimensions, and the characteristics of subwavelength variations contained in the far-field are coupled and thus will be difficult to separate. To retrieve multi-dimension subwavelength

dynamics signatures of diffraction structure, the approach to separate the coupled far-field characteristics containing multi-dimension subwavelength variation information was explored.

A three-detector, embedded-aperture interferometer configuration was proposed to record the far-field irradiance information that containing coupled 2-D subwavelength variation information of a rectangular aperture. The coupled far-field irradiance information was then separated by a blind source separation method, time-delayed correlation method. The precision of the proposed measurement was numerically demonstrated could successfully characterize the two-dimensional dynamical signatures of subwavelength variations information with error ratio below 1%.

6.1.4 One-dimension Marginal Roughness Measurement

A constructed-aperture measurement system behaving as an optical ruler was proposed to measure the marginal roughness of the test sample. The precision of the proposed method of roughness measurement is only depending on the relation between optical ruler width b and the roughness spatial variation scale

T

_d. It has been numerically demonstrated that with the proposed method while the width of the

optical ruler is 1/10 of the spatial variation scale

T

_d, the maximum error percentage of the retrieving subwavelength-scale marginal roughness could be below 3%. From this discussion, it emphasizes that not only subwavelength-scale temporal variation but also subwavelength-scale spatial variation could be retrieved from far-field characteristics measurement.

6.2 Future works

In this thesis, we have demonstrated the feasibility of retrieving dynamic signature of 1-D subwavelength-scale variation, enhancing 1-D subwavelength variation measurement intensity, and decoupling 2-D subwavelength dynamic variation with far-field optical measurement. Besides, an extension application to identify the 1-D spatial subwavelength-scale marginal roughness from only far-field irradiance measurement has also been demonstrated to be possible.

Owing to the preliminary results we obtained as described in this thesis, we

Owing to the preliminary results we obtained as described in this thesis, we

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