Spectral noise reduction and temporal coherence control using a phase unsynchronized wave synthesizing method

Hideo Ando; Hideo Ando; Juichiro Ukon; Toshiaki Iwai; Izumi Nishidate

doi:10.1364/OE.441562

1. Introduction

Near infrared (NIR) light has a wavelength between 800 and 2500 nm. Both the scattering and absorption cross-sections of NIR light are smaller than those of visible light [1], so that NIR light can more deeply penetrate an organism than visible light. Moreover, the water content of an organism absorbs less NIR light and thus allows NIR light to more deeply penetrate the organism compared with infrared light. Therefore, NIR light is suitable for measuring the status or change of status inside an organism.

Jöbsis utilized the NIR spectrum to measure the concentration of oxygen in blood [1]. The NIR spectrum is also useful for detecting the first and second overtones and the combination of group vibrations related to hydrogen atoms [2]. However, NIR light tends to generate greater spectral noise than visible light. To date, spectral noise arising from optical interference has obstructed accurate measurements of the overtones and combinations of group vibrations because the detection signal relating to the overtones and combinations is too small in the NIR spectrum. It is therefore important to reduce spectral noise.

It is known that coherent light, such as laser light, can exhibit speckle noise. Researchers and engineers have made efforts to reduce speckle noise. Mckenchie [3] has demonstrated the removal of speckle noise, as follows:

A] Reducing the temporal coherence of the light source and
B] Reducing the spatial coherence of the light source.

With respect to item [A], it is known that a white light source (panchromatic light source) is slightly effective in reducing temporal coherence compared with a monochromatic light source. However, it is known that both thermal light and white light have both spatial and temporal coherence [4,5], so that adopting only white light is not enough to sufficiently reduce temporal coherence. For item [B], a diffuser is effective in reducing spatial and temporal coherence [6–9]. Suzuki and Hioki [10] have shown that the amount of speckle noise is proportional to the square of the degree of coherence.

In response to item [B], Zernike developed the spatially partial coherence theory [11] and Mandel and Wolf defined a complex spectral degree of spatial coherence [12]. With respect to item [A], the spectral degree of temporal coherence was recently calculated based on a Gaussian pulse [9,13].

In this paper, we propose a phase unsynchronized wave synthesizing (PuwS) method [14] that provides different optical path lengths for different wave elements obtained from a division of a wavefront and then synthesizes the respective wave elements to have the same propagation direction. To describe the experimental data, we also propose a series of analytical models that extend the traditional wave train model [15]. We first show that, using the ensemble average effect, PuwS reduces the spectral noise. Then, as an example, we use the spectral profile variation of a silk sample to demonstrate that a combination of PuwS and an appropriate diffuser reduces the optical noise level and that multiple absorption bands appear, that were previously hidden by the optical noise. Finally, we demonstrate spectral visibility control regarding temporal coherence.

2. Experimental optical system and PuwS method

Figure 1(a) shows the experimental optical system, comprising a light source area LS, a target sample setting area SS, and a detection area DT. A bundled optical fiber BF and a single-core optical fiber SF connect the areas.

Fig. 1. Optical experimental system using PuwS: (a) schematic representation of the experimental optical system with light source area LS, target sample setting area SS, detection area DT, concave mirror CM, halogen lamp HL, φ3 mm aperture A3, lenses L1–L4, optical phase unsynchronizer PU, removable diffusers RD1 and RD2, mask pattern MP, wave synthesizing function WS, bundled optical fiber BF, φ10 mm aperture A10, target sample TS, single-core optical fiber SF, and spectrometer SM; and (b) structure of the PU.

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In the DT, a spectrometer SM provides the spectral signals. The spectrometer is a C11482GA produced by Hamamatsu Photonics Co. and has 512 detection cells. The wavelength resolution Δλ of one detection cell is 7 nm.

The SF, having a core diameter of 0.6 mm, connects the SS to the DT. In the SS, a combination of two lenses L3 and L4 forms an image of the exit surface of the BF on the entrance surface of the SF. The focal lengths of L3 and L4 are respectively 50 mm and 250 mm, giving a lateral magnification of 5. Therefore, the image of a part of only one core area on the exit surface of the BF is formed on the entrance surface of the SF because one core diameter in the BF is 0.23 mm. The numerical aperture NA of the SF is 0.22. Parallel light between L3 and L4 irradiates a target sample TS, and an aperture A10 with a diameter of 10 mm, located before the TS, restricts the irradiated area of the TS. Moreover, a removable diffuser RD2 can be inserted before the TS.

A halogen lamp HL emits white light. A part of the white light propagating backward is reflected by a concave mirror CM and then passes through the inside of the HL and propagates toward a lens L1. Between two lenses L1 and L2, the white light is parallel and an optical phase unsynchronizer PU, a mask pattern MP, and a removable diffuser RD1 can be inserted. The PuwS method comprises the PU and a wave synthesizing function WS (where “Pu” refers to the PU and “wS” to the WS). A combination of L2 and the BF realizes the WS.

Figure 1(b) shows the structure of the PU. It consists of two sets of glass pairs, comprising two semicircular glass plates with thicknesses of 2 mm and 3 mm. In the glass pairs, the two semicircular glass plates are rotated 90° relative to each other and are bonded together (stacked). The glass pairs are then stacked with a rotation angle between the two glass pairs of 45°. With this structure, the PU comprises eight divisional components with different glass thicknesses, shown in the figures as components A-H with thicknesses of 0, 2, 4, 7, 10, 8, 6, and 3 mm.

Consequently, the PU functions to divide the wavefront, with each wave element passing through a divisional component controlled to have a different optical path length. The interval of the glass thickness is greater than or equal to 1 mm, which is greater than the coherence length ΔL₀ [15], given below in formula (10). Therefore, according to the explanation of Born and Wolf [15], each phase of the wave elements is unsynchronized with the other wave elements, so that each element is free from optical interference with the other elements. The specific mask pattern MP extracts one or more desirable wave elements passing through the corresponding components.

L2 condenses all the wave elements passing through the PU and the MP into core areas of the BF. Each of the combined (synthesized) wave elements is controlled to have the same propagation direction within the BF. The combination of the BF and L2 generates the WS. Therefore, all the wave elements passing through the BF are synthesized to form temporally partial coherence light.

In brief, PuwS [14] provides the following optical control:

1. An optical phase unsynchronizer effects division of a wavefront;
2. After division, the wave elements have different optical path lengths;
3. The difference between optical path lengths is greater than or equal to the coherence length, so that each phase of the divided wave elements is unsynchronized with the other wave elements;
4. A waves synthesizing function (a combination a lens L2 and a bundled optical fiber) synthesizes the divided wave elements;
5. Each synthesized wave element is controlled to have the same propagation direction within the bundled optical fiber.

Figure 2 shows a schematic diagram of the fundamental concept of PuwS. According to the traditional wave train model of Born and Wolf [15], the phases of propagating wave trains are not synchronized, as shown in Fig. 2(a).

Fig. 2. Schematic diagram indicating the fundamental concept of PuwS: (a) original wave train propagation, (b) one wave element provided by a wavefront division of the original wave train, (c) another wave element divided and temporally delayed, (d) all synthesized wave elements accounting for the ensemble average effect on the intensities.

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When the original wave train passes through the PU shown in Fig. 1(b), the wavefront is divided to form eight wave elements, each with a reduced amplitude. Figure 2(b) shows a wave element passing through component A with glass thickness of 0 mm. As shown below in Eq. (6), the other divided wave elements in Fig. 2(c) are temporally delayed because of the glass thicknesses of other components. Figure 2(d) shows the WS accounting for the ensemble average effect on the intensities.

The MP extracts one or more wave elements passing through the components. Conventional light can be obtained when the MP extracts only the wave element passing through component A. In this paper, PuwS_M and M represent the total number of wave elements extracted by the MP. M = 1 corresponds to conventional light and hence PuwS_M and M are always greater than or equal to 1.

As explained in section 3.2 and given by Eq. (55), wave front aberration in the LS not only generates spectral noise but also distorts the spectral visibility. Therefore, the optical experimental system shown in Fig. 1(a) includes a φ3 mm aperture A3 to avoid coma-aberration of L1. Moreover, the optical system uses “optical contact technology” [16], which provides accurate optical flatness to the PU compared with conventional UV curing resin, to ensure the PU does not add extra wave front aberration. UV curing resin tends to contract when used to bond to semicircular glass plates, which would distort the optical flatness of the PU. Before conducting the experiments mentioned, we measured the optical flatness of the PU and obtained a maximum root-mean-square value for the transmitted wave front aberration of one wave element of approximately 160 nm.

3. Analytical model

3.1 Fundamental interference model based on a traditional wave train model

Figure 3 shows the fundamental interference model. Panchromatic plane wave light with amplitude 1 passes through a transparent and optically parallel film (plate) as a TS and converges at the “P” position. The plate has a refractive index n and thickness d = d₀ + δd, where d₀ indicates the average thickness and δd indicates the thickness deviation. The maximum thickness is defined as d₀ + Δd/2 and the minimum thickness is d₀ - Δd/2.

Fig. 3. Fundamental analytical model generating a spectral interference fringe.

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The amplitude transmittance is T₁ when the panchromatic light passes through an entrance surface from air into the plate. The amplitude transmittance is Tn when the light passes through an exit surface toward a convergent lens L4. The parameter R represents the reflectance when an inner wall (surface) of the plate reflects the panchromatic light. Figure 3 shows directly propagating light of amplitude A₁ = T₁Tn and dual-reflected light of amplitude A₂ = T₁TnR².

Born and Wolf [15] give the following formulae:

(1)$${T_1} =\frac{2}{{n + 1}}\;;$$

(2)$$Tn =\frac{{2n}}{{n + 1}}\;;$$

(3)$${R^2} =\frac{{{{({n - 1} )}^2}}}{{{{({n + 1} )}^2}}}\; .$$

The following equations are obtained from Eqs. (1)–(3):

(4)$${A_1} ={T_1}Tn =\frac{{4n}}{{{{({n + 1} )}^2}}} =1 - {R^2}\;;$$

(5)$${A_2} ={T_1}Tn{R^2} =\frac{{4n{{({n - 1} )}^2}}}{{{{({n + 1} )}^4}}}\;.$$

The optical path length difference between light passing through air and passing through the plate is

(6)$$\{{({2j + 1} )\,n - 1} \}\,d =\{{({2{\kern 1pt} j + \;1} )\,n - 1} \}\,({{d_0} + \delta {\kern 1pt} d} )\;{\kern 1pt} .$$

where j = 0 gives the difference for directly propagating light and j = 1 gives the difference for doubly-reflected light. The optical path length difference is related to the light propagation time delay τ_j which, with the light velocity c is represented as

(7)$${\tau _j} = \;{\tau _{0j}} + \delta \,{\tau _j} =\;{{\{{({2{\kern 1pt} j + \;1} )\,n - 1} \}\,({{d_0} + \delta {\kern 1pt} d} )} / c} =\;{\kern 1pt} {{\{{({2{\kern 1pt} j + \;1} )\,n - 1} \}\,d} / c}\;.$$

As shown in Fig. 3, the propagating direction for both directly propagating light A₁ and doubly-reflected light A₂ is parallel to the r-axis. The focal length and pupil radius of L4 are represented as F and a. P is located on the principal axis and the focal plane of L4. Accordingly, at P, A₁ and A₂ converge and generate a spectral fringe pattern.

Panchromatic light comprises multiple different wavelengths, each detected by a detection cell in the SM. Each wavelength of the SM generally has the same detectable wavelength range Δλ. In other words, each wavelength has a detectable wavelength range of λ₀ – Δλ/2 to λ₀ + Δλ/2, where λ₀ indicates the mean wavelength of the wavelength region. Incidentally, the corresponding frequency range Δν changes with λ₀ when Δλ is constant. The following equation is thus obtained:

(8)$$c ={\lambda _0}{\nu _0} =({{\lambda_0} + {{\Delta \lambda } / 2}} )\,({{\nu_0} - {{\Delta \nu } / 2}} ) \approx {\lambda _0}{\kern 1pt} {\nu _0} + {\nu _0}{\kern 1pt} {{\Delta \lambda } / 2} - {\lambda _0}{\kern 1pt} {{\Delta \nu } / 2}\;.$$

Equation (8) leads to

(9)$$\Delta \nu =({{{\Delta \lambda } / {{\lambda_0}}}} )\,{\nu _0}\;.$$

Born and Wolf [15] provide the following formula for ΔL₀:

(10)$$\Delta {L_0} ={{{\lambda _0}^2} / {\Delta \lambda }}\;.$$

Substituting Eq. (10) in Eq. (9),

(11)$$\Delta \nu ={c / {\Delta {L_0}}}$$

is obtained. The wavelength region has a wave number k:

(12)$$k ={{2\pi } / {{\lambda _0} ={{2\pi {\nu _0}} / c}\;.}}$$

According to the Huygens–Fresnel principle [15], the wavelength portion forms a complex amplitude U(Q) at position Q on the focal plane of L4:

(13)$$U(Q ) =- {\kern 1pt} \,\frac{{i{\kern 1pt} B{e^{ - ikF}}}}{{{\lambda _0}{\kern 1pt} F}} \int\!\!\!\int {{\kern 1pt} \frac{1}{s}\;\sum\limits_{j{\kern 1pt} = {\kern 1pt} 0}^1 {\,{A_j}{\kern 1pt} {e^{i\,k\,({s\, + \,c{\kern 1pt} {\tau_j}} )}}{\kern 1pt} dS} } \;.$$

Here, B is a normalization constant. For Q = P, s = F, so that Eq. (13) becomes

(14)$$U(Q ) =- {\kern 1pt} \,\frac{{i{\kern 1pt} B}}{{{\lambda _0}{\kern 1pt} {F^2}}} \int\!\!\!\int {{\kern 1pt} {\kern 1pt} \sum\limits_{j{\kern 1pt} = {\kern 1pt} 0}^1 {\,{A_j}{\kern 1pt} \exp \{{{\kern 1pt} \,i{\kern 1pt} \,2{\kern 1pt} \pi {\kern 1pt} \,{\nu_0}{\kern 1pt} \,{\tau_j}} \}\,{\kern 1pt} dS} } \;.$$

Moreover, for τ₀ = τ₁ = A₁ = 0 and A₀ = 1, Eq. (14) becomes

(15)$${U_0} =- {\kern 1pt} \,\frac{{i{\kern 1pt} B}}{{{\lambda _0}{\kern 1pt} {F^2}}}\;\,\pi \,{a^2}\;.$$

All the relative spectral information described in Chapter 4 corresponds to the ratio of the spectral data of the TS to the reference data. Therefore, an amplitude ratio Φ_R(ν₀) is defined based on Eqs. (14) and (15):

(16)$${\Phi _R}({{\nu_0}} )\; \equiv \;{{U(P )} / {{U_0}}}\; = \;{\kern 1pt} \frac{1}{{{\kern 1pt} \pi {\kern 1pt} {a^2}}} \int\!\!\!\int {{\kern 1pt} {\kern 1pt} \sum\limits_{j{\kern 1pt} = {\kern 1pt} 0}^1 {\,{A_j}{\kern 1pt} \exp \{{{\kern 1pt} \,i\,{\kern 1pt} 2{\kern 1pt} \pi \,{\kern 1pt} {\nu_0}\,{\kern 1pt} {\tau_j}} \}\,{\kern 1pt} dS} } \;.$$

Although Φ_R(ν₀) has a fixed frequency value ν₀ in Eq. (16), each detection cell of the SM detects light in a frequency range Δν described by Eq. (9) or Eq. (11). Therefore, Eq. (16) should be integrated over Δν. Moreover, substituting Eq. (7) in Eq. (16), Eq. (16) should be integrated within the thickness deviation range Δd described above. Therefore, Eq. (16) can be transformed to

(17)$${\Psi _R}({{\nu_0}} )\; = \;\alpha \;\sum\limits_{j\; = \;0}^1 {{A_j}\,} \int_{{d_0} - {{\Delta d} / 2}}^{{d_0} + {{\Delta d} / 2}} {\int_{{\nu _0} - {{\Delta \nu } / 2}}^{{\nu _0} + {{\Delta \nu } / 2}} {\exp \{{ - i\,2\pi \,\nu {\kern 1pt} \,({t - {r / c} - {\tau_j}} )} \}} \,d\nu \,dd\;.}$$

Here, α is a normalization constant. The inner integration of Eq. (17) can be transformed as

(18)$$\begin{aligned} {\varphi _R}({{\nu_0}} )\; &\equiv \;\int_{{\nu _0} - {{\Delta \nu } / 2}}^{{\nu _0} + {{\Delta \nu } / 2}} {\,\exp \{{ - \,i\,2\pi \,\nu ({t - {r / c} - {\tau_j}} )} \}d\nu } \\ & = \;\Delta \nu \;\textrm{sinc} \{{\pi \,\Delta \nu \,({t - {r / c} - {\tau_j}} )} \}\;{e^{ - i2\pi {\nu _0}\;({t - {r / c} - {\tau_j}} )}}\;. \end{aligned}$$

Under the condition τ_0j >> δτ_j in Eq. (7), the following approximate equation can be obtained:

(19)$$\Delta \nu \times \delta {\tau _j} \approx 0\;.$$

Substituting Eqs. (7) and (19) for Eq. (18), Eq. (18) changes to

(20)$${\varphi _R}({{\nu_0}} ) \approx \Delta \nu \textrm{sinc} \{{\pi \,\Delta \nu {\kern 1pt} \,({t - {r / c} - {\tau_{0j}}} )} \} \exp \{{ - \,i\,2\pi \,{\nu_0}({t - {r / c} - {\tau_j}} )} \}\;.$$

With respect to Eqs. (20) and (11), this paper defines

(21)$$\begin{aligned} {S_j}({{\tau_{0j}},\,\Delta {L_0},\,t} )\; \equiv \;\textrm{sinc} &\{{\pi {{({ct - r - c{\tau_{0j}}} )} / {\Delta {L_0}}}} \} \\ &when |\;ct - r - c{\tau _{oj}}| \; \le \;\Delta {L_0} \end{aligned}, $$

(22)$${S_j}({{\tau_{0j}},\,\Delta {L_0},\,t} )\; \equiv \;0\;\quad \;\;\;when\quad |\;ct - r - c{\tau _{oj}}| \; > \;\Delta {L_0}\;.$$

This condition is suitable for the experimental data shown later in Fig. 17. If the condition of Eqs. (21) and (22) is not set, the sinc function may have a negative value. Equation (28) shows that the spectral fringe pattern inverts when S_j(τ_0j, ΔL₀, t) < 0. The spectral fringe pattern shown in Fig. 17 does not show an inverted area. Under the condition of Eqs. (21) and (22), Fig. 17 indicates a good consistency between the measured spectrum information Fig. 17(a) and calculated information in Fig. 17(b) obtained from Eq. (28) given below.

Substituting Eqs. (20)–(22) and (7) in Eq. (17), the following integration result is obtained:

(23)$${\Psi _R}({{\nu_0}} ) =\alpha \,\Delta \nu \,\Delta d\;\sum\limits_{j\; = \;0}^1 {{A_j}\,{D_j}\,{S_j}} \exp \{{ - \,i\,2\pi \,{\nu_0}({t - {r / c} - {\tau_{0j}}} )} \}\;.$$

Here,

(24)$${D_j}({\Delta d,\,{\lambda_0}} ) \equiv \textrm{sinc} \{{\,\pi \,[{({2j + 1} )\;n - 1} ]{{\,\Delta d} / {{\lambda_0}}}} \}\;.$$

Based on Eq. (23), using Eqs. (4) and (5), Born and Wolf [15] give the light transmittance < I_R > in the form

(25)$$\begin{aligned} \left\langle {{I_R}} \right\rangle &=\left\langle {{\Psi _R}({{\nu_0}} )\cdot \,{\Psi _R}^ \ast ({{\nu_0}} )} \right\rangle \\ \;& ={\{{\alpha \,\Delta \nu \,\Delta d({1 - {R^2}} )} \}^2}\left\langle {{D_0}^2{S_0}^2 + {R^4}{D_1}^2{S_1}^2 + 2{R^2}{D_0}{S_0}{D_1}{S_1}\cos ({{{4\pi n{d_0}} / {{\lambda_0}}}} )} \right\rangle \;. \end{aligned}$$

Equation (24) included in Eq. (25) is time-independent, though Eq. (21) is time-dependent. Therefore, utilizing the condition of Eqs. (21) and (22), this paper defines

(26)$$\left\langle {{S_j}\,{S_l}} \right\rangle \equiv {({\alpha \,\Delta \nu \,\Delta d} )^2}\int_{{\tau _{0j}} + {r / c} - {{\Delta {L_0}} / c}}^{{\tau _{0j}} + {r / c} + {{\Delta {L_0}} / c}} {{S_j}({{\tau_{0j}},\Delta {L_0},t} )\,{S_l}({{\tau_{0l}},\Delta {L_0},t} )\,dt\;.}$$

When α is adjusted,

(27)$$\left\langle {{S_j}\,{S_j}} \right\rangle =1\;.$$

Substituting Eqs. (26) and (27) in Eq. (25), Eq. (25) can be rewritten as

(28)$$\left\langle {{I_R}} \right\rangle ={({1 - {R^2}} )^2}\left\{ {{D_0}^2 + {R^4}{D_1}^2 + 2{R^2}{D_0}{D_1}\left\langle {{S_0}\,{S_1}} \right\rangle \cos ({{{4\pi n{d_0}} / {{\lambda_0}}}} )} \right\}\;.$$

Figures 13(c) and 17(b), given later, show calculation results based on Eq. (28). Equations (21) and (22) correspond to the wave train described by Born and Wolf [15]. Figure 4 shows profiles of propagating wave trains based on Eqs. (21) and (22). S₀ indicates a wave train of directly propagating light. The series of S₁ indicates wave trains of doubly-reflected light. In Fig. 4, d₀ = 138.40 µm and Δλ = 7.5 nm. This is the same condition as that in Fig. 17 (b). Figure 4 expresses the optical distance (delayed position) based on ΔL₀, which depends on λ₀ (Eq. 10), so that the optical distance between directly propagating light and doubly-reflected light depends on λ₀. Each S₁ is positioned based on λ₀ = 0.9, 1.1, 1.3, 1.5, and 1.7 µm.

Fig. 4. Wave train profiles and an overlapped area < S₀S₁> between different wave trains.

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Fig. 5. Schematic diagram of the diffuser analyzing model showing a slightly distorted Gaussian distribution considering the diffuser surface roughness being approximated to a rectangular distribution.

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The size of the overlapped area (blue-slashed area) corresponds to < S₀S₁> described in Eqs. (26) and (27). Equation (28) shows that the spectral visibility regarding the temporal coherence depends on the size of < S₀S₁>. It is important that the < S₀S₁> vanishes when λ₀ decreases.

3.2 Wave train model extension (major and dispersed wave train model)

This section extends the wave train model described in the previous section to propose a generalized model that can explain many optical interference phenomena and temporally partial coherence phenomena. According to Eq. (23), the synthesized light Ψ_R(ν₀) comprises two propagating wave trains. Equation (28) shows that the optical distance between two propagating wave trains is 2nd₀. This section defines the following optical distance between two propagating wave trains:

(29)$${\chi _j} \equiv c\,({{\tau_{0j}}\, - \;{\tau_{00}}} )\;.$$

Although a combination of Eqs. (7) and (29) gives χ₀ = 0 and χ₁ = 2nd₀, this paper uses Eq. (29) for both the analytical model described in Fig. 3 and general optical interference analysis. Ψ_R(ν₀) can be represented as

(30)$${\Psi _R}({{\nu_0}} ) =\sum\limits_{j\; = \;0}^1 {\,{\phi _j}} = \;\sum\limits_{j\; = \;0}^1 {\,\sqrt {{I_j}} } \;{S_j}\,\exp \{{ - \,i\,2\pi \,{\nu_0}({t - {r / c} - {{{\chi_j}} / c}} )} \}\;.$$

When Eq. (29) is used in this section, cτ₀₁ = χ₁ or cτ₀₀ = 0 should substituted into Eq. (21). Therefore, using Eq. (30), the spectral degree of temporal coherence SDTC |μ^τ₀₁| is

(31)$$|{\mu ^\tau }_{01}| \;\, \equiv \;\,\frac{{\left|{\;\left\langle {{\phi_0} \cdot \;{\phi_1}^ \ast } \right\rangle \;} \right|}}{{\sqrt {{I_0}} \;\sqrt {{I_1}} }}\;\, = \;\,\left\langle {{S_0}\,{S_1}} \right\rangle \;.$$

Substituting Eqs. (29)–(31) for Eq. (28), the following well-known formula is obtained:

(32)$$\left\langle {{I_R}} \right\rangle \; = \;{I_0}\; + \;{I_1}\; + \;2\,\sqrt {{I_0} \cdot \;{I_1}} |{{\mu^\tau }_{01}} |\;\cos ({{{2\pi {\chi_1}} / {{\lambda_0}}}} )\;.$$

Equation (31) shows that the pair of Eqs. (26) and (27) corresponds to the SDTC. According to Born and Wolf [15], an essential metric for the partial coherence property is the visibility, so this section defines the spectral visibility SV as

(33)$$SV\; \equiv \;\frac{{{{\left\langle {{I_R}} \right\rangle }_{\max }} - \;\;{{\left\langle {{I_R}} \right\rangle }_{\min }}}}{{{{\left\langle {{I_R}} \right\rangle }_{\max }} + \;\;{{\left\langle {{I_R}} \right\rangle }_{\min }}}}\; = \;\frac{{2\;\sqrt {{I_0} \cdot {I_1}} \;|{{\mu^\tau }_{01}} |}}{{{I_0} + \;\,{I_1}}}\;\;.$$

Section 3.5 explains that a diffuser can reduce SV. To distinguish the SV considered in this section from that discussed in section 3.5, this section refers to the ‘original spectral visibility’ SVorg. Comparing Eq. (32) with Eq. (28), Eq. (33) can be transformed to

(34)$$SVorg{\kern 1pt} ({{\lambda_0}} )\; = \;\,\frac{{2\,{\kern 1pt} {R^2}{\kern 1pt} {D_0}{\kern 1pt} ({{\lambda_0}} )\,{D_1}{\kern 1pt} ({{\lambda_0}} )\;\left\langle {{S_0}\,{S_1}} \right\rangle }}{{{D_0}{\kern 1pt} {{({{\lambda_0}} )}^2} + \;{R^4}\,{D_1}{\kern 1pt} {{({{\lambda_0}} )}^2}}}\;\;.$$

It is important to note that SVorg relating to Eqs. (24) and (26) varies with λ₀.

Based on Eq. (30), this paper defines a ‘major wave train’ Φ₀ and a ‘dispersed wave train’ Φ₁, where there may be multiple dispersed wave trains indicated by a subscript j. If there are multiple dispersed wave trains of low intensity, Eq. (32) approximately transforms to

(35)$$\left\langle {{I_R}} \right\rangle \; \approx \;{I_0}\; + \;\sum\limits_{j\; \ne \;0} {\;2\sqrt {{I_0} \cdot \;{I_j}} \left\langle {{S_0}\,{S_j}} \right\rangle \cos ({{{2\pi {\chi_j}} / {{\lambda_0}}}} )} \;.$$

An example of the relation between a major wave train and a dispersed wave train is shown in Fig. 3. After the initial major wave train having amplitude 1 passes through the plate, the directly propagating light having amplitude A₁ = T₁Tn corresponds to the next major wave train Φ₀ and the dual-reflected light having amplitude A₂ = T₁TnR² corresponds to a dispersed wave train Φ₁.

Comparing Eq. (30) with Eq. (23), this section defines the amplitude efficiency:

(36)$${E_j} \equiv {{\sqrt {{I_j}} } / {{D_j} =\alpha \,\Delta \nu \,\Delta d\,{A_j}\;.}}$$

Only the particular case described in Fig. 3 satisfies Eq. (24), because general optical interference phenomena rarely result from dual reflections. Therefore, Eq. (24) should be replaced with

(37)$${D_j}({\Delta d,\;{\lambda_0}} ) \equiv \textrm{sinc} \{{\,\pi \,({n - 1} ){{\,\Delta d} / {{\lambda_0}}}} \}\;.$$

This equation shows that D_j(Δd, λ₀) varies with the surface roughness Δd of the TS or an optical noise generating object. Meanwhile, if the total intensity of all of the wave trains is normalized to 1, the following condition is satisfied:

(38)$$\sum\limits_j {{E_j}^2 = \;\,1\;.}$$

Substituting Eq. (36) in Eq. (35), the following approximate equation is obtained:

(39)$$\left\langle {{I_R}} \right\rangle \;\, \approx \;\,{({{E_0}{\kern 1pt} {D_0}} )^2}\; + \;\sum\limits_{j\; \ne \;0} {{\kern 1pt} 2\,{E_0}{\kern 1pt} {E_j}{\kern 1pt} {D_0}{\kern 1pt} {D_j}\;\left\langle {{S_0}\,{S_j}} \right\rangle \;\cos ({{{2\pi {\chi_j}} / {{\lambda_0}}}} )} \;.$$

According to the extended wave train model, Eq. (39) describes the spectral noise characteristics. The second term of the right side in Eq. (39) gives the spectral noise characteristics in detail. When an initial major wave train having amplitude 1 passes through a prescribed optical path, the initial intensity is dispersed to generate one or more wave trains having optical distances (optical path length differences) χ_j. The period of the spectral noise varies with the χ_j. We presume that the spectral noise results from wavefront aberration, a transparent step, the surface roughness of a transparent object (including the TS), partial reflection, or local absorption. Therefore, within the prescribed optical path, at least one of these factors may generate dispersed wave trains. After the initial intensity has dispersed, the amplitude efficiency of the next major wave train E₀ is reduced (E₀ < 1). The amplitude efficiency E_j of the dispersed wave trains is generally small enough compared to E₀ that the amplitudes of the spectral noise (the amplitude values in the second term of the right side in Eq. (39)) are relatively small. Each term in the second term of the right side in Eq. (39) includes D_j. Equation (37) shows that D_j is reduced when λ₀ decreases and Δd is unchanged. It should be emphasized that the amplitude of the spectral noise is reduced when λ₀ decreases. Figure 6(a) shows this trend. Therefore, the spectral noise of NIR light is greater than that of visible light, so Fig. 6(a) suggests that this trend prevents accurate measurements, as mentioned in the Introduction.

Fig. 6. Spectral noise examples generated by an Ra = 2.08 µm diffuser as the TS.

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The first term on the right side in Eq. (39), which includes a function D₀, represents the characteristics of the next major wave train. Equation (37) shows that the characteristics of next major wave train depend on λ₀.

3.3 Ensemble average effect of PuwS

As shown in Fig. 2, the phase of a divided wave element is unsynchronized with the phases of other wave elements and there is no optical interference between different wave elements. Therefore, the optical properties of the synthesized light result only from a sum of the intensity properties of the wave elements.

The total number of wave elements divided by PuwS is M. With respect to Eq. (39), this section defines a new subscript m to represent the individual wave elements. Referring to Eq. (39), the intensity property of the m-th wave element is:

(40)$$\left\langle {{I_{Rm}}} \right\rangle \;\, \approx \;\,\frac{1}{M}\;\left\{ {{{({{E_{m0}}{\kern 1pt} {D_{m0}}} )}^2}\; + \;\sum\limits_{j\; \ne \;0} {{\kern 1pt} 2\,{E_{m0}}{\kern 1pt} {E_{mj}}{\kern 1pt} {D_{m0}}{\kern 1pt} {D_{mj}}\;\left\langle {{S_{m0}}\,{S_{mj}}} \right\rangle \;\cos ({{{2\pi {\chi_{mj}}} / {{\lambda_0}}}} )} } \right\}\;\,.$$

E_m₀, D_m₀, E_mj, D_mj, S_m₀, and S_mj are approximately equal to E₀, D₀, E_j, D_j, S₀, and S_j. Therefore, Eq. (40) approximately transforms to

(41)$$\left\langle {{I_{Rm}}} \right\rangle \;\, \approx \;\,\frac{1}{M}\;\left\{ {{{({{E_0}{\kern 1pt} {D_0}} )}^2}\; + \;\sum\limits_{j\; \ne \;0} {{\kern 1pt} 2\,{E_0}{\kern 1pt} {E_j}{\kern 1pt} {D_0}{\kern 1pt} {D_j}\;\left\langle {{S_0}\,{S_j}} \right\rangle \;\cos ({{{2\pi {\chi_{mj}}} / {{\lambda_0}}}} )} } \right\}\;\,.$$

After passing through the prescribed optical path, the intensity of the synthesized light, based on Eq. (41), is approximately:

(42)$$\begin{aligned} \left\langle {{I_R}} \right\rangle \; &= \;\sum\limits_{m\; = \;1}^M {\,\left\langle {{I_{Rm}}} \right\rangle } \\ &\approx \;\,{({{E_0}{\kern 1pt} {D_0}} )^2}\; + \;\,\frac{2}{M}\;\,\left\{ {\sum\limits_{m\; = \;1}^M {\sum\limits_{j\; \ne \;0} {{\kern 1pt} {E_0}{\kern 1pt} {E_j}{\kern 1pt} {D_0}{\kern 1pt} {D_j}\;\left\langle {{S_0}\,{S_j}} \right\rangle \;\cos ({{{2\pi {\chi_{mj}}} / {{\lambda_0}}}} )} } } \right\}\;\;. \end{aligned}$$

It should be emphasized that each optical distance χ_mj is different from other optical distances χ_m′j (m′ ≠ m) and each optical noise period is different. Therefore, the second term of the right side in Eq. (42) gives the ensemble average effect reducing the spectral noise. The second term of the right side in Eq. (42) indicates that the spectral noise is reduced when M increases. More specifically, the following condition is satisfied:

(43)$$\mathop {\lim }\limits_{M\; \to \;\infty } \;\frac{2}{M}\;\left\{ {\sum\limits_{m\; = \;1}^M {\sum\limits_{j\; \ne \;0} {{E_0}\,{E_j}\,{D_0}\,{D_j}\left\langle {{S_0}\,{S_j}} \right\rangle \;\cos ({{{2\pi {\chi_{mj}}} / {{\lambda_0}}}} )} } } \right\}\; = \;0\;.$$

Utilizing the condition of Eq. (43), Eq. (42) can be changed to

(44)$$\mathop {\lim }\limits_{M\; \to \;\infty } \;\left\langle {{I_R}} \right\rangle \; \approx {({{E_0}\,{D_0}} )^2}\;.$$

Equation (44) shows the spectral noise-free state. Figure 7 below shows the spectral noise reduction effect indicated by Eq. (43).

Fig. 7. Standard deviation showing spectral noise magnitude based on M.

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3.4 Diffuser action

Weiner [17], Frumker, Silberberg [18], and Torres-Company et al. [9] found that a diffuser (or a phase modulator) forms two or more temporally shifted pulses when transmitting one femtosecond pulse, which suggests the reliability of the next extended wave train model described in this section.

Figure 5 shows a schematic diagram of a diffuser analyzing model based on the next extended wave train model. Equation (17) presumes a rectangular distribution of δd on the surface. In other words, δd should be uniform between δd = −Δd/2 and δd = Δd/2. Meanwhile, we predict that the surface roughness of a diffuser has a slightly distorted Gaussian distribution, shown by the broken black line in Fig. 5(b). The horizontal axis of the center figure of Fig. 5 indicates the local δd resulting from the surface roughness of the diffuser, and the vertical axis indicates the probability of the corresponding local δd. The extended wave train model approximates the slightly distorted Gaussian distribution shown in Fig. 5(b), composed of the blue rectangular distributions shown in Figs. 5(c), 5(e), and 5(g). Figure 5(c) shows a fundamental rectangular distribution having a deviation range Δd₀. Figures 5(e) and 5(g) show the first and second rectangular distributions with Δd₁ and Δd₂, respectively. We predict that a diffuser will almost always have a slightly distorted Gaussian distribution. Therefore, each of the central positions of the first and second rectangular distributions in Figs. 5(e) and 5(g) shift slightly from the central position of the fundamental rectangular distribution shown in Fig. 5(c), represented by χ₁ and χ₂, respectively.

Based on Eq. (17), the diffuser analyzing model shown in Fig. 5 requires three integrations over the ranges −Δd₀/2 to +Δd₀/2, χ₁ − Δd₁/2 to χ₁ + Δd₁/2, and χ₂ − Δd₂/2 to χ₂ + Δd₂/2. The three integrations generate the modified major wave train shown in Fig. 5(d), the 1st order dispersed wave train shown in Fig. 5(f), and the 2nd order dispersed wave train shown in Fig. 5(h) after the initial major wave train shown in Fig. 5(a) passes through the diffuser. In other words, according to Fig. 5, the fundamental rectangular distribution shown in Fig. 5(c) forms the modified major wave train shown in Fig. 5(d) after the initial wave train (Fig. 5(a)) passes through the diffuser (Fig. 5(b), corresponding to a composition of Figs. 5(c), 5(e), and 5(g)). Moreover, the first rectangular distribution in Fig. 5(e) forms the 1st order dispersed wave train shown in Fig. 5(f), and the second rectangular distribution part in Fig. 5(g) forms the 2nd order dispersed wave train shown in Fig. 5(h).

Equation (45) below shows that each wave train amplitude is equal to E_lD_l (l = 0, 1, 2), and Fig. 5 shows that Δd₂ >> Δd₀. Therefore, Eq. (45) shows that the amplitude of the 2nd order dispersed wave train shown in Fig. 5(h) is small compared to the amplitude of the modified major wave train shown in Fig. 5(d), although the area of the second rectangular distribution in Fig. 5(g) is greatest.

Referring to Eqs. (21), (30), (36), (37), and (41), the complex amplitude of the m-th wave element after passing through the diffuser is approximately

(45)$${\Psi _{Rm}}({{\nu_0}} ) \approx \frac{{S({0,\Delta {L_0},t} )}}{{\sqrt M }}\;\exp \{{ - \,i\,2\pi \,{\nu_0}({t - {r / c}} )} \}\;\left\{ {\sum\limits_l {{E_l}\,{D_l}\,\exp ({\,i\,{{2\pi \,{\chi_{ml}}} / {{\lambda_0}}}} )} } \right\}\,\;.$$

For the sake of simplifying Eq. (45), χ_m₀ = 0 is assumed. An approximate equation τ_ol = 0 can be simultaneously substituted in Eq. (21) because Fig. 5 shows that χ_m₀ << ΔL₀. It is important to note that each wave element has a different χ_ml. In other words, χ_m_′l ≠ χ_ml when m′ ≠ m and l ≠ 0.

Using Eqs. (27) and (45), the intensity of the synthesized light is

(46)$$\begin{aligned} \left\langle {{I_R}} \right\rangle \; &= \;\sum\limits_{m\; = \;1}^M {\,\left\langle {{\Psi _{Rm}}({{\nu_0}} )\cdot {\Psi _{Rm}}^ \ast ({{\nu_0}} )} \right\rangle } \\&\approx \;\sum\limits_l {\,\left\{ {{{({{E_l}\,{D_l}} )}^2}\, + \;\frac{2}{M}\,\sum\limits_{m = 1}^M {\sum\limits_{j\; \ne \;l} {\,({{E_l}\,{D_/}} )({{E_j}\,{D_j}} )\;\cos [{{{2\pi ({{\chi_{ml}} - {\chi_{mj}}} )} / {{\lambda_0}}}} ]} } } \right\}} \;\,. \end{aligned}$$

The second term of the right side in Eq. (46) indicates the spectral noise. Therefore, the diffuser may generate spectral noise when M = 1, as shown in Fig. 6(a). In contrast, the second term of the right side in Eq. (46) may cancel the external optical noise when the prescribed optical path has an external optical noise source. Moreover, based on the second term of the right side in Eq. (46), the following condition is satisfied:

(47)$$\mathop {\lim }\limits_{M\; \to \;\infty } \;\frac{2}{M}\,\sum\limits_{m\; = \;1}^M {\sum\limits_l {\sum\limits_{j\; \ne \;l} {\,({{E_l}\,{D_l}} )({{E_j}\,{D_j}} )\,\cos \{{\,{{2\pi ({{\chi_{ml}} - \;{\chi_{mj}}} )} / {{\lambda_0}}}\,} \}\;\, = \;\,0\;\,.} } }$$

The condition of Eq. (47) reduces the optical noise when M increases. Substituting Eq. (47), Eq. (46) can be transformed to

(48)$$\mathop {\lim }\limits_{x \to \infty } \;\left\langle {{I_R}} \right\rangle \;\; \approx \;\;\sum\limits_l {\;{{({{E_l}\,{D_l}} )}^2}} \;.$$

Equation (48) suggests that the diffuser characteristics E_l and D_l can be measured when PuwS is used. A profile characterized by Eq. (48) is shown in Fig. 11(b) by fitting to the measured data shown in Fig. 11(a).

3.5 Temporal coherence control (spectral visibility ratio)

This section explains how the diffuser controls the visibility based on the temporal coherence and how PuwS contributes to the visibility control. This section supposes that synthesized light expressed by Eq. (46) passes through a plate as shown in Fig. 3. Referring to Eqs. (4), (5), (23), and (45), the complex amplitude of the m-th wave element after passing through the plate is approximately

(49)$${\Psi _m}\; \approx \;\frac{{({1 - {R^2}} )}}{{\sqrt M }}\;\,{e^{ - \,i\,2\pi \,{\nu _0}({t - {r / c}} )}}\left\{ {\sum\limits_{j\; = \;0}^1 {{R^{2j}}\,D{p_j}\;{S_j}\,{e^{i\,2\pi \,{\nu_0}{\tau_{0j}}}}} } \right\}\;\left\{ {\sum\limits_l {{E_l}\,{D_l}\,{e^{i\,{{2\pi \,{\chi_{ml}}} / {{\lambda_0}}}}}} } \right\}\;.$$

Here, Dp_j corresponds to Eq. (24) and D_l corresponds to Eq. (37). Using Eqs. (27) and (45), the intensity of the synthesized light is

(50)$$\begin{array}{l} \left\langle I \right\rangle \; = \;\sum\limits_{m\; = \;1}^M {\,\left\langle {{\Psi _m}({{\nu_0}} )\cdot {\Psi _m}^ \ast ({{\nu_0}} )} \right\rangle } \;\; = \;\;{({1 - {R^2}} )^2}({D{p_0}^2 + \;{R^4}{\kern 1pt} D{p_1}^2} )\,\sum\limits_l {{{({{E_l}\,{D_l}} )}^2}} \\ \quad \;\; + \;\;{({1 - {R^2}} )^2}({D{p_0}^2 + \;{R^4}{\kern 1pt} D{p_1}^2} )\;\frac{1}{M}\,\sum\limits_{m\; = \;1}^M {\sum\limits_l {\sum\limits_{j\; \ne \;l} {\;({{E_l}\,{D_/}} )({{E_j}\,{D_j}} )\;{G_{mlj}}} } } \\ \quad \;\; + \;\,2\,{({1 - {R^2}} )^2}{R^2}\,D{p_0}\,D{p_1}\,\left\langle {{S_0}\,{S_1}} \right\rangle \;\cos ({{{4\pi \,n\,{d_0}} / {{\lambda_0}}}} )\;\sum\limits_l {\;{{({{E_l}\,{D_l}} )}^2}} \\ \quad \;\; + \;\;{({1 - {R^2}} )^2}{R^2}\,D{p_0}\,D{p_1}\,\left\langle {{S_0}\,{S_1}} \right\rangle \;\frac{1}{M}\sum\limits_{m\; = \;1}^M {\sum\limits_l {\sum\limits_{j\; \ne \;l} {({{E_l}\,{D_/}} )({{E_j}\,{D_j}} )\;{H_{mlj}}\;,} } } \end{array}$$

(51)$${G_{mlj}}\; \equiv \;\cos \{{{{\,2\pi \,({{\chi_{ml}} - \;{\chi_{mj}}} )} / {{\lambda_0}}}\,} \}\;, \quad \textrm{and}$$

(52)$${H_{mlj}}\; \equiv \;\cos \{{{{\,2\pi \,({\,2n{d_0} + \;{\chi_{ml}} - \;{\chi_{mj}}} )} / {{\lambda_0}}}\,} \}\;.$$

Here, χ_m₀ = 0. The second term of the right side in Eq. (50) satisfies the following condition:

(53)$$\mathop {\lim }\limits_{M\; \to \;\infty } \;\frac{1}{M}\,\sum\limits_{m\; = \;1}^M {\sum\limits_l {\sum\limits_{j\; \ne \;l} {\,({{E_l}\,{D_l}} )({{E_j}\,{D_j}} )\,\cos \{{{{\,2\pi ({{\chi_{ml}} - \;{\chi_{mj}}} )} / {{\lambda_0}}}} \}\; = \;0\;.} } }$$

This section defines the relative spectrum information < I_R > as the ratio of the spectral data < I > represented by Eq. (50) to the reference data obtained from the synthesized light passing through the plate. The reference data corresponds to Eq. (48) when M is large. Therefore, using Eqs. (48), (50), and (53):

(54)$$\left\langle {{I_R}} \right\rangle \; = \;{({\,1 - \;{R^2}} )^2}({\,D{p_0}^2 + \;{R^4}\,D{p_1}^2} )+ \;2\,{({1 - {R^2}} )^2}{R^2}\,D{p_0}\,D{p_1}\left\langle {{S_0}\,{S_1}} \right\rangle \;{V_R}{\kern 1pt} ({{\lambda_0}} )\;,$$

(55)$$\begin{aligned} {V_R}\; &= \;\cos ({{{4\pi n{d_0}} / {{\lambda_0}}}} )\\ &+ \;\frac{1}{{2M\sum\limits_l {{{({{E_l}{D_l}} )}^2}} }}\sum\limits_{m\; = \;1}^M {\sum\limits_l {\sum\limits_{j\; \ne \;l} {({{E_l}\,{D_l}} )({{E_j}\,{D_j}} )\,\cos \{{{\kern 1pt} {{2\pi ({{\kern 1pt} 2n{d_0} + {\chi_{ml}} - {\chi_{mj}}} )} / {{\lambda_0}}}} \}} } } \;. \end{aligned}$$

The first term of the right side in Eq. (55) corresponds to the original spectral fringe pattern obtained from coherent light. The second terms of the right side in Eq. (55) form multiple spectral fringe patterns having additional phases χ_ml–χ_mj, which combine with the original spectral fringe pattern to reduce SV. Moreover, the SV reduction effect increases when M increases.

Referring to Eqs. (34), (54), and (55), when the optical experimental system includes a diffuser, the reduced spectral visibility SVdif is

(56)$$SVdif{\kern 1pt} ({{\lambda_0}} )\; = \;\,\frac{{{R^2}{\kern 1pt} D{p_0}{\kern 1pt} ({{\lambda_0}} )\,D{p_1}{\kern 1pt} ({{\lambda_0}} )\;\left\langle {{S_0}\,{S_1}} \right\rangle }}{{D{p_0}{\kern 1pt} {{({{\lambda_0}} )}^2} + \;{R^4}\,D{p_1}{\kern 1pt} {{({{\lambda_0}} )}^2}}}\;({{V_R}{\kern 1pt} \max \; - \;{V_R}{\kern 1pt} \min } )\;\;.$$

We define the spectral visibility ratio SVR to be the ratio of SVdif to SVorg. Using Eqs. (34) and (56), SVR is expressed as

(57)$$SVR{\kern 1pt} ({{\lambda_0}} )\; = \;\frac{{SVdif({{\lambda_0}} )}}{{SVorg{\kern 1pt} ({{\lambda_0}} )}}\; = \;\frac{1}{2}\,({{V_R}{\kern 1pt} \max \; - \;\,{V_R}{\kern 1pt} \min } )\;.$$

Equation (57) indicates that SVR depends only on Eq. (55). Equation (55) shows how PuwS contributes to reducing SVR. If M increases, the total number of visibility reduction terms in the second term of the right side in Eq. (55) increases, increasing the ensemble average and reducing SVR. This suggests that PuwS enhances the visibility reduction effect. Figure 16(b) below shows this effect.

According to Eq. (55), SVR is reduced when the absolute values of the visibility reduction terms in the second term of the right side in Eq. (55) increase. The denominators of the visibility reduction terms include Σ(E_lD_l)². Therefore, SVR is reduced when Σ(E_lD_l)² decreases, as shown in Fig. 16(c).

According to Eq. (55), the visibility reduction terms include E_lE_j (l ≠ j), so that SVR is reduced when E_lE_j (l ≠ j) increases. Equation (38) indicates that E_l (l > 0) increases when E₀ decreases. Therefore, SVR tends to be reduced when E₀ decreases.

With respect to Eq. (55), each E_l shows a diffuser characteristic. If there is no diffuser, the diffuser analyzing model shown in Fig. 5 gives E₀ = 1. Then, Eq. (38) gives E_l = 0 (l > 0), so that the visibility reduction terms in the second term of the right side in Eq. (55) are simultaneously 0 because E₀E_l = 0 (l > 0). This means that PuwS does not have a visibility reduction effect without a diffuser. It should be noted that some kinds of wave aberration generating objects can increase E_l (l > 0) and form additional phases χ_ml–χ_mj. Therefore, a combination of PuwS and a wave aberration generating object can reduce SV.

In conclusion, Eq. (55) gives the following appropriate conditions which effectively reduce SVR:

A. [Diffuser condition]
- - Σ(E_lD_l)² should be reduced,
- - E₀ should be smaller than 1;
- - E_l (l > 0) should not be smaller than “E₀”;
B. [PuwS method]
- - M should be large.

As explained above, the ensemble average effect is highly effective for periodic signals (spectral noise or visibility of fringe pattern). Also, when a TS is inserted within the optical path shown in Fig. 1(a), the SM obtains many kinds of absorption band information included in the TS. The absorption band generally has a sharp projection figure in the spectrum and the bandwidth (wavelength range) of the absorption band is generally narrow. Therefore, we suppose that the ensemble average effect does not affect the absorption band information.

3.6 Slope variation of spectral baseline

Using two diffusers (or a combination of a diffuser and a rough TS), PuwS can change the slope of the spectral baseline obtained by the SM. This phenomenon occurs when the experimental optical system shown in Fig. 1(a) has a removable diffuser RD1 added into the optical path and another diffuser (or a rough TS) at the position of the TS.

When RD1 is in the optical path, Eq. (45) shows the complex amplitude of the m-th wave element after passing through RD1. After the m-th wave element passes through the second diffuser or rough TS in the SS, the complex amplitude changes from Eq. (45) to

(58)$$\begin{aligned}{\Psi _{Rm}}({{\nu_0}} )\; &\approx \;\frac{{S({0,\,t} )}}{{\sqrt M }}\;\,{e^{ - \,i\,2\pi \,{\nu _0}({t - {r / c}} )}}\left\{ {\sum\limits_p {{E_p}\,{D_p}\,{e^{{{i\,2\pi {\chi_p}} / {{\lambda_0}}}}}} } \right\}\;\left\{ {\sum\limits_l {{E_l}\,{D_l}\,{e^{i\,{{2\pi \,{\chi_{ml}}} / {{\lambda_0}}}}}} } \right\}\\ &= \;\frac{{S({0,\,t} )}}{{\sqrt M }}\;\,{e^{ - \,i\,2\pi \,{\nu _0}({t - {r / c}} )}}\left\{ {\sum\limits_p {\sum\limits_l {({{E_p}\,{D_p}} )({{E_l}\,{D_l}} )} \,{e^{i\,{{2\pi ({{\chi_p} + {\chi_{ml}}} )\,} / {{\lambda_0}}}}}} } \right\}\;\;. \end{aligned}$$

Here, the subscript p indicates the characteristics of the diffuser or rough TS in the SS. Based on Eq. (58), χ₀ = χ_m₀ = 0. Using Eqs. (27) and (58), the intensity of the synthesized light is

(59)$$\begin{aligned} \left\langle {{I_R}} \right\rangle \;\;& = \;\;\sum\limits_{m\; = \;1}^M {\,\left\langle {{\Psi _{Rm}}({{\nu_0}} )\cdot {\Psi _{Rm}}^ \ast ({{\nu_0}} )} \right\rangle } \\ &= \;\frac{1}{M}\;\sum\limits_{m\; = \;1}^M {\sum\limits_p {\sum\limits_q {\sum\limits_l {\sum\limits_j {({{E_p}\,{D_p}\,{E_l}\,{D_l}\,{E_q}\,{D_q}\,{E_j}\,{D_j}} )\,\cos ({{{\,2\pi \,{\theta_{mpqlj}}} / {{\lambda_0}}}{\kern 1pt} } )} } } } } \;. \end{aligned}$$

Here,

(60)$${\theta _{mpqlj}}\; \equiv \;\;({\,{\chi_p} - \;{\chi_q}{\kern 1pt} } )\; + \;({\,{\chi_{ml}} - \;{\chi_{mj}}{\kern 1pt} } )\;\,.$$

With respect to Eq. (60), this section presumes that a wave element passing through component A has the following phase factor θ_1pq00:

(61)$${\theta _{1pqlj}}\; \equiv \;\;({\,{\chi_p} - \;{\chi_q}{\kern 1pt} } )\; + \;({\,{\chi_{1l}} - \;{\chi_{1j}}{\kern 1pt} } )\;\,.$$

Then, the difference between Eqs. (60) and (61) represents Δχ_mlj. Here, Δχ_m₀₀ = 0. Substituting Eq. (61) in Eq. (60), Eq. (60) transforms to

(62)$${\theta _{mpqlj}}\; = \;\;{\theta _{1pqlj}}\; + \,\;\Delta {\chi _{mlj}}\;\,.$$

Substituting Eq. (62) in Eq. (59), Eq. (59) transforms to

(63)$$\begin{array}{l} \left\langle {{I_R}} \right\rangle \;\; \approx \;\;\sum\limits_p {\sum\limits_q {\sum\limits_l {\sum\limits_j {({{E_p}\,{D_p}\,{E_l}\,{D_l}\,{E_q}\,{D_q}\,{E_j}\,{D_j}} )\,\cos ({{{\,2\pi \,{\theta_{1pqlj}}} / {{\lambda_0}}}{\kern 1pt} } )} } } } \\ \; - \;\frac{1}{M}\;\sum\limits_{m\; = \;1}^M {\sum\limits_p {\sum\limits_q {\sum\limits_l {\sum\limits_j {({{E_p}\,{D_p}\,{E_l}\,{D_l}\,{E_q}\,{D_q}\,{E_j}\,{D_j}} )\left( {\frac{{2\pi \,\Delta {\chi_{mlj}}}}{{{\lambda_0}}}} \right)\;\sin \left( {\frac{{2\pi \,{\theta_{1pqlj}}}}{{{\lambda_0}}}} \right)} } } } } \;. \end{array}$$

For two diffusers or a diffuser and a rough TS, there is a high probability that the value of Eq. (61) is small. For example, 2π|θ_1pqlj|/λ₀ = 2π/3 when |θ_1pqlj| = 0.3μm and λ₀ = 0.9μm, and 2π|θ_1pqlj|/λ₀ = π/3 when |θ_1pqlj| = 0.3μm and λ₀ = 1.8μm. If there is a high probability that Δχ_mlj θ_1pqlj > 0 because of the transmitted wave aberration of the PU, there is a high probability that the terms in the second term of the right side in Eq. (63) are positive when |θ_1pqlj| = 0.3μm and λ₀ is between 0.9 and 1.8 μm. Meanwhile, Eq. (37) shows that D_pD_l and D_qD_j in Eq. (63) decrease when λ₀ decreases. With respect to the second term of the right side in Eq. (63), when Δχ_mlj θ_1pqlj > 0, the slope formed by the first term of the right side in Eq. (63) is cancelled. Therefore, Eq. (63) suggests that the slope variation of the spectral baseline results from the second term of the right side in Eq. (63).

Together, Eqs. (38) and (63) suggest that a slope variation effect of the spectral baseline occurs when E₀ decreases. Because Eq. (38) indicates that both E_q (q > 0) and E_j (j > 0) increase when E₀ decreases, and the absolute value of the second term of the right side in Eq. (63) increases. E₀ tends to decrease when the averaged roughness Ra of the diffuser increases. Therefore, Ra of the diffuser controls the slope variation of the spectral baseline, as shown Fig. 10.

In the meantime, we suppose the condition |θ_1pqlj| = 0.6μm. This gives 2π|θ_1pqlj|/λ₀ = 4π/3 for λ₀ = 0.9 μm, and 2π|θ_1pqlj|/λ₀ = 2π/3 for λ₀ = 1.8 μm. This condition leads to negative terms in the second term of the right side in Eq. (63) when λ₀ approaches 0.9 μm, which increases the spectral baseline, as shown Figs. 10(b) and 10(c).

In conclusion, the ensemble average effect of PuwS affects the slope of the spectral baseline, changing the slope.

4. Data acquisition and processing

The SM shown in Fig. 1(a) requires an exposure time between 15 ms and 1.25 s to obtain single scanned spectral signals. The SM outputs an average signal of 100 scanned spectral signals for a silk scarf as the TS and an average signal of 250 scanned spectral signals for a TS other than a silk scarf.

The detection sensitivities of adjacent detection cells in the SM were varied. To correct for sensitivity deviation, we carried out the following transformation to obtain accurate spectral data:

(64)$${x_s}^{\prime}\;\, = \;\,{{({{x_{s - 1}} + \;\,2\,{x_s}\; + \;\,{x_{s + 1}}} )} / 4}\;\;.$$

Here, x_s represents the original detection signal of the s-th detection cell and x_s′ represents the s-th corrected datum, taken as the accurate datum.

First, we measured the original reference signals without the TS. Next, utilizing Eq. (64), we obtained the accurate reference data from the original reference signals. We then set the TS within the optical path and obtained the accurate spectral data of the TS after the correction described above. Finally, we applied the ratio of the spectral data of the TS to the reference data to generate relative spectrum information, shown in Figs. 6, 8, 10, 13, 17, and 18.

Fig. 8. Slope variation of spectral baseline based on M.

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Fig. 9. NIR spectrum comparison of the same silk scarf measured with different kinds of light.

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Fig. 10. Spectral profile variation based on Ra of the diffuser with PuwS_8 light.

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5. Spectral properties

5.1 Spectral noise reduction

Section 3.4 explained that a standard diffuser can generate spectral noise. Therefore, we artificially generate spectral noise with a diffuser and confirm the spectral noise reduction effect of PuwS.

The red line in Fig. 6(a) shows an example of artificially generated spectral noise. The blue line in Fig. 6(c) shows spectral noise reduced by PuwS. The horizontal axis indicates λ₀ and the vertical axis shows the transmittance of an Ra = 2.08 µm diffuser as a TS.

A diffuser having Ra = 2.08μm is placed as the TS to artificially generate spectral noise, and the transmittance corresponds to the relative spectrum information described in section 4. In other words, the transmittance is the ratio of the detected intensity after the light passes through the diffuser in the SS to the initial intensity without the diffuser.

Conventional light generates the greatest spectral noise, as shown in Fig. 6(a). As described in section 3.2, the amplitude of the spectral noise at long wavelengths (right side in Fig. 6) is clearly larger than that at short wavelengths (left side in Fig. 6). The second term of the right side in Eq. (39) shows that the spectral noise is a sum of cosine functions having different periods.

The purple line in Fig. 6(b) represents the condition with a diffuser of Ra = 1.51 µm in place of RD1 in the LS without PuwS applied. In this case the spectral noise is slightly smaller than the case in Fig. 6(a) and that the period of the spectral noise is smaller than that in Fig. 6(a). Therefore, using a diffuser in the LS alone only reduces the spectral noise slightly.

Figure 6(c) shows the spectral profile obtained from PuwS_8 light passing through an Ra = 1.51 µm diffuser as RD1 in the LS. In this case the spectral noise is drastically reduced because of the ensemble average effect described in section 3.3.

Figure 7 shows spectral noise magnitudes measured with 12 different kinds of light. The standard deviation values are used as metrics for the spectral noise magnitudes. The notation PuwS_M indicates which MP is used. Specifically, PuwS_2 light comprises two wave elements passing through components A and B in the PU shown in Fig. 1(b); PuwS_3 comprises three wave elements passing through components A, B, and F; PuwS_4 light comprises four wave elements passing through components A, B, E, and F; and PuwS_6 light comprises six wave elements passing through components A, B, D, E, F, and H.

Figures 6(a)-(b) and 8(a)–(e) show that all the spectral profile outlines are almost linear for wavelengths between 1.45 and 1.65 µm. Therefore, based on Fig. 7, we calculated each standard deviation based on the deviations from ideal straight lines within the wavelength area by the following procedure:

1. Measure the transmittance properties of the Ra = 2.08 µm diffuser with PuwS_2 light to PuwS_6 light without the Ra = 1.51 µm diffuser and plot the results;
2. Measure the transmittance properties of the Ra = 2.08 µm diffuser with PuwS_2 light to PuwS_6 light using the Ra = 1.51 µm diffuser for RD1 in the LS and plot the results;
3. Measure the transmittance property of the Ra = 2.08 µm diffuser with PuwS_8 light without the Ra = 1.51 µm diffuser and plot the results;
4. Calculate average values of the transmittance within the wavelength range;
5. Set the respective central positions from the average values;
6. Provisionally generate ideal slope lines through the central positions;
7. Calculate the deviation between the measured transmittance and the ideal slopes;
8. Optimize the slopes to minimize the average of the deviation values within the wavelength range;
9. Recalculate the deviation values based on the optimized slopes;
10. Statistically analyze each distribution of recalculated deviation values and compute the standard deviations.

Figure 7(a) shows the standard deviations for the six kinds of light for the case where the light did not pass through the Ra = 1.51 µm diffuser. Figure 7(b) shows the case where the light passed through the Ra = 1.51 µm diffuser as RD1 in the LS. The horizontal axis shows the total numbers of wave elements, where 1 corresponds to conventional light without PuwS.

Figure 6(a) provides the standard deviation for M = 1 in Fig. 7(a) and Fig. 6(b) provides the standard deviation for M = 1 in Fig. 7(b). Figure 6(c) provides the standard deviation for M = 8 in Fig. 7 (b).

According to section 3.3, Fig. 7 shows that the ensemble average effect of PuwS decreases the standard deviation of the spectral noise and the standard deviation decreases when M increases, consistent with Eqs. (43) and (44).

5.2 Spectral baseline changing

The slope of the spectral baseline of Fig. 6(c) is clearly different from that of Fig. 6(a) although the same TS was measured. Section 3.6 explained that PuwS changes the slope of the spectral baseline because of the ensemble average effect when the experimental optical system includes a pair of diffusers or a combination of a diffuser RD1 and a rough TS.

Figure 8 shows the slope variation of the spectral baseline based on the total number M of wave elements in the PuwS method. For Fig. 8, the experimental optical system included an Ra = 2.08 µm diffuser at the TS position and an Ra = 1.51 µm diffuser for RD1 in the LS. The red line in Fig. 8(a) corresponds to Fig. 6(b) and the blue line in Fig. 8(e) corresponds to Fig. 6(c). Figures 8(b)–(d) are respectively obtained with PuwS_2 light to PuwS_6 light.

Figure 8(a), corresponding to conventional light, shows the biggest slope. The slope is reduced when M increases. Therefore, Fig. 8(e), corresponding to PuwS_8, shows the smallest slope. Because the total number of summed terms in the second term of the right side in Eq. (63) increases as M increases, the ensemble average effect described in section 3.6 changes the slope of the spectral baseline.

5.3 Measured NIR spectrum indicating highly accurate spectrometry

To confirm that PuwS is effective for obtaining highly accurate spectrometry, we measured the NIR transmittance spectrum of a silk scarf having a thickness of 0.1 mm (as TS) using different kinds of light. The vertical axis in Fig. 9 is the absorbance Ab, defined by the following formula:

(65)$$Ab\; \equiv \; - \;{\log _{10}}\;\left\{ {{{\,\left\langle {\;Isilk\;} \right\rangle \;} / {\;\left\langle {\;Iref\;} \right\rangle }}\,} \right\}\;\;.$$

Here, < Isilk > represents the spectral transmittance data of the silk scarf and < Iref > represents the reference data without the silk scarf.

The left side of Fig. 9 shows the first condition where no diffuser is included. The red line in Fig. 9(a) shows the conventional light case, exhibiting high spectral noise. The blue line in Fig. 9(b) shows the case for PuwS_8 light illuminating the same silk scarf with no diffuser, exhibiting reduced spectral noise.

The right side of Fig. 9 corresponds to the experimental optical system shown in Fig. 1(a) with a removable RD2 before the same silk scarf (the TS) and RD1 removed. Ra for RD2 is 2.08 µm. The red line in Fig. 9(c) corresponds to conventional light passing through RD2 and illuminating the same silk scarf, exhibiting high spectral noise, regardless of RD2 inserted in the optical path. The blue line in Fig. 9(d) corresponds to PuwS_8 light passing through RD2 and illuminating the same silk scarf.

Figure 9(d) shows spectral noise reduction and suggests absorption bands at the positions indicated by arrows. This indicates that a combination of PuwS and an appropriate diffuser is effective for obtaining highly accurate spectrometry.

In Fig. 9, the slope direction of the spectral baselines in the left side is opposite to that in the right side, even though the same silk scarf was measured. Equation (44) indicates that the slope of the spectral baseline results from D₀, expressed by Eq. (37), because E₀ is independent of λ₀. We can regard the silk scarf as a rough TS, described in section 3.6. Equation (44) suggests that local thickness deviations of the silk scarf form the slope of the spectral baseline in Fig. 9(b).

More specifically, when the local thickness deviation of the silk scarf is included as Δd, the slope of the spectral baseline in Fig. 9(b) approximates to Eq. (44), which includes Eq. (37). When λ₀ decreases, Eq. (37) decreases and the corresponding Ab increases, as shown in Fig. 9 (b).

In other words, we can regard the slope of the spectral baseline in Fig. 9 (b) as a kind of optical noise resulting from the local thickness deviation of the silk scarf (or the surface roughness deviation of the TS). According to section 3.6, a combination of PuwS and an appropriate diffuser achieves an ensemble average effect that reduces optical noise, thus changing the slope of the spectral baseline from that in Fig. 9(b) to that in Fig. 9(d), in which previously hidden absorption bands appear. Therefore, Fig. 9(d) seems to show the original absorbance characteristics of the silk scarf without some kinds of optical noise.

With respect to Fig. 9, section 3.6 explained that a slope variation of the spectral baseline arises when Ra of the corresponding RD2 increases. To confirm this prediction, we investigated the slope variation of the spectral baseline based on Ra for the corresponding RD2, as shown in Fig. 10. Note that the vertical axis in Fig. 10 is the light transmittance of the silk scarf, whereas the vertical axis in Fig. 9 is the absorbance.

The red line in Fig. 10(a) corresponds to PuwS_8 light directly illuminating the same silk scarf without passing through RD1 and RD2, corresponding to Fig. 9(b). The slope of the spectral baseline exhibits a decrease with decreasing λ₀.

The green line in Fig. 10(d) and the blue line in Fig. 10(e) correspond to PuwS_8 light passing through RD1 with Ra = 0.77 µm or Ra = 1.51 µm, respectively, in the LS, illuminating the same silk scarf without passing through RD2 in the SS.

The slopes of these spectral baselines increase with decreasing λ₀. Referring to Fig. 9(d), it is considered that the slope direction results from the original absorbance characteristics of the silk scarf. Figure 9(d) shows the light absorption characteristics for wavelengths between 1.4 and 1.7 µm. Figures 10(d) and 10(e) show the same light absorption characteristics because they both show concave profiles from the straight broken lines (f) and (f’) in the wavelength ranges [B] and [B’]. Therefore, it is considered that Figs. 10(d) and 10(e) approach the original absorbance characteristics of the silk scarf.

As shown below in Fig. 12, a diffuser with greater Ra tends to reduce E₀. Equation (38) suggests that E_j (j > 0) tends to increase and the absolute value in the second term of the right side in Eq. (63) tends to increase when E₀ decreases. Therefore, as shown in Fig. 10, a slope variation of the spectral baseline arises when Ra of the corresponding diffuser increases.

We slightly changed Ra of RD1 continuously in the range 0.48 to 0.54 µm to obtain the transmittance profiles shown in Figs. 10(b) and 10(c). The brown line in Fig. 10(b), corresponds to an Ra near 0.48μm. The purple line in Fig. 10(c) corresponds to an Ra near 0.54 μm. Figures 10(b) and 10(c) shows that Ra can control the slope of the spectral baseline. Moreover, Figs. 10(b) and 10(c) show a greater transmittance than that of conventional light in the wavelength ranges [A] and [A’]. As described in section 3.6, it is considered that these phenomena result from the second term of the right side in Eq. (63).

6. Temporal coherence property

6.1 Diffuser characteristics measurement

Equation (48), described in section 3.4, indicates that we can estimate the diffuser characteristics using PuwS-M light with a large M. Therefore, we measured the spectral transmittance of a diffuser using PuwS_8 light and analyzed the diffuser characteristics. We took the experimental optical system shown in Fig. 1(a) and removed both RD2 and the TS. First, we measured the reference data without RD1. Next, we placed RD1 in the LS and measured the spectral data. We then obtained the spectral transmittance of RD1, which is the relative spectrum information corresponding to the ratio of the spectral data to the reference data, as explained in section 4.

The red line in Fig. 11(a) shows the spectral transmittance with RD1 with Ra = 1.51 µm. Using the profile in Fig. 11(a) and the rule in Eq. (38), we adopt a fitting method to obtain Figs. 11(b)–(d) based on Eq. (48).

Fig. 11. Diffuser transmittance and analyzing characteristics of the corresponding diffuser.

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For Fig. 11(a), the vertical axis is the transmittance of the Ra = 1.51 µm diffuser. According to section 3.4, after passing through a diffuser, the intensity of the initial major wave train is distributed among the intensities of the modified major wave train and a few dispersed wave trains based on the intensity distribution rule in Eq. (48). Therefore, for Figs. 11(b)–(d), the vertical axis indicates the distributed intensities.

Equation (48) indicates that the total transmittance of RD1 approaches a summation of (E_lD_l(λ₀))², shown in Fig. 11(b) as the purple line. The modified major (0th-order) wave train area (E₀D₀(λ₀))² is between the 0% line and the profile shown in Fig. 11(d) as the blue line. Moreover, the 1st-order dispersed wave train area (E₁D₁(λ₀))² is between the two profiles shown in Figs. 11(d) and 11(c) as the green line, and the 2nd-order dispersed wave train area (E₂D₂(λ₀))² is between the two profiles shown in Figs. 11(c) and (b).

Figure 11 shows that the distributed intensity of the modified major (0th-order) wave train E₀² is 35.5% and the distributed intensities of the 1st-order and 2nd-order dispersed wave trains E₁² and E₂² are 28.0% and 36.5% respectively. From Eq. (37), Δd for D₀ to D₂ is respectively 0.33, 1.52, and 2.20 µm. Here, we calculated Eq. (37) by substituting the refractive index n of BK7, considering n to vary with λ₀.

Figure 11 shows that the Ra = 1.51 µm diffuser has the following characteristics:

A. The modified major (0th-order) wave train component (E₀D₀(λ₀))² is almost independent of λ₀;
B. The 2nd-order dispersed wave train component (E₂D₂(λ₀))² vanishes when λ₀ is near 1.15 µm;
C. The 1st-order dispersed wave train component (E₁D₁(λ₀))² is small when λ₀ is near 0.9 µm and monotonously increases when λ₀ increases.

Equation (55) indicates that these characteristics affect the SVR of the spectral fringe pattern.

To compare the Ra = 1.51 µm diffuser characteristics with diffusers with different Ra, we measured the spectral transmittance of other diffusers and applied a fit. Figure 12 shows the results.

Fig. 12. Different kinds of diffuser characteristics obtained from analyzing transmittance measurements.

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As shown in Fig. 12, the distributed intensity of the modified major (0th-order) wave train E₀² monotonously decreases as Ra increases. Section 3.5 showed that the spectral visibility of a spectral fringe pattern tends to decrease when E₀ decreases. Therefore, a diffuser with a high Ra can effectively reduce the spectral visibility.

6.2 Spectral fringe pattern obtained from a thin film

The blue line in Fig. 13(a) shows the spectral fringe pattern measured for a transparent thin film having a thickness of approximately 10 μm made from pure polyethylene resin. We measured both the spectral data and reference data, as explained in section 4, using PuwS_8 light without RD1 and RD2. The vertical axis of Fig. 13 is the transmittance of the transparent thin film and the horizontal axis is λ₀. We confirmed that SVorg (section 3.2) obtained from Fig. 13(a) was the same as that obtained with conventional light, as explained in section 3.5.

Fig. 13. Spectral fringe pattern comparison obtained from a thin film having thickness 10.705 µm.

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The black line in Fig. 13(c) shows the calculated data obtained from Eq. (28). For the calculation, the average thickness d₀ of the transparent thin film was 10.705 μm and Δd = 0.225 μm. With respect to Eq. (3), we also considered the wavelength dependence of n.

Both Figs. 13(a) and 13(c) show that the amplitudes of the spectral fringe patterns decrease when λ₀ decreases and that the central lines of the spectral fringe patterns increase when λ₀ increases. These phenomena result from D₀(Δd, λ₀) and D₁(Δd, λ₀) in Eq. (28). In other words, Δd for the transparent thin film is slightly larger than that for an ideal thin film.

The red line in Fig. 13(b) shows another spectral fringe pattern measured from the same transparent thin film. The amplitude of pattern is clearly smaller than that in Fig. 13(a). PuwS_8 light passed through a removable Ra = 2.08 µm diffuser RD1 positioned in the LS generates Fig. 13(b). In other words, Fig. 13(b) shows that the spectral visibility SVdif expressed by Eq. (56) is clearly smaller than SVorg expressed by Eq. (34). Moreover, the phase in Fig. 13(b) is slightly shifted compared with that in Fig. 13(a). We expect that the phase shift phenomenon results from the additional phases χ_ml–χ_mj in Eq. (55).

Comparing Fig. 13(b) with Fig. 13(a), SVR can be calculated. Based on Eq. (57), SVR is approximated by the ratio of the amplitude of the spectral fringe pattern shown in Fig. 13(b) and that shown in Fig. 13(a).

Figure 14 shows SVR with Ra for an inserted RD1 in the LS. Based on Fig. 14, the original reference data without the 10.7 µm thick polyethylene film TS were obtained with PuwS_8 light passing through a different RD1 having a different Ra. We previously calculated SVorg from Fig. 13(a). Therefore, we can obtain SVR based on Eq. (57). The green line in Fig. 14(a) shows the SVR profile when the Ra = 0.35 µm diffuser is inserted. The blue and purple lines in Figs. 14(b) and 14(c) show the SVR profile when the Ra = 0.48 µm and Ra = 0.54 µm diffusers are inserted, respectively. The red line in Fig. 14(d) shows the SVR profile when the Ra = 2.08 µm diffuser is inserted.

Fig. 14. SVR obtained with a combination of PuwS_8 and a diffuser.

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Figure 14 shows that SVR tends to decrease when Ra increases. Each SVR profile is a minimum when λ₀ is close to 0.9 µm (area [B]) and a maximum value when λ₀ is close to 1.3 µm (area [A]). Equations (55) and (57) indicate that the SVR profiles are based on the corresponding diffuser characteristics.

Figure 15 shows the relation between the SVR profile and the analyzed results detailed in section 6.1. The left hand vertical axis in Fig. 15 is SVR and the right hand vertical axis is the distributed intensity. The horizontal axis is λ₀.

Fig. 15. Relation between SVR profile and analyzed results for diffuser.

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The deep blue line in Fig. 15(a) corresponds to Fig. 14(d), with PuwS_8 passing through a Ra = 2.08 µm diffuser RD1 in LS. The purple line in Fig. 15(b) shows the diffuser transmittance (including distributed intensities Σ(E_lD_l(λ₀))²) when the Ra = 2.08 µm diffuser RD1 is in LS. As described in section 6.1, the profile in Fig. 15(b) is analyzed to obtain the characteristics of the Ra = 2.08 µm diffuser.

The modified major (0th-order) wave train area (E₀D₀(λ₀))² is between the 0% line and the blue line in Fig. 15(d).

A 1st-order dispersed wave train area (E₁D₁(λ₀))² exists between the blue line and the green line in Fig. 15(c). A 2nd-order dispersed wave train area (E₂D₂(λ₀))² exists between the green line and the purple line in Fig. 15(b). According to Fig. 15, the distributed intensities E₀² to E₂² are respectively 32.0%, 32.2%, and 35.8%, and Δd of D₀ to D₂ are respectively 0.36, 1.52, and 2.33 µm.

According to Eq. (48) in section 3.4, the diffuser transmittance shown in Fig. 15(b) is approximately Σ(E_lD_l)², which is included in the denominator of Eq. (55). A combination of Eqs. (55) and (57) indicates that SVR decreases when Σ(E_lD_l)² decreases. Figure 15 (b) shows that Σ(E_lD_l)² is minimum when λ₀ approaches 0.9 µm (area [B’]). Therefore, SVR is minimum when λ₀ is close to 0.9 μm (area [B]). Moreover, Figs. 15(b) and 15(c) show that (E₂D₂)² approaches 0 when λ₀ is close to 1.2 µm (area [A’]). The absolute values of the terms including (E₂D₂) in the second term of the right side in Eq. (55) decrease so as to not cancel the first term of the right side when (E₂D₂)² approaches 0. Therefore, SVR is maximum when λ₀ is close to 1.2 µm (area [A]).

Figure 16 shows how PuwS contributes to controlling the temporal coherence. Born and Wolf [15] used visibility as a metric of partial coherence, so this paper uses SVR defined in Eq. (57) as a metric of the temporally partial coherence. It should be emphasized that SVR decreases when PuwS effectively contributes to reducing the temporal coherence.

Fig. 16. SVR comparison between PuwS_8 light and conventional light with Ra of RD1 in LS.

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In the experimental optical system, RD2 was removed in the SS. In the LS, four different kinds of diffuser were used for RD1, each with a different Ra. The four diffusers had an Ra of 0.35, 0.48, 0.77, and 2.08 μm. In Fig. 16, the horizontal axis is the Ra of RD1. Here, Ra = 0 μm indicates the experimental optical system with no RD1 in the LS.

In Figs. 16(a) and 16(b), the same transparent thin film as used in Figs. 13–15 was used as the TS. We measured the transmittance of the transparent thin film with 10 kinds of light. SVR was calculated with the same method as used in Fig. 14. The left side vertical axis in Figs. 16(a) and 16(b) is SVR. The right side vertical axis in Fig. 16(c) is the transmittance of the corresponding RD1, as the diffuser transmittance relates to SVR when PuwS_8 light is used. The diffuser transmittance was measured with the same method as used for Figs. 11 and 12.

The purple squares in Fig. 16(a) show the case of conventional light passing through RD1 and irradiating the transparent thin film made from pure polyethylene resin. The blue circles in Fig. 16(b) show the case of PuwS_8 light passing through RD1 and irradiating the same transparent thin film.

As shown in Fig. 12, E₀² decreases when Ra increases. Section 3.5 explained that SVR decreases when E₀² decreases. Therefore, both Figs. 16(a) and 16(b) show that SVR decreases when Ra increases.

Figures 16(a) and 16(b) shows that PuwS_8 light effectively decreases SVR compared with conventional light. This suggests that PuwS light enhances the visibility reduction effect. Meanwhile, Eq. (48) indicates that when PuwS_8 light passes through RD1, the diffuser transmittance is approximately Σ(E_lD_l)². A combination of Eqs. (55) and (57) indicates that Σ(E_lD_l)² effectively affects SVR. The black triangles in Fig. 16 (c) indicate the diffuser transmittance. It is especially notable that the profile shown in Fig. 16(b) is similar to that shown in Fig. 16(c). Therefore, the profile similarity between Figs. 16(b) and 16(c) suggests that we can experimentally confirm the reliability of at least part of Eq. (55).

Figure 16 shows that a combination of PuwS_8 light and an appropriate diffuser achieves a bigger visibility reduction effect than conventional light and a diffuser when |μ^τ₀₁| approaches 1.

6.3 Spectral fringe pattern obtained from thick glass

In Fig. 13, the thickness of the TS (transparent thin film with an approximate thickness of 10 μm) was small enough that the spectral degree of temporal coherence |μ^τ₀₁|, defined in Eq. (31), approaches 1. Meanwhile, as shown in Fig. 4, increasing d₀ of the TS has the effect of reducing the amplitude of the spectral fringe pattern to zero. Figure 17 illustrates this.

Fig. 17. Spectral fringe patterns calculated and experimentally obtained from a thick transparent parallel glass.

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The blue line in the top part of Fig. 17(a) shows the measured spectral fringe pattern obtained from an optically flat and parallel glass plate. The glass plate is made from BK7 and the experimental optical system used PuwS_8 light with no RD1 and RD2, that is, PuwS_8 light directly irradiated the optically flat and parallel glass plate as the TS. We measured the transmittance of the optically flat and parallel glass plate with the method explained in section 4.

Figure 17(a) shows a small strain in the envelope lines in the spectral fringe pattern when λ₀ approaches 1.4 µm, while the strain does not appear in Fig. 17(b). We expect that the strain results from an absorption band of hydroxyl group in BK7.

The black line in the top part of Fig. 17(b) shows the calculated result based on Eq. (28) for d₀ of the glass plate of 138.40 μm and Δd set to 0 μm. Note that Δd = 0 μm means the glass is optically flat. The wavelength range Δλ of the SM was set to 7.5 nm to match a wavelength resolution of 7 nm of one detection cell in the SM, as described in chapter 2. With respect to Eq. (3), we also considered the wavelength dependence of n for BK7.

Both Figs. 17(a) and 17(b) shows that SVorg is slowly reduced to zero as λ₀ decreases. As described in sections 3.1 and 3.2 and shown in Fig. 4, this phenomenon results from a change in the spectral degree of temporal coherence |μ^τ₀₁|. The black line in the bottom part of Fig. 17(c) shows a calculated result based on Eq. (26). The vertical axis is SDTC. Figure 17(c) shows that SDTC decreases when λ₀ decreases.

As predicted in section 3.5, we experimentally confirmed that conventional light with no diffuser shows the same profile as in Fig. 17(a).

Figure 18 shows the visibility reduction effect for a small |μ^τ₀₁|. The blue line corresponds to Fig. 17(a). For Fig. 18(b), the reference data was obtained from PuwS_8 light passing through a Ra = 2.08 μm diffuser RD1 in the LS and an optically flat and parallel glass plate with d₀ = 138.45 μm as the TS, predicted in Fig. 17(b). We then measured the transmittance of the optically flat and parallel glass plate with the method presented in section 4.

Fig. 18. Comparison of spectral fringe patterns obtained from a thick glass plate of thickness 138.45 µm.

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Comparing Fig. 18(b) with Fig. 18(a) shows that a combination of PuwS_8 light and an appropriate diffuser can reduce the visibility, as Fig. 18(b) clearly shows a smaller amplitude of the spectral fringe pattern than that shown in Fig. 18(a). Figure 18(b) shows a slight phase shift compared with Fig. 18(a). We consider that this is related to additional phases χ_ml–χ_mj expressed in Eq. (55).

Figure 18 shows smaller values of SVR than Fig. 13. It is possible that the small |μ^τ₀₁| makes SVR sensitive to χ_ml–χ_mj in Eq. (55).

7. Conclusion

In this paper, we proposed the PuwS method, which transforms conventional light to PuwS light. PuwS light is appropriate for optically accurate measurements because PuwS effectively reduces optical noise.

To theoretically demonstrate the benefits of PuwS light, we proposed a series of analytical models which form an extended wave train model based on a traditional wave train model [15] describing a temporal coherence property.

In the traditional wave train model, one wave train does not interfere with another when the two wave trains are separated by more than the coherence length ΔL₀. PuwS divides the conventional light into multiple wave elements separated by more than ΔL₀. After separating the wave elements, PuwS synthesizes the separated wave elements such that they propagate in the same direction simultaneously.

The synthesizing operation generates an ensemble averaging effect that reduces optical noise. We theoretically and experimentally confirmed that the optical noise reduction effect increases when the total number of wave elements M increases.

After the synthesizing operation, the temporal coherence of each wave element remains, so that each wave elements can form a spectral fringe pattern. We propose that a combination of PuwS and an appropriate diffuser in the light source area LS can reduce the spectral visibility.

According to the extended wave train model, when a wave train passes through the diffuser, the diffuser forms multiple wave trains having different phases . Then, the ensemble average effect reduces the spectral visibility because the reduction terms in the model form phase shifted spectral fringe patterns in the spectrum. Moreover, when multiple wave elements divided by PuwS pass through the diffuser, they generate multiple phases of the spectral patterns in the spectrum. Therefore, PuwS enhances the ensemble average effect and reduces the spectral visibility more effectively, as experimentally confirmed.

The surface roughness of the target sample TS affects the transmittance profile of the spectral baseline, and the transmittance profile based on the surface roughness of the TS should be recognized as a kind of optical noise when we detect multiple absorption bands from the TS. We theoretically and experimentally confirmed that a combination of PuwS and an appropriate diffuser decreases the transmittance profile of the spectral baseline and that multiple absorption bands appear, previously hidden by the optical noise. The transmittance profile change of the spectral baseline results from the ensemble average effect reducing the optical noise. According to the extended wave train model, the first phase distribution generated by the surface roughness of the TS interacts with the second phases generated by the combination PuwS and the appropriate diffuser, to decrease the transmittance profile of the spectral baseline in the spectrum. PuwS light (after passing through the appropriate diffuser) provides an optimal environment to achieve highly accurate spectrometry.

Acknowledgements

We would like to thank Mr. Hayata and Mr. Endo of Japan Cell Limited Company for measuring the transmitted wave front aberration for the optical phase unsynchronizer. Mrs. Chizuru Ando (Ando’s wife) offered her silk scarf to experimentally confirm the benefits of PuwS light, and the author would also like to express his appreciation for her devoted help in other areas of this study.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Spectral noise reduction and temporal coherence control using a phase unsynchronized wave synthesizing method

Abstract

1. Introduction

2. Experimental optical system and PuwS method

3. Analytical model

3.1 Fundamental interference model based on a traditional wave train model

3.2 Wave train model extension (major and dispersed wave train model)

3.3 Ensemble average effect of PuwS

3.4 Diffuser action

3.5 Temporal coherence control (spectral visibility ratio)

3.6 Slope variation of spectral baseline

4. Data acquisition and processing

5. Spectral properties

5.1 Spectral noise reduction

5.2 Spectral baseline changing

5.3 Measured NIR spectrum indicating highly accurate spectrometry

6. Temporal coherence property

6.1 Diffuser characteristics measurement

6.2 Spectral fringe pattern obtained from a thin film

6.3 Spectral fringe pattern obtained from thick glass

7. Conclusion

Acknowledgements

Disclosures

Data availability

References

Data availability

Cited By

Figures (18)

Equations (65)

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