Apertureless Scanning Near-field Optical Microscopy Using Heterodyne Detection Method

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Apertureless Scanning Near-field Optical Microscopy Using Heterodyne

Detection Method

Chin-Ho Chuang and *Yu-Lung Lo

Department of Mechanical Engineering, National Cheng Kung University,

No.1, Ta-Hsueh Road, Tainan 701, Taiwan

*Correspond author:

[email protected]

ABSTRACT

Heterodyne detection is most common technique in apertureless scattering near-field optical microscopy (A-SNOM), because it can get better signal-to-noise ration (S/N ratio) than the homodyne detection. Fore-researchers provided a rough way to express the heterodyne detection signals in order to avoid the complicated interference signal expressions. Therefore, the explanation is not clear and missing much information in signal processing. In this study, we analyze and verify the amplitude and phase of heterodyne detection signals in complete interference model which including the tip enhancement phenomena and tip reflective background electric field in different harmonics of tip vibration frequency. From our analytical results, we propose meaningful concepts: (1) the high order harmonic tip scattering noise decays faster with high order Bessel function in small phase modulation depth than the near-field interaction signal; (2) longer wavelength and smaller incident angle of incident electric field has better signal contrast, and it opposites traditional optical microscope about shorter wavelength has better resolution.

1. Introduction

Aperture scanning near-field optical microscopes (SNOM) are most common instruments with resolution below diffractive limits [1,2]. The tapered metal-coated optical fiber aperture confines the illumination or collection electric field range in the near-field area, therefore, the background noise isn’t serious issue. However, the aperture SNOM has some drawbacks [3]: (1) the inherent resolution is limited to about 20 nm in the visible light because of the finite skin depth of the metal defining the aperture; and (2) the light throughput for such small apertures is scarce and the incident power level is limited by thermal damage of the probes.

With attain sub-10 nm resolution, the development of apertureless-SNOM (A-SNOM) has been demonstrated that the use of a sharp tip ending with a nanometer-scale radius of curvature can pride a local enhancement of the field by several orders of magnitude [4-6]. However, the A-SNOM has serious and complicated background interference electric field with the near-field electric field, since the phase of the stray fields depends on the relative distance between these sources, the background fields is not a constant plateau. One of crucial technique of A-SNOM is to eliminate the background-scattering contributions from the detector signal. The heterodyne [7-8] or homodyne [8-9] detection techniques are the most common technique using in A-SNOM. In order to simplify the signals, they treat the modulation signal in a rough way, because they consider all the lock-in order frequency electric fields mix the near-field and background. Therefore, they get simple intensity signal formula instead of complicated interference expressions. The interpretation doesn’t express the properties of the signal and background noise clearly. Recently, some studies begin to analyze the complicated background noise in transmission or total reflection type A-SNOM [10-12]. However, they didn’t analyze the signal by heterodyne Proceedings of the XIth International Congress and Exposition

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technique in different harmonics of tip vibration frequency and discuss the signal contrast relation between signal phase modulation depth and phase difference. Therefore, they didn’t discuss the effects about wavelength, incident angle, and tip vibration amplitude in heterodyne detection. We had proposed an analytical study about A-SNOM homodyne signal analysis [13], and the homodyne signals are more complicated and weaker. However, the near-field signal can be magnified by the heterodyne detection [9]. Therefore, it is the best method to measure the near-field complex dielectric constant.

In this study, we focus on the reflective type of A-SNOM and heterodyne detection method as ref. [7-9], because it’s the most common A-SNOM and real nano-scale surface material parameters measurement. We study the complicated modulation signal in a detail interference mathematical model and simulate the signal contrast with the view of heterodyne detection in difference modulation frequencies. We propose some new important concepts which will help us to understand and improve signal contrast through the simulation results and compare with the previous research works.

2. Electric fields in heterodyne A-SNOM

Fig. 1 presents a schematic illustration of a Mach-Zehnder interferometer-type A-SNOM. As shown, a frequency shift

Δω

is added to the reference beam by a generic frequency shifting device such as an acousto-optic modulator (AOM). Therefore, the reference beam can be expressed analytically as

=

JJK

Reference

E

i(( )t R) R

E e

ω Δω+ +φ (1)

where

E

_R is the amplitude of the reference beam and

ω

and are the frequency and initial phase of the incident light, respectively. As shown in Fig. 1, the measurement beam,

JJK

E

i, is focused on the AFM tip by an objective lens and produces a corresponding enhancement of the near-field electric field. Fig. 2 presents an enlarged view of the near-field region. As shown, an assumption is made that all of the incident light and detected light passes through the objective lens. The incident electric field,

JJK

E

i, strikes the sample with an angle

θ

and produces three discrete electromagnetic wave sources, namely (a) an interaction signal between the AFM tip and the sample; (b) a scattering electric field from AFM tip; and (c) a scattering electric field from the sample. The field of interest in A-SNOM is that produced by the interaction between the AFM tip and the sample. According to the general model of quasi-electrostatic theory [4-6], the interaction (or tip enhancement) electric field between the AFM tip and the sample can be formulated as

T S

E

−

=

JK

( TS) ( TS) i t i t eff

E e

i

E

T S

e

ω φ ω φ

α

+ + −

≡

(2) where

α

_eff is the effective polarizability,

E

_iis the amplitude of the incident electric field, and

ω

and

φ

_TS are the frequency and initial phase of the interaction light, respectively. The

α

_eff is a highly important parameter since it contains everything necessary to predict the relative constants observable in the A-SNOM technique. In practice, its value is determined by the tip radius, the dielectric constants of the AFM tip and the sample, respectively, and the tip-sample distance [4-6]. Generally speaking, A-SNOM is performed using a commercial AFM operating in tapping mode.

In the A-SNOM procedure, the AFM drives the probe, as illustrated in Fig. 2, with a vertical cosine vibration around a mean position

Z

₀. Assuming that the amplitude and frequency of the vibration of the probe are denoted by

A

and

ω

₀, respectively, the variation of the tip position over time can be written as

0 0

( )

cos(

)

Z t

=

Z

+

A

ω

t

(3) In the present analysis, it is assumed that the AFM tip does not perturb the near-field region. Consequently, the scattering electric field from the tip can be formulated as

=

JJK

Tip

E

i( t T) i(2 sin( ) ( ))K Z t T

E e

ω φ+

e

θ (4) R

φ

(3)

where

E

_T and

φ

_T are the amplitude and initial phase of the scattering electric field,

ω

is the frequency of the incident light, and K is the wave number of the incident light and is given by

2 /

π λ

. In addition, i(2Ksin( ) ( ))Z t

e

θ

represents the phase vibration caused by the probe’s vertical dither.

The third electric field in the near-field region is that of the scattering light from the sample surface. Since this electric field is not modulated by the AFM tip, it can be expressed simply as

=

JJK

Sample

E

i( t S) S

E e

ω φ+ (5)

where

E

_S and

φ

_S are the amplitude and initial phase of the scattering light.

Fig. 1 Schematic illustration of heterodyne A-SNOM.

Fig. 2 Detailed view of near-field region in A-SNOM.

Laser BS Detector Δω±nω0 Objective Lens Lock-in ω0 Sample BS Frequency shifter Mirror ω+Δω ω BS

JJK

Reference

E

i

E

JJK

θ

JJK

i

E

( )

b E

JJK

Tip −

JJK

T S

a E

( )

0 0

( )

cos(

)

Z t

=

Z

+

A

ω

t

AFM Tip Sample

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3. Signal analysis in heterodyne A-SNOM

The total electric field entering the detector is given by the sum of the three electric fields in the near-field region and the reference beam, respectively, i.e.

−

=

+

JJK

total T S Tip Sample Reference

E

(6) Therefore, the corresponding intensity signal

I t

( )

is given by

( )

hom

( )

het

( )

I t

=

I

t

+

I

t

(7) in which the homodyne intensity component may be further derived as

hom

( )

cos(

)

cos[

sin( )

sin( ) cos(

)]

cos[

sin( )

sin( ) cos(

)]

2 2 2 T S T S T S S TS S T S T S 0 0 T S T T TS 0 0

I

t

E

2E

E

2E E

2K

Z

2K

A

t

2E

E

2K

Z

2K

A

t

φ

φ φ

θ

ω

φ φ

θ

ω

− − −

=

+

−

+

−

+

−

+

(8)

while the heterodyne intensity component has the form

( )

cos(

)

cos(

)

cos(

sin( )

sin( ) cos(

))

2 het R T S R R TS S R R S R T R T 0 0

I

t

E

2E

E

t

2E E

t

2E E

t

2K

Z

2K

A

t

Δω φ φ

θ

ω

−

=

+

−

+

−

+

−

(9)

The component of interest in the current analysis is the heterodyne intensity,

I

_het

( )

t

. Applying the Fourier Bessel series expansion and defining the phase modulation depth as

ψ

₁

=

2

K

sin( )

θ

A

and the phase difference as

2 R T

2

K

sin( )

Z

0

ψ

=

φ φ

−

θ

, Eq. (9). Since the amplitude of the interaction electric field is nonlinear, an assumption is made that

E

_{T S}− can be expressed as the sum of the individual components oscillating at different

harmonics of the AFM probe modulation radian frequency [14], i.e.

0 0 0 0

0 1 2 3

0 0 0

cos(

)

cos(2

)

cos(3

) ...

T S T S T S T S T S

E

₋

=

E

₋ω

+

E

ω₋

ω

t

+

E

ω₋

ω

t

+

E

ω₋

ω

t

+

(10) The series coefficients, n 0

T S

E

ω₋ , can be obtained from the Fourier components of

E

_i

α

_eff . Substituting Eq.(10) into Eq.(9), the heterodyne intensity

I

het

( )

t

.

Rearranging Eq.(12) according to the order of the modulation frequency, i.e.

(

Δω

+

n

ω

₀

)

t

, the heterodyne intensity signal can be separated into the following components:

( )

2

...

het R

I

t

=

E

DC

(

) cos(

)

cos(

)

cos(

)...

0 0 R T 0 1 2 T S R R T S S R R S

2E E J

t

2E

E

t

2E E

t

ω

ψ

Δω ψ

φ

Δω

φ

Δω

− −

+

−

+

−

+

(5)

(

) cos(

) sin(

)

cos(

) cos(

)...(

)

0 R T 1 1 0 2 1 T S R 0 R TS 0

4E E J

t

2E

ω

E

t

ψ

ω

Δω ψ

ω

φ

Δω

Δω ω

−

+

−

+

±

(11)

(

) cos(

)

cos(

) cos(

)...(

)

0 R T 2 1 0 2 2 T S R 0 R TS 0

4E E J

2

t

2E

ω

E

2

t

2

t

ψ

ω

Δω ψ

ω

φ

Δω

ω

−

+

−

+

±

(

) cos(

) sin(

)

cos(

) cos(

)...(

)

0 R T 3 1 0 2 3 T S R 0 R TS 0

4E E J

3

t

2E

ω

E

3

t

3

t

ψ

ω

Δω ψ

ω

φ

Δω

ω

−

+

−

+

±

+ Higher Order Heterodyne Modulation Frequency Terms

Eq.(11) provides a basic insight into the fundamental characteristics of the heterodyne intensity signal. For example, it can be seen that it is impossible to acquire the absolute interaction electric field

JJK

E

T S− with the tip scattering field. Secondly, the intensity of the background electric field

E E

_R _T has a coefficient of

J

_n

(

ψ

₁

)

, and thus if the higher-order coefficients decay more rapidly than the n 0

T S

E

ω− signal, the lock-in detection signal will exhibit

an improved signal contrast between different samples at higher-order harmonic modulation radian frequencies.

4. Comparisons Between Simulations and Previous Experimental Results

In Eq.(13), we get the lock-in detection signals from DC to the

(

Δω

+

3

ω

₀

)t

order harmonic radian frequency. In order to prove it and get the reasonable relative amplitude of electric fields,

E

_R,

E

_S, and

E

_T₋_S, we simulate the different order harmonics of heterodyne detection signal and compare with fore-experimental results [7,15]. Firstly, we assume the amplitude of DC (

0

ω

₀) component of the interaction electric field 0 0

T S

E

ω₋

∝

−

JJK

T S

E

∝

α

_eff

E

i

JJK

. Therefore, we can simulate the amplitude of electric field 0 0

T S

E

ω₋ by using the effective polarizability formula [9], and let the amplitude of long tip-sample distance approach to 1 by dividing a constant. The 0 0

T S

E

ω− amplitude

versus distance

Z

₀ is shown in Fig. 4. The wavelength is 633 nm HeNe laser, the sample is Si with dielectric constant 15 [ no unit], and the tip sphere to be Au with radius a = 20 nm and dielectric value -10 + 2 I [15]. From Fig.4, in the short distance

Z

₀ region (0 ~ 40 nm), the amplitude of interaction electric filed decays steeply, however, it keeps constant 1 after 40 nm. It is because that the electric dipole enhancement effect is enormous in the near-field region but keeps constant in the long distance

Z

₀.

Fig. 3 The calculated DC component of near-field interaction electric field amplitude versus the distance Z between the tip and the sample.

Amplit ude ( arb. unit s) 0

Z

(nm)

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Secondly, the n-th order interaction amplitude is given

E

T S

n

n S T

/

3

0 0 0ω ω −

−

=

that ideally approximates to the

Fourier components with

A

=

0 5

. a

in ref. [9] and the phase difference

φ

_R

−

φ

_TS is equal to

π

4

. The phase

φ

_TS of near-field electric field is assumed independent of distance

Z

₀, because the phase

φ

_TS is only relative to the effective polarizability

α

_eff in ref. [9]. Thirdly, we set the amplitude of tip reflective

E

_T and reference

E

_Relectric fields are 0.5 and 1000, respectively. The phase difference can be written as

ψ

₂

=

φ φ

R

−

T

+

2

K sin( )Z

θ

₀, where

the K is wave number and the initial phase

φ

_R

−

φ

_T is different form n=1 to n=3. Besides, the dynamic range of lock-in amplifier (i.e. Mod. 844 Stanford Research Systems) is from

−

π

to

+

π

, therefore, the wavelength period becomes

1 2

λ

. The incident angle

θ

is set for

π

6

and

A

=

20

nm, so the

ψ

₁

=

2

K sin( )A

θ

=

0 199

.

. Then we can get the detection amplitude versus distance

Z

₀ using Eq.(13) for order n = 1 to 3 as shown in Figs. 5 (a)~(d).

Fig. 4(a) The calculated n=1 of detection signal intensity versus the distance Z0 between the tip and the

sample with

ψ

₂

=

π

2 2

+

K sin( )Z

θ

₀.

Fig. 4(b) The calculated n=2 of detection signal intensity versus the distance Z0 between the tip and the

sample with

ψ

₂

= +

π

2

K sin( )Z

θ

₀.

Intensit y (arb. unit s) Intensit y (arb. unit s) 0

Z

(nm) 0

Z

(nm)

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Fig. 4(c) The calculated n=3 of detection signal intensity versus the distance Z0 between the tip and the

sample with

ψ

₂

=

2

K sin( )Z

θ

₀.

Fig. 4(d) The calculated n=3 of detection signal intensity versus the distance Z0 (0~80 nm) between the tip

and the sample with

ψ

₂

=

2

K sin( )Z

θ

₀.

From above simulation results, we compare them with experimental results in Fig. 6 of ref. [15] and Fig. 3 of ref. [7]. In the Figs. 4 (a) ~ (d), we find out the intensity of heterodyne signals decay fast in the near-field region and variation with the distance Z0 because of the phase difference

ψ

2 in the far field region. Those simulation results

consist with experimental results although their tip is not Au, therefore, the model of Eq.(13) and the assumptions are reasonable.

. From Fig. 4 (a), we know that the background tip reflective amplitude is similar to near-field amplitude when n = 1, but it decays faster with Bessel function. Therefore, higher harmonics detection signal has better signal to noise ratio (S/N ratio). However, the Bessel function has a coefficient

ψ

₁ to affect the S/N ratio. Next section, we’ll focus on the influence of phase modulation depth

ψ

₁ and its components in wavelength, incident angle and tip vibration amplitude.

5. Incident Wavelength λ of

JJK

E

_i in

ψ

₁

From

ψ

₁

=

2

K sin( )A

θ

, we know the modulation depth

ψ

₁ consists of wavelength λ (

K

=

2

π

λ

), incident angle

θ

, and tip vibration amplitude A. Firstly, we study the relation between wavelength λ and signal contrast. We set the incident angle

θ

=

π

6

and tip vibration amplitude A = 20 nm, and we substitute those parameters into Eq.(13) then get Fig.8, which shows that (1) the signal contrast of

I

Δ_ω_t approximates to 1; (2) in the short

wavelength region, the signal contrast is not correct; (3) in the long wavelength region, the high order contrasts approach to 1.2; and (4) higher order harmonic radian frequency has better signal contrast in short wavelength

Int e nsity ( arb. unit s) In tensity ( ar b. unit s) 0

Z

(nm) 0

Z

(nm)

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region. The longer wavelength of incident electric field has better signal contrast is opposite to traditional microscope concept. In traditional, the resolution of an optical microscope is inverse proportion the wavelength, even the A-SNOM in previous studies proposed that one advantage is wavelength independent [14]. Therefore, this is a new and interesting ideal about wavelength in optical measurement. In the visible light region, the higher order lock-in detection radian frequency has more correct signal contrast.

Fig. 5 The signal contrast vs. wavelength of incident electric field in heterodyne detection of A-SNOM.

5. Conclusions and discussions

In this article, we study the heterodyne detection signals of reflective type A-SNOM with complete interference electric fields expression. Although the signal seems complicated, we provide a physical model and analyze it according to the harmonic frequency. We have compared these simulation results with previous studies, and get the signal formula of these electric field sources. Then we discuss the influences on signal contrast about the two parameters

ψ

₁ and

ψ

₂ . From the phase modulation depth

ψ

₁ , we know longer wavelength and smaller incident angle of incident electric field has better signal contrast. It opposites traditional optical microscope about shorter wavelength has better resolution.

Recently, some studies begin to analyze the complicated background noise in transmission or total reflection type A-SNOM [10-12]. However, the authors didn’t analyze the signal by heterodyne technique in different harmonics of tip vibration frequency and discuss the signal contrast relation between signal phase modulation depth and phase difference. Therefore, the authors didn’t discuss the effects about wavelength, incident angle, and tip vibration amplitude in heterodyne detection. As compared to the previous study conducted by our groups [13], the homodyne signals are more complicated and weaker than the heterodyne signals. Because of the heterodyne signals can be magnified by the reference beam and only the tip scattering electric field disturbs the measurement. Therefore, the heterodyne detection technique is the best method for near-field complex dielectric constant measurement.

From above, this study gives us more meanings and the methods to deal with about heterodyne signal process of reflective type of A-SNOM measurement, and it makes the A-SNOM more practicable. However, the method can also apply to other type A-SNOM, and offers more new thoughts to eliminate background electric field sources.

Wavelength (nm) Visible Region Contr ast ｜ S1/S2 ｜ 0 2 ( )t

I

_{Δω ω}₊ t

I

_Δω 0 ( )t

I

_{Δω ω}₊ 0 3 ( )t

I

_{Δω ω}₊

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Acknowledgments

The authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under Grant No. NSC 95-2221-E-006-049-MY3, 2007.

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