Organization of this thesis

1.3 Organization of this thesis

The thesis is organized as following: The possibility verification and an approach to retrieve 1-D subwavelength feature variation from far-field optical measurement will be shown in Chapter 2. In Chapter 3, a tunable asymmetrically embedded-aperture interferometer configuration was proposed to enhance detection sensitivity of 1-D subwavelength variation measurement. A three-detector, embedded-aperture interferometer configuration accompany blind signal separation method was proposed to recover 2-D subwavelength variation information of a rectangular aperture with far-field irradiance measurement was shown in Chapter 4.

In Chapter 5, a constructed-aperture measurement system behaving as an optical ruler to measure the marginal roughness of the test sample with error ratio below 3%

was proposed. Finally, a summary of this dissertation, and the future works are presented in Chapter 6.

Chapter 2

Retrieving of 1-D subwavelength variation information

2.1 Introduction

Recently, to detect subwavelength signal in the far-field is experimentally demonstrated to be possible while the diffraction structure size are fulfill with the assumptions of scalar diffraction theory^∗. It highlighted the possibility of measuring optical signals in the far field with sufficient sensitivity to show variations that are orders of magnitude below the wavelength of light. Could we retrieve it only by measuring far-field characteristics? With the academic aim of clarifying this point, we investigated to retrieve subwavelength dynamic variations of the most simple diffraction structures, slit and rectangular aperture. In this chapter, it will be shown that the 1-D subwavelength variation information is retrievable from far-field characteristics measurement.

2.2 Retrieving 1-D subwavelength signature from far-field irradiance

measurement

In section 2.2, a method was proposed to retrieve subwavelength variation of the diffraction structure by measuring far-field irradiance variation. The 1-D dynamic signature of the subwavelength variation of the simplest geometric structures, a slit and rectangular aperture, will be shown to be determinable from far-field irradiance with a precision of better than 1 nm ^[19].

Fig. 2-1. Schematic diagram of the variation diffraction structures

2.2.1 Basic assumptions

A physical quantity, derivative intensity, of these two simple diffraction situations was deduced to retrieve the subwavelength variation of these two diffraction structures. Considering the most simple situation that a monochromatic

plane wave of amplitude A and a wavelength λ is assumed to orthogonally illuminate on a rectangular aperture, with dimensions 2a and 2b . The observation plane was positioned at a distance

z away from the aperture. The

diffracted intensity was collected over a rectangular detector with dimensions 2

X

and 2 centered at the origin of the observation plane. The far-field intensity

Y

distribution on the detector is given by the expression ^[20]

( ) ( )

The overall power P_z , flowing through the collection region is given by

∫ ∫ ( )

P

, . And the physical quantity that derivative intensity was defined

as the derivative of P_z with respect to aperture width a,

da dP

_z

. The derivative intensities of these two situations could be derived as:

⎪⎪

where Si is the sine-integral function, and the function

f was defined and could

_p be evaluated by Leibniz Integral Rule ^[21] as

( ) ( ) ( ) ( )

The derivative intensities of the two diffraction structure are both proportional to the function ⎟

, which was shown in the Fig. 2-2 (In this demonstrated

figure, X is 100 m

μ

, z is 100 mm, and light source wavelength of 632.8nm was used.). It could be found that the fluctuations of derivative intensities were vary small while the width of the diffraction structure is varying in the subwavelength scale. We make the assumption that while the width of the diffraction structure is

varying in subwavelength scale, the variation of derivative intensity is small enough to be estimated as a constant value.

Fig. 2-2. (a) Schematic diagram of the variation of function Si

(

2kaX /z

)

(b) enlargement of part of the Fig. 2-2 (a)

Thus, the equation will approximately hold

t

The derivative intensity

a0

⎛ can be evaluated analytically and is a constant,

whereas the temporal variation of power

t

Therefore, Eq. (2-3) could be used to deduce the rate of variation of the width of the

slit

dt

t

da ⎟

⎠

⎜ ⎞

⎝

⎛ , so does the aperture width variation:

t dt

a da

t

⎟ Δ

⎠

⎜ ⎞

⎝

≈⎛

Δ * .

2.2.2 Simulation verification with subwavelength variation

Three different typical cases - periodic, quasi-periodic, and random variations - were used to investigate the feasibility of the proposed retrieving method. In the simulation, the amplitude of the vibration was taken to be 10nm at a frequency of 100 Hz . The sampling rate of detector was 1.0 KHz , which is higher than the frequency of the vibration of the slit. The half-width of the slit a was 50 m

μ

. The half width of the detector, X and Y were both 100 m

μ

; the detector was placed behind the lens with a focal length of 30mm. A light source wavelength of 632.8nm was used.

The sine function was used to represent the periodic subwavelength variation:

the slit variation then would be,

a ( ) t

=

a

₀+

α

×sin

(

2

π ft )

, where α was the amplitude of the vibration of the slit. The quasi-periodic subwavelength variation of the slit is given by ^a

( )

^t =^a0+α×^sin

(

²π^ft

)

+α×^sin

(

²π ^2ft

)

. The random fluctuation is specified by a

(

t+Δt

) ( )

=at +α×η, where

η

is a randomly selected value ranging from -1 to 1. Numerically, we know the exact width of the aperture

along the x direction from time to time, which is referred to as the simulation-setting value of the width of the slit. Figure 2-3 shows the relation between the deduced value and the simulation-setting value by the method we used. The curves on the left plot the deduced variation of the width of the slit, a*, and the curves on the right plot the difference between the deduced value and the simulation-setting value,

) ( ) (

*

t a t

a

− . The difference between the deduced and the simulation-setting values is about 10^-6nm. It means that the method of deduction is with very high precision, and the inaccuracy is only about 10^-7 of the vibration amplitude of 10nm.

Fig. 2-3. Deduced subwavelength variation, a*, and the difference between the deduced value and the simulation-setting value, a*−a, for a slit. (a), (b): periodic, (c), (d): quasi-periodic, and (e), (f): random fluctuation.

In the simulation of the rectangular diffraction aperture, all simulation settings were same as that of slit case, except the aperture widths a and b were both set to 50 m

μ

. As shown in Fig. 2-4, the difference between the deduced value and the simulation-setting value remained far below 1 nm (specifically, about 10^-5nm). In other words, in the case of a general rectangular aperture, subwavelength variation can be retrieved precisely from the far-field irradiance. Extensive simulations revealed that in the general case of a light diffracting rectangular aperture, even when the vibration amplitude is 1 m

μ

, the inaccuracy remains bellow 1nm.

Fig. 2-4. Subwavelength variation for a rectangular aperture. (a): deduced a*, (b):

the difference a*−a, where the dotted curve refers to the periodic case, the solid-line curve refers to the quasi-periodic case and the bold solid-line curve refers to the random case.

2.3 Retrieving 1-D subwavelength signature from far-field diffraction pattern measurement

In the section 2.2, the 1-D dynamic subwavelength variation signature is shown to be determinable a precision of better than 1 nm from far-field irradiance measurement by using the deduced quantity, derivative intensity. We may ask if there is any other quantity that could be used to retrieve the subwavelength variation of diffraction structure, besides the quantity of derivative intensity. Truly, derivative intensity is not the only far-field optical quantity of characterization. In addition, we have an alternative characterization with far-field diffraction pattern. The subwavelength variation of diffractive structure causes the variation in the far-field diffraction pattern. Hence, one can retrieve the information contained in the far-field diffraction pattern and use it to trace the scale of subwavelength variation. In the section 2.3, it will be shown that the associated shifting of the dark line of diffraction pattern, caused by subwavelength variation, had good linear correlation to that and will be magnified about hundred times. Hence, an alternative method of detecting subwavelength variation from far-field measurement, based on pattern measurement could be achieved.

2.3.1 Basic formalism

The basic formalism of associated shifting of the dark line of diffraction pattern to the corresponding 1-D subwavelength variation in two situations: (1) Direct observing the diffraction pattern, and (2) Observing interference pattern with an embedded aperture, will be addressed in this section.

(a) Directly observing the diffraction pattern

Consider the optical wave diffracted by a single aperture with the dimensions a

2 and 2b. The intensity distribution on the detector that positioned at a distance

z away from the aperture is

The analytical result of the dark line locations on the detector along the x direction is

, where m is the order of dark line pattern. Expanding the width at a certain specified aperture width , e.g.,

a

= , we have

a

₀

Apparently, when the aperture has a subwavelength variation and the diffraction aperture is much larger compared to the wavelength of light, then under the first order approximation, the relation between the pattern shift on the screen, Δ , and the

x

_d

subwavelength variation Δa is

a a z x_d ≅−m Δ

Δ ₂

2 0

λ . (2-7)

The inaccuracy ratio of the approximation is ~ a−a₀ / a₀.

(b) Observing interference pattern with an embedded aperture

In this section, we consider the diffraction pattern variation under embedded-aperture interferometer configuration. As shown in Fig. 2-5, it is a common Mach-Zehnder interferometer but with an embedded aperture. To be specific, a monochromatic plane wave of amplitude A and wavelength λ was assumed to be orthogonally illuminated on a beam splitter B1. The beam was split into two after passing through the beam splitter. One beam was reflected by a mirror M_1, and passed through a rectangular observation aperture S_1, with dimensions 2a and 2b. Another beam was reflected by mirror M_2, and passed through another rectangular embedded aperture S_2, whose dimensions were 2a' and 2b. (For simplicity, we set the two apertures to be different in one direction only.) Then the two beams were passed through another beam splitter B2 and recombined into one

beam. The diffracted intensity was collected over a rectangular detector. The detector was with dimensions X2 and Y2 centered at the beam width, and was positioned at a path distance z away from each aperture.

Fig. 2-5. Schematic diagram of embedded-aperture interferometer configuration

The interference intensity distribution of these two beams on the detector is

( )

Assuming that there is subwavelength variation along the x direction for the observation aperture with dimensions 2a and 2b , where

a

=

a

₀+Δ

a

and the symbol Δa denotes the subwavelength variation of the half aperture-width. The other beam passes through another embedded rectangular aperture, with dimensions

'

2a and 2b, where

a

' a= . Explicitly, from Eq. (2-8), the interference intensity ₀ distribution of these two beams on the detector can be rewritten as:

( ) ( ) ( )

where the symbol C denotes the function which has no relationship to the widths along x-direction of the two apertures. Therefore, one can use Eq. (2-9) to solve the numerical result of the dark line locations of the interference pattern. The condition follows that

Assuming that the variation scale, compared to the aperture width for the observing subwavelength variation was very small, we can deduce the linear relation between the subwavelength variation of the half aperture-width and the dark-line pattern shift as follow. The function ^sin

[ ^k ( ^a

0+Δ

^a ) ^x

^/

^z ]

was expanded to the half aperture in the subwavelength scale, with substitutions cos(

k

Δ

ax

/

z

)≈1 and

z

sin( Δ ≈ Δ , the interference intensity can be rewritten as

( )

x,y C

{

2sin

(

ka₀x/z

) (

k ax/z

)

cos

(

ka₀x/z

) }

²

I = + Δ × . The relations between the

dark-line position

x axis and the subwavelength variation

_d Δa is

( ka x

_d /

z ) k ax

_d /2

z

tan ₀ =− Δ . (2-11)

The pattern shift

a

da

( )

_⎟

⎛ . Because the variation of the aperture was in the

subwavelength scale, tan^{2 0} ⎟<<1

⎠ diffracted by the single aperture with half width

a , with a substitution

₀

x from Eq.

_d (2-7). Overall we can estimate the relation between the subwavelength variation of

the half observation aperture-width and the dark-line pattern shift in linear form a a

2.3.2 Simulation verification with subwavelength variation

To verify the feasibility of the relationship in Eq. (2-7) and (2-13) in detail, we carried out numerical evaluation for the variation of the first dark-line, while the observation aperture varying in subwavelength scale. In the case where the wavelength of the incident light was 632.8nm, the detector was at a distance of 100mm from the aperture, and the half aperture width along the x direction was 10 m

μ

; the results are shown in the lighter lines in Fig. 2-6. The analytical result is depicted using a solid line, while the first-order approximation is shown with a dotted

line. The shifting of the diffraction pattern, associated with subwavelength variation, had good linear correlation and was magnified about 300 times. In this case, when the aperture variation was under 0.5

μ m

, the inaccuracy of the first order approximation was under 5%. Taking the second order approximation, the inaccuracy would be less than 0.5%. Meanwhile, a direct numerical examination based on the Fraunhofer approximation ^[22], i.e., using the Fraunhofer diffraction integral, was used to evaluate the dark line position shift; this is shown with hollow triangular symbols. The diffraction pattern shift, with the half observing aperture with 100nm variation, is shown in Figs. 2-7(a) and (b). The half observation aperture was 10 m

μ

and the detector, with dimensions 100 m

μ

and 100 m

μ

, was

centered at the first order dark-line position at a distance of 100mm from the aperture.

The pattern shift was 31.82 m

μ

, considering the subwavelength variation of the aperture was magnified about 300 times.

Again, for a comparison of section 2.3.1, the numerical evaluation of observing interference pattern with an embedded aperture are shown in Fig. 2-6 in the thicker lines. The numerical result from Eq. (2-11) is shown with a solid line, and the linear approximation result (Eq. (2-13)) is shown with a dotted line. The examination of the Fraunhofer approximation for the dark line position shift is shown with solid

triangular symbols. The diffraction pattern shift, caused by a 100nm variation of the half observing aperture, is shown in Figs. 2-7(c) and (d). The parameters used in the simulation were as follows: the half observation aperture was 10 m

μ

; the detector (size: 100 m

μ

X100 m

μ

) was centered at the first order of the dark-line position on the focal plane; and the focal plane was located at a distance of 100mm from the observing aperture. From the figures, one can see that the pattern shift was 15.49 m

μ

; comparing the subwavelength variation of the aperture, the scale was magnified about 150 times.

Fig. 2-6. The dark-line position shift versus the half aperture variation. The thicker line denotes the interferometer configuration, while the lighter line denotes the single aperture.

Fig. 2-7. The diffraction patterns before and after 100nm half aperture variation in two different situations. Directly detected method: (a) and (b); embedded-aperture interferometer configuration: (c) and (d).

2.3.3 Discussion on contrast influence in pattern measurement

The shifting of the diffraction pattern associated with subwavelength variation held a good linear correlation; however, under embedded-aperture interferometer configuration, the scale was magnified about 150 times, which is only half of the directly detected method. The shifting amount of the dark line, however, is not the only factor in taking a good measurement. Contrast of the diffraction pattern is also

crucial in detecting the signal. To demonstrate the influence of the contrast, we first calculated the intensity difference between the maximum and the minimum diffraction patterns within the area that was centered at the dark line position with a finite width; this was 30 m

μ

, for both cases. We then normalized the intensity

difference; the results are shown in Figs. 2-8 (a) and (b). The parameters used in the simulation were as follows: the half observing aperture was 10 m

μ

; the detector was 30 m

μ

X 100 m

μ

and was centered at the first order dark-line position on the focal plane; and the focal plane was located at a distance of 100mm from the observing aperture. The cross-section along the x axis of the first dark-line of the two cases is shown in Fig. 2-8 (c). It is obvious that the diffraction pattern of the embedded-aperture interferometer configuration is sharper than that of directly detected method, which implies that it is easier to confirm the detection of subwavelength variation, using the embedded-aperture interferometer configuration.

Fig. 2-8. The diffraction patterns centered at the first dark-line position: (a) directly detected method and (b) embedded-aperture interferometer configuration. (c): the cross sections along the X axis of (a) and (b), where the thicker line represents the embedded-aperture interferometer configuration and the lighter line represents the directly detected method.

2.4 Summary

In summary, it is shown that 1-D subwavelength signature of diffraction structure can be retrieved from the far-field characterization, such as irradiance and diffraction pattern. The 1-D dynamic subwavelength variation signature of a slit and a rectangular aperture is shown to be determinable from its far-field irradiance with a precision of better than 1nm. Another feasible method of detecting subwavelength variation with diffraction pattern variation was also proposed. The variation of the subwavelength scale was verified contained in dark-line pattern shift and was

enlarged in an order about hundred times to be easily measured in the far-field.

Form the results of this chapter, it implies the possibility to extract much useful information, such as an object’s thermal characteristic, vibration, deformation and others in the precision of subwavelength scales, form only far-field optical measurement.

Chapter 3

Enhancement of detection sensitivity of 1-D subwavelength variation measurement

3.1 Introduction

In the chapter 2, a scheme to retrieve the dynamic signature of the subwavelength variation from far-field irradiance with an appreciable quantity- derivative intensity with a precision of better than 1 nm was proposed. However, while measuring the structure variation in subwavelength scale, what we retrieved in the far-field is usually a weak optical signal 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 a higher sensitivity. The enhancement of detection sensitivity is certainly possible to simply increase the light power that is transmitted through the test sample. However, in most situations, the test sample may suffer saturation and/or damage; hence, incident power must be limited. This means that enhancing detection sensitivity, via a direct increase of the incident power, simply may not work.

In this chapter, two embedded-aperture interferometer configurations were proposed, which could enhance the detection sensitivity of 1-D subwavelength variation measurement of a rectangular aperture with arbitrary aperture width ^[23,24]. In these configurations, an aperture (named the reference aperture) was posited symmetrically or asymmetrically relative to the aperture with the subwavelength variation was to be identified (named the test aperture). In symmetrical configuration, to enhance the detection sensitivity at any specific detection width, we have to modify the configuration and width size of the reference aperture. In asymmetrical configuration, the detection sensitivity could be enhanced at any specific detection width by only shifting the relative position of the reference aperture with fixed width size. On the other hand, with these two embedded-aperture interferometer configurations, the detection sensitivity is directly in proportion to the power of the reference beam. By increasing the power of the light beam transmitting through the reference aperture, detection sensitivity can be increased to a desired order without damaging the test sample owing to increase the incident power on the test aperture.

3.2 Symmetrically-embedded-aperture interferometer

3.2.1 Basic formalism and general features

Although it is not necessary to be limited to one specific interferometer configuration,

for simplicity, we have demonstrated a typical scheme. The proposed configuration, which is similar to the Mach-Zehnder Interferometer structure, is shown in Fig. 3-1.

A common Mach-Zehnder interferometer was used, but with an embedded aperture, for which the associated subwavelength variation can be detected. In addition, another aperture was embedded for reference and optimization control.

Fig. 3-1. Schematic diagram of interferometer configuration.

To be specific, a monochromatic plane wave of amplitude A and a wavelength λ was assumed to be orthogonally illuminated on a beam splitter B1. The beam

was split into two after passing through the beam splitter. One beam was reflected by a mirror M_1, and passed through a rectangular observation aperture S_1, with dimensions 2a and 2b. Another beam was reflected by mirror M2, and passed through another rectangular embedded aperture S2, whose dimensions were 2a' and

b

2 . (For simplicity, we set the two apertures to be different in one direction only.)

Then the two beams were passed through another beam splitter B₂ and recombined into one beam. The diffracted intensity was collected over a rectangular detector.

The detector was with dimensions X2 and Y2 centered at the beam width, and was positioned at a path distance z away from each aperture.

In the far field region, the diffraction field is the Fourier transform of the

In the far field region, the diffraction field is the Fourier transform of the

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