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Nanomanufacturing and Metrology

, Volume 2, Issue 4, pp 187–198 | Cite as

A New Optical Angle Measurement Method Based on Second Harmonic Generation with a Mode-Locked Femtosecond Laser

  • Hiraku MatsukumaEmail author
  • Shuhei Madokoro
  • Wijayanti Dwi Astuti
  • Yuki Shimizu
  • Wei Gao
Original Articles
  • 119 Downloads

Abstract

This paper proposes a new optical angle measurement method based on second harmonic generation with a mode-locked femtosecond laser source by taking use of the unique characteristic of high peak power and high-intensity electric field of the femtosecond laser pulses. The angle is measured from the second harmonic wave intensity, which is a function of the angle between the laser beam axis and the optic axis of a nonlinear crystal attached to the target of interest. It is found that the beta barium borate (BBO) crystal is suitable as the nonlinear crystal for this purpose through theoretical simulation. As the first step of research, an experimental demonstration is also carried out in such a way that a change of the second harmonic wave intensity due to the angular displacement of BBO crystal is measured to verify the feasibility of the proposed principle of the angle measurement method.

Keywords

Femtosecond laser Angle measurement Second harmonic generation 

1 Introduction

Angle is one of the fundamental quantities to determine the geometry or motion of an object. Angle measurement has therefore played an important role in various applications such as scanning probe microscopy [1], machine tools [2, 3], and so on. Among the angle measurement methods, optical methods are characterized as non-contact and non-destructive measurement methods, and have been applied to straightness measurement [4, 5, 6], surface profile measurement [7, 8, 9], multi-axis position measurement [10, 11, 12, 13], spindle error measurement [14], etc. Especially, measurement methods employing a laser source, such as the laser autocollimation [15], the method based on internal-reflection effect [16], and the method based on interference [17] have been widely used in a large variety of applications. A continuous wave (CW) laser source is employed in these conventional optical methods, where the characteristics of single frequency and coherence of the CW laser are utilized for the angle measurement.

Meanwhile, mode-locked femtosecond pulse lasers, or simply called femtosecond lasers, have been employed for precision dimensional measurement in recent years [6, 18, 19, 20, 21, 22, 23]. Similar to a CW laser, a femtosecond laser has the advantage of high directivity of light beam, which makes it suitable for angle measurement compared with a filament lamp or a LED. On the other hand, compared with a CW laser, a femtosecond laser has two unique characteristics. The first unique characteristic is the optical frequency comb over a wide spectral range with optical frequency comb modes equally spaced with a mode spacing (repetition rate). The second unique characteristic is the high peak power and high-intensity electric field of the femtosecond laser pulses [24]. Such unique characteristics of a femtosecond laser are expected to bring new possibilities for angle measurement. On the other hand, development of new measurement principles is required when a femtosecond laser is applied to replace a CW laser for angle measurement so that the specific characteristics of a femtosecond laser can be effectively utilized.

Based on the first unique characteristic of a femtosecond laser, i.e., the wide spectral range with optical frequencies equally spaced by the repetition rate, the authors have proposed a series of new angle measurement principles. In the newly proposed femtosecond laser optical lever method [25], a collimated laser beam from a femtosecond laser, after passing through a Fabry–Pérot etalon for effective repetition frequency modulation, is projected onto a grating reflector to generate a group of first-order diffracted beams from the grating reflector to be received by a photodiode (PD). Differing from a conventional PD-type optical lever employing a laser beam with a single frequency as the light source, the angle measurement range can be significantly expanded with the group of widely spread first-order diffracted beams. A similar concept has been applied to a new femtosecond laser autocollimator in which the group of first-order diffracted beams from the grating reflector are received by an autocollimation unit where the beams are focused by a collimator objective (CO) onto the PD located at the focal position of the CO [26]. Compared with the femtosecond optical lever, the femtosecond laser autocollimator is not influenced by the distance from the grating reflector. Then a new optical frequency domain angle measurement method has been proposed to increase the visibility of output signal of the femtosecond laser autocollimator, which is limited by the overlap of the focused diffracted light spots [27]. The output visibility of a prototype femtosecond laser autocollimator has been increased by the proposed method to approximately 100% over a large range of 21,600 arc-seconds. A new absolute angular position measurement method has also been proposed by utilizing the stabilized optical frequency comb of a femtosecond laser based on the fact that each of the first-order diffracted beams from the grating reflector has a deterministic angular position corresponding to the optical frequency of each comb mode of the femtosecond laser.

The motivation of this paper is to explore the possibility of taking use of the second unique characteristic of a femtosecond laser, i.e., the high peak power and high-intensity electric field, for angle measurement. This characteristic is conventionally employed to generate second harmonic waves based on the nonlinear optical phenomenon [28, 29], which is well known as the second harmonic generation (SHG). SHG has usually been applied as a wavelength conversion method of lasers, where maximization of the output power of the second harmonic wave has attracted for various applications [30, 31]. Meanwhile, it is also known that the output power of the second harmonic wave correlates with an angle between wave vectors and a crystal optic axis [24, 28, 31, 32], which is expected to be employed for angle measurement.

In this paper, a new principle of angle measurement based on SHG with a femtosecond laser is proposed. As the first step of research, fundamental characteristics of the proposed method are at first investigated through theoretical analysis, followed by basic experiments of angle measurement for demonstrating the feasibility of the proposed method. In the theoretical analysis, since the incident angle between the nonlinear optical element and the laser beam axis is an important parameter, the incident angular characteristics of second harmonic generation are calculated for several nonlinear optical elements. In addition, the dependence of the SHG intensity on the angular displacement is also simulated. For the basic experiments of angle measurement, a prototype optical setup employing a beta barium borate (BBO) crystal as the nonlinear crystal is designed and constructed. A ray-tracing calculation is carried out to further investigate the validity of the experimental results.

2 Fundamental Principle of the Angle Measurement Based on the Second Harmonic Generation

Figure 1 shows a schematic of the optical setup for the proposed optical angle measurement method based on SHG with a femtosecond laser. The optical setup is mainly composed of a femtosecond laser source, a photodetector, and a nonlinear crystal mounted on a measurement target, which rotates about the Y-axis in the figure. The optic axis of the nonlinear crystal is aligned to be parallel to the XZ plane in the figure. A collimated beam from a femtosecond laser with a beam diameter of ϕ as the fundamental wave from the laser source, which is also aligned to be parallel to the XZ plane, is made incident to the nonlinear crystal to generate second-harmonic waves. Since the power of the generated second harmonic wave P2 depends on the angle θ between the optic axis of the nonlinear crystal and the axis of the incident laser beam, θ can be obtained by detecting P2 with the photodetector.
Fig. 1

Schematic of the optical setup for the proposed optical angle measurement method based on SHG with a collimated femtosecond laser beam

Now we assume that the ordinary and extraordinary axes of the nonlinear crystal are aligned to be parallel to the Y- and X-axes, respectively, while the incident fundamental wave and the second harmonic wave are polarized in the Y- and X-directions, respectively. Denoting the power of the incident fundamental wave as P1, the relationship between P2 and θ can be expressed by the following equation [32]:
$$P_{2} = \frac{{8\pi^{2} d_{\text{eff}}^{2} }}{{n_{\text{o}} \left( {\lambda_{1} } \right)^{2} n_{\text{e}} \left( {\theta ,{\kern 1pt} \lambda_{2} } \right)\varepsilon_{0} c\lambda_{1}^{2} }}P_{1}^{2} \frac{{L^{2} }}{S}{\rm{sinc}}^{2} \frac{\Delta k(\theta )L}2,$$
(1)
where deff is the effective nonlinear coefficient [32], ne and no are refractive indices of a negative uniaxial crystal (ne < no) for extraordinary and ordinary rays, respectively, λ1 and λ2 are wavelengths of the fundamental wave and the second harmonic wave, respectively, ε0 is the vacuum permittivity, c is the speed of light in vacuum, S is the cross-sectional area of the collimated laser beam, and L is the crystal length. Δk is a phase mismatching that can be represented as follows:
$$\Delta k\left( \theta \right) = 2k_{1} - k_{2} = \frac{4\pi }{{\lambda_{1} }}\left[ {n_{\text{o}} \left( {\lambda_{1} } \right) - n_{\text{e}} \left( {\theta ,\,\lambda_{2} } \right)} \right],$$
(2)
where k1 and k2 are the magnitude of the fundamental wave vector and the second harmonic wave vector, respectively. As can be seen in Eq. (1), P2 is proportional to sinc2k(θ)L/2], and becomes maximum when Δk(θ) is zero. In the case with a uniaxial birefringent crystal such as beta barium borate (BBO), regarding the refractive index ellipse shown in Fig. 2a, the angle θm referred to as the matching angle that satisfies no(λ1) = ne(θm, λ2), as well as Δk(θm) = 0, can be found; this procedure is known as the index matching [31].
Fig. 2

Schematic of the second harmonic generation. a Refractive index ellipse of a negative uniaxial crystal. b Case of phase mismatching. c Case of phase matching

It should be noted that P2, which is the sum of the second harmonic waves generated in the nonlinear crystal, is also proportional to the square of P1, as can be seen in Eq. (1). At a position where the incident fundamental wave generates the second harmonic wave, the two are coherent with each other. The fundamental wave continues to generate additional contributions of the second harmonic wave while propagating through the nonlinear crystal. In the case where the phase is mismatched (no(λ1) ≠ ne(θm, λ2)), as shown in Fig. 2b, effective SHG cannot be accomplished due to the different propagating speeds of the fundamental and second harmonic waves in the crystal. Meanwhile, in the case where the phase is matched (no(λ1) = ne(θm, λ2)), as shown in Fig. 2c, all the second harmonic waves are combined totally constructively, and P2 can be maximized. As described above, efficient SHG can be accomplished by the index matching. Although the angle-dependence of SHG is a well-known phenomenon, in this paper an attempt is made to utilize the phenomenon for detecting small angular displacement of the nonlinear crystal.

It is noted that, in addition to mode-locked femtosecond lasers, other pulsed laser sources such as Q-switched Nd: YAG lasers can also generate second harmonic waves. However, femtosecond lasers are more stable in output powers, and therefore more suitable for angle measurement based on SHG than other pulsed laser sources. Therefore, a femtosecond laser is more suitable for stable angle measurement based on SHG, which is a nonlinear process associated with optical power.

3 Theoretical Calculations on the Angle Measurement Sensitivity

To estimate the sensitivity of angle measurement by the proposed method based on SHG, theoretical calculations are carried out based on Eq. (2). As the first step of theoretical calculations, an attempt is made to select a nonlinear crystal appropriate for the proposed optical angle measurement method with a specific mode-locked femtosecond laser source.

The refractive index of a medium ne for an extraordinary ray is a function of two variables θ and λ, and can be expressed by the following equation:
$$n_{\text{e}} \left( {\theta ,\,\lambda } \right) = \left[ {\frac{{\sin^{2} \theta }}{{N_{\text{e}}^{2} \left( \lambda \right)}} + \frac{{\cos^{2} \theta }}{{n_{\text{o}}^{2} \left( \lambda \right)}}} \right]^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} ,$$
(3)
where Ne(λ) = ne(90°, λ) and no(λ) = ne(0°, λ). It is known that the refractive index Ne(λ) or no(λ) of a transparent medium can be expressed by the following empirical equation [32]:
$$N_{\text{e}} \left( \lambda \right),{\kern 1pt} n_{\text{o}} \left( \lambda \right) = \sqrt {A + \frac{B}{{\lambda^{2} - C}} - D\lambda^{2} } ,$$
(4)
where coefficients A, B, C, and D are medium-specific constants. These parameters for the refractive indices no and Ne for BBO, lithium-iodate (LiIO3), and lithium-niobate (LiNbO3) whose phase matching angles have relatively smaller dependences on the wavelength around 1560 nm are summarized in Table 1 [32].
Table 1

Parameters for calculation of the empirical equation of refractive indices

Crystal

A (−)

B (−)

C (µm2)

D (µm−2)

BBO

    

 for no (θ = 0°)

2.7539

0.01878

0.01822

0.01354

 for Ne (θ = 90°)

2.3753

0.01224

0.01667

0.01516

LiIO3

    

 for no (θ = 0°)

3.4132

0.0476

0.0338

0.0077

 for Ne (θ = 90°)

2.9211

0.0346

0.0320

0.0042

LiNbO3

    

 for no (θ = 0°)

4.91296

0.116275

0.048398

0.0273

 for Ne (θ = 90°)

4.54528

0.091649

0.046079

0.0303

According to Eqs. (2) and (3), the phase matching angle θm of a medium can be obtained as follows:
$$\theta_{\text{m}} = \arcsin \left( {\sqrt {\frac{{\left[ {n_{\text{o}} \left( {\lambda_{1} } \right)} \right]^{ - 2} - \left[ {n_{\text{o}} \left( {\lambda_{2} } \right)} \right]^{ - 2} }}{{\left[ {N_{\text{e}} \left( {\lambda_{1} } \right)} \right]^{ - 2} - \left[ {n_{\text{o}} \left( {\lambda_{2} } \right)} \right]^{ - 2} }}} } \right)$$
(5)
By using the parameters summarized in Table 1, phase matching angles of BBO, LiIO3, and LiNbO3 are calculated based on Eq. (5). Figure 3 shows the results. To investigate the wavelength-dependence of the phase matching angles, calculations are carried out in a wavelength range of 1500–1620 nm, which corresponds to the spectral width of the erbium-doped fiber laser employed in the following experiments. As can be seen in the figure, the dispersion of the phase matching angle of BBO is found to be smaller than those of LiIO3 and LiNbO3 in the wavelength range; this wavelength-independent characteristic of the phase matching angle of BBO is ideal for the proposed optical angle measurement method based on the SHG.
Fig. 3

Wavelength dependence of phase matching angles for several uniaxial crystals

By using the calculated phase matching angles shown in Fig. 3, sinc2k(θ)L/2] term in Eq. (1) for each crystal is also calculated. Figure 4a shows the results, which are referred to as the response curves showing the relationship between the power of the second harmonic wave P2 and the angle θ [Eq. (1)]. In the figure, sinc2k(θ)L/2] term is calculated for the cases with wavelengths of 1500, 1560, and 1620 nm, respectively. As can be seen in the figure, peaks of the curves for several wavelengths are almost overlapped with each other for the case with BBO crystal, while the peaks of the curves for the cases with LiIO3 and LiNbO3 are not. In the proposed angle measurement method, the photo detector detects the power of second harmonic waves regardless of the wavelength. The integrated values of sinc2k(θ)L/2] in the spectral range are therefore calculated to simulate the photodetector output. Figure 4b shows the result. The peak of the response curve for BBO crystal is several times higher than those for LiIO3 and LiNbO3. In addition, the variation of the photodetector output with BBO crystal is similar to the sinc2k(θ)L/2] curve shown in Fig. 4a. Meanwhile, those for LiIO3 and LiNbO3 are significantly different from the sinc2k(θ)L/2] curves. Here, the sensitivity of the angle measurement system is defined as the slope of second harmonic wave power to angular displacement. The relative sensitivities of the second harmonic wave to angular displacement in the range between 45 and 55% of the peaks at the left shoulders of the curves in Fig. 4b are shown in Table 2. BBO crystal is therefore suitable for the optical setup with the erbium-doped fiber laser employed in the following experiments.
Fig. 4

Calculated response curves of second harmonic wave power with respect to angular displacement for a collimated femtosecond laser beam. a sinc2k(θ)L/2] for wavelengths of 1500 nm (blue lines), 1560 nm (green lines), and 1620 nm (red lines). b The integrated value of sinc2k(θ)L/2] with respect to a wavelength between 1500 and 1620 nm

Table 2

Relative sensitivity of simulation results

Crystal

Relative sensitivity arbitrary units

BBO

1.0

LiIO3

0.16

LiNbO3

0.091

Ideally, a collimated laser beam is expected in angle measurement. This can be realized in an SHG-based angle sensor when a femtosecond laser source with a high output power is available. Since it is difficult to make a collimated laser beam with a small beam diameter, which is typically an order of 1 mm, when the output power of a femtosecond laser is not enough, the intensity of a collimated femtosecond laser beam cannot reach the threshold intensity for SHG. In this case, it is effective to focus the laser beam in a nonlinear crystal to generate the second harmonic wave although the applications of such an angle sensor with a focused laser beam are limited. As the second step of theoretical calculations, the response curve of the second harmonic wave power with respect to angle under the condition of such a focused beam is investigated in the following analysis.

Figure 5 shows a schematic of the focused laser beam inside a nonlinear crystal. The laser spot diameter 2w0 at the beam waist is expressed as follows:
$$2w_{0} = \frac{{4f\lambda_{1} }}{\pi \phi },$$
(6)
where ϕ is the diameter of a collimated laser beam made incident to an objective lens, f is the focal length of the objective lens, and λ1 is a light wavelength of the laser beam. The Rayleigh length b/2, which is known as the distance along the propagation direction of the beam from the beam waist to the place where the area of the cross section is doubled, can be expressed as follows:
$$b = \frac{{8f^{2} \lambda_{1} }}{{\pi \phi^{2} }}$$
(7)
Fig. 5

Schematic of second harmonic generation and focusing parameters

For the case of focusing a fundamental wave into a nonlinear crystal, the length of nonlinear crystal as L, the angle between the optic axis of the nonlinear crystal, and the laser beam axis as θ, Eq. (1) can be modified as follows [33, 34]:
$$P_{2} = KP_{1}^{2} Lk_{1} h(\sigma ,\,\xi ),$$
(8)
where
$$\begin{aligned} & K = \frac{{8\pi d_{\text{eff}}^{2} }}{{n_{1}^{2} n_{2} \varepsilon_{0} c\lambda_{1}^{2} }},\;k_{1} = \frac{2\pi }{{\lambda_{1} }},\;\sigma = \frac{1}{2}b\Delta k\left( \theta \right),\;\xi = \frac{L}{b}, \\ & h(\sigma ,\,\xi ) = \frac{{2\pi^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} }}{\xi }\int_{ - \infty }^{\infty } {\left| {H(\sigma ,\,\xi )} \right|^{2} e^{{ - 4s^{2} }} {\text{d}}s} = \frac{{\pi^{2} }}{\xi }\left| {H(\sigma ,\,\xi )} \right|^{2} , \\ & H(\sigma ,\,\xi ) = \frac{1}{\pi }\int_{0}^{\xi } {\frac{\cos \sigma \tau + \tau \sin \sigma \tau }{{1 + \tau^{2} }}{\text{d}}\tau } \;{\text{and}}\;\tau = \frac{2(z - f)}{b} \\ \end{aligned}$$
(9)

The above equations are valid under the condition that the influences of the walk-off effect and the light absorption in the crystal are negligibly small [33, 34]. As can be seen in Eqs. (8) and (9), the power of second-harmonic wave depends not only on the phase mismatching Δk(θ) but also on double of the Rayleigh length b, which is a function of three variables: λ, f, and ϕ.

Based on the above equations, calculations are carried out to obtain the response curves for several f. The parameters used in the calculations are summarized in Table 3. Figure 6 shows the results. It can be seen that the response curves showed a similar trend to the result of BBO shown in Fig. 4b where a collimated beam with a large power is employed, indicating the possibility of angle measurement based on SHG with a focused laser beam. On the other hand, the full width at half maximum (FWHM) of the response curve in Fig. 6 is wider than that of BBO in Fig. 4b. This means the sensitivity of angle measurement based on SHG with a focused laser is lower than that with a collimated beam. In Fig. 6, the second harmonic wave power is found to be increased as the decrease of f. This was mainly due to the smaller laser spot diameter with smaller f, since the irradiance of second harmonic wave is proportional to the square of irradiance of the fundamental wave. This can be utilized to improve the sensitivity of angle measurement based on SHG with a focused laser beam.
Table 3

Parameters used in the simulation of SHG

K (W−1)

P1 (W)

λ1 (nm)

L (mm)

ϕ (mm)

f (mm)

3.66 × 10−9

1000

1560

2.0

3.6

40, 75, or 150

Fig. 6

Calculated response curves of second harmonic wave power with respect to angular displacement for several focal length of focusing lenses under the condition of focused laser beam

4 Experiments and Results

To demonstrate the feasibility of the proposed angle measurement method based on SHG, experiments were carried out by using the developed prototype optical setup, a schematic of which is shown in Fig. 7. A commercial mode-locked femtosecond laser (C-fiber, Menlo Systems) having a central wavelength of 1560 nm with 15 mW output power was employed as the light source for the setup. The pulse repetition rate was approximately 100 MHz, while the pulse duration was approximately 150 fs. The resultant peak power of the laser pulse was approximately 1000 W. The femtosecond laser emitted from an edge of a single-mode optical fiber was collimated to be a collimated beam with a beam diameter of ϕ by a collimating lens. ϕ is 3.6 mm. Since the power of the femtosecond laser source was low and it was difficult to make a collimated beam with a small diameter, the intensity of the collimated beam is too small to make SHG. The technique of using a focused laser beam for SHG shown in Fig. 5 was employed in the experiment. For this purpose, the collimated laser beam whose polarization direction was aligned to be parallel to the Y-axis in Fig. 7 was focused into a nonlinear crystal mounted on a rotary stage by an achromatic focusing lens having a focal length of 40 mm. A type I BBO crystal, in which the angle between its optic axis and the crystal surface was designed to be its matching angle, was employed as the nonlinear crystal. By using an objective lens, both the generated second harmonic wave and the unconverted fundamental wave were coupled into a multimode fiber having a core diameter of 50 µm. These coupled light waves were observed by an optical spectrum analyzer (AQ6370C-20, Yokogawa Electric) having a wavelength resolution of 0.02 nm.
Fig. 7

Schematic of the experimental setup for observing spectra of fundamental and second harmonic waves

Firstly, the spectrum of second harmonic wave was verified. The BBO crystal was rotated by the rotary stage in a step of 48 arc-seconds, which corresponds to 500 steps of the rotary stage, to investigate the angle-dependence of the spectrum of the second harmonic wave. Figure 8a and b show the observed spectra of the fundamental wave and the second harmonic wave, respectively. In each figure, the integrated laser power over the observed spectrum was also plotted. As can been seen in the figure, second harmonic waves were successfully generated. In Fig. 8b, a strong dependence of the power of the generated second harmonic wave on the angular displacement of the BBO crystal was observed, indicating the possibility of angle measurement based on SHG with a focused beam for a femtosecond laser source with a small power. The maximum conversion efficiency of the fundamental wave to the second harmonic wave was evaluated to be approximately 4%. The angle in which the intensity of the second harmonic wave reaches the maximum was considered to be that the incidence angle of the fundamental wave into the BBO crystal was matching angle. It should be noted that the observed second harmonic spectra were Gaussian-like ones, while the spectrum of the fundamental wave was flat; the root cause of this difference is mainly due to the chromatic aberration of the focused fundamental light wave, the details of which are explained in the following of this paper.
Fig. 8

a Observed fundamental wave spectra. b Observed second harmonic spectra. The integrated power of the spectra intensity is also shown in both figures

For the proposed angle measurement method, the power of the second harmonic wave P2 is preferred to be as high as possible. Regarding Eq. (8), a higher power P1 of the fundamental wave contributes to increasing P2. It is therefore required for the optical setup to optimize the beam focusing of the fundamental light in the BBO crystal. According to Eqs. (6) and (7), the focused spot diameter of the fundamental wave, which is one of the important parameters to describe the condition of beam focusing, can be changed by the focal length f of the focusing lens. The influence of focused laser beam diameter was therefore evaluated in experiments by using several focusing lenses having different focal lengths.

Figure 9 shows the optical setup for the proposed angle measurement method based on SHG. In the setup, the BBO crystal mounted on a precision rotary table was placed at the focal plane of the focusing lens. To reduce the influence of chromatic aberration, achromatic doublet lenses were employed as the focusing lens. A polarizer was placed just behind the BBO crystal in such a way that the transmission axis of the polarizer was set to be parallel to X-axis so that the second harmonic light wave could transmit the polarizer, while the unconverted fundamental light wave could be absorbed. The transmitted second harmonic light wave was detected by a photodiode with a cutoff frequency of 30 MHz. The photocurrent signal from the photodiode was converted into the corresponding voltage signal by a trans-impedance amplifier, and was captured by an oscilloscope. The angular displacement was given to the BBO crystal by the rotary table. A flat mirror was also mounted on the rotary table so that the angular displacement given to the BBO crystal could be verified by a commercial autocollimator employed as a reference.
Fig. 9

a Schematic of the experimental setup. b Photograph of the experimental setup

Experiments were carried out to evaluate the sensitivity of the developed optical setup. Figure 10 shows the variations of signal from the photodiode due to the angular displacement given to the BBO crystal. In the figures, the photodiode signals obtained by the setups with different focal lenses having different focal lengths were plotted. As predicted in the theoretical calculation results, the power of the second harmonic wave was found to be higher with the decrease of focal length f. In addition, the observed powers of the second harmonic waves showed similar angular dependence compared with the simulation results. Meanwhile, the measurable angle range, in which the variation of the output signal from the PD can be observed, were found to be wider by several times compared with the range predicted in the simulation results. The sensitivity was evaluated as the steepest region of these curves. The results are summarized in Table 4. Figure 11 shows a typical noise component in the output signal from the PD in the case with the focusing lens having a focal length f of 75 mm. The results of the noise level (2σ) are also summarized in Table 4 From the obtained sensitivities and noise components, in each case, a resolution of the developed angle measurement method was evaluated as the ratio of the noise component (2σ) to the sensitivity. The results are summarized in Table 4. From the above results, a resolution of approximately 0.4 arc-second was estimated to be achieved by the developed setup.
Fig. 10

Experimental results of the second harmonic generation with angular displacements for several focal lengths of focusing lenses

Table 4

Specification of the presented angle measurement system

Focal length of the focusing lens (mm)

Sensitivity (mV/arc-second)

Noise level (mV)

Resolution (arc-second)

40

0.97

0.44

0.45

75

1.18

0.43

0.36

150

0.79

0.37

0.47

Fig. 11

PD output stability for detection of SHG with the focusing lens of f = 75 mm. (Cutoff frequency of the photodiode: 30 MHz)

Experiments were further extended to verify the resolution of the developed setup. The BBO crystal was rotated by the rotary table in a step of 0.4 arc-second, while the output signal from the PD was monitored. Figure 12 shows the variation of PD output observed during the experiment. In the figure, the output signal from the laser autocollimator is also plotted. As can be seen in the figure, it was verified that the developed angle measurement system could distinguish the given angular displacement. Meanwhile, as shown in Figs. 6 and 10, the sensitivity of the developed angle measurement system observed in the above experiments was lower than that predicted in theoretical calculation results. One of the reasons for this difference is considered to be due to the chromatic aberration of the focusing lens.
Fig. 12

Variation of the angle measurement system output due to the applied angular displacement with the focusing lens of f = 75 mm. The reference sensor output is also shown

To investigate the influence of the chromatic aberration, ray tracing is carried out for the developed optical setup. The optical model used for the ray tracing is shown in Fig. 13. In the model, an achromatic doublet lens composed of a N-LAK22 substrate and a N-SF6 substrate is employed as the focusing lens, where the data of ref [35] are used for Sellmeier equations of these glasses. The origin of the ray tracing is set at the center of the surface S1. The lens parameters are summarized in Table 5. A ray whose incidence axis is x0 from Z-axis is refracted at surfaces 1, 2, and 3 in the figure, and is then is refracted at surface 4 of the nonlinear crystal. The result of the ray tracing for the cases with a lens focal length of 40 mm is shown in Fig. 14a for the case with Δθ = 0°. In the figure, the results with wavelengths of 1500, 1560, and 1620 nm are plotted. Ray tracing is also carried out for the case with Δθ = 1°. The result is shown in Fig. 14b. To clarify the influences of refraction at surface 4, ray tracing is also carried out for the case without BBO crystal. The result is shown in Fig. 14c. To evaluate the influences of chromatic aberration of the focusing lens, a parameter ΔZ, which is the Z-directional distance between the focal positions of the rays with wavelengths of 1500 and 1620 nm, is introduced.
Fig. 13

The system for calculation of ray tracing

Table 5

Specification of the lens and the position of the BBO

Focal length of the focusing lens (mm)

R1 (mm)

R2 (mm)

R3 (mm)

t1 (mm)

t2 (mm)

d (mm)

40

22.81

42.11

84.66

4.5

2.5

36.4

75

21.91

42.30

84.66

3.0

2.5

72.6

150

250.49

565.11

1071.9

6.0

4.0

145.4

Fig. 14

Results of ray tracing for the focal length of 40 mm of focusing lens; (blue lines) 1500 nm, (green lines) 1560 nm, and (red lines) 1620 nm. a The case for Δθ = 0°, b that for Δθ = 1°, and c that without the BBO crystal

Table 6 summarizes ΔZ for the cases of f = 40, 75, and 150 mm. The difference between the sensitivities observed in the experiments and those predicted in theoretical calculation results can be explained by the influences of the chromatic aberration summarized in the table: in the experiments, second harmonic wave was effectively generated around the beam waist position of the light with a wavelength of 1560 nm (Z1560 nm), since the spectrum of second harmonic wave had its central peak around 780 nm. This was because the irradiance of the laser around Z1560 nm was larger than those in other areas due to the chromatic aberration, and the resultant SHG occurred effectively around Z1560 nm; in other words, the chromatic aberration made the effective crystal length Leff shorter.
Table 6

Specification of the lens and the position of the BBO

Focal length of the focusing lens (mm)

ΔZ without BBO crystal (mm)

Angle (°)

ΔZ with BBO crystal (mm)

40

0.09

0

0.14

  

1

0.14

75

0.16

0

0.26

  

1

0.26

150

0.32

0

0.52

  

1

0.52

The full width at half maximum of PD output voltage dependence on angular displacement in Fig. 10 is summarized in Table 7. FWHM for shorter f is found to be wider than that for longer f. The ratio of ΔZ to b is also summarized in the table. The ratio of ΔZ to b for shorter f is also found to be larger than that for longer f. Therefore, the effective crystal length Leff is shorter for shorter f due to the stronger influence of the chromatic aberration. Thus, the resolution using f = 75 mm is higher than that using f = 40 mm. For shorter Leff, FWHM of sinc2k(θ)L/2] becomes wider. FWHMs of sinc2k(θ)L/2] calculated for various L are plotted in Fig. 15. The experimental results are found to take closer value for L = 0.25, 0.5, and 0.8 mm for f = 40, 75, and 150 mm, respectively. These Leff are several times longer than ΔZ; this result indicates that SHG is localized around Z1560 nm within Leff. Therefore, the SHG spectra in Fig. 8b are not similar to the fundamental wave spectra. Surface 4 of the BBO crystal makes the chromatic aberration ΔZs 1.6 times longer than those without the BBO crystal as shown in Table 6.
Table 7

FWHM of second harmonic power dependence on angular displacement and ΔZ/b

Focal length of focusing lens (mm)

FWHM (arc-second)

ΔZ/b

40

6500

0.29

75

3900

0.15

150

2300

0.075

Fig. 15

FWHM of SHG power dependence on angular displacement for various crystal length

5 Conclusions

A new optical angle measurement method has been proposed by taking use of the unique characteristic of high peak power and high-intensity electric field of the femtosecond laser pulses, which can generate second harmonic waves based on the nonlinear optical phenomenon. Results of the theoretical analysis have clarified that BBO crystal is suitable for the proposed angle measurement method when a mode-locked femtosecond laser source having the spectrum ranging from 1500 to 1620 nm is employed. It has been demonstrated by theoretical analysis and experiment that a focused femtosecond laser beam is effective to realize SHG-based angle measurement for a femtosecond laser source with a small power where the intensity of a collimated beam from such a femtosecond laser source is too small to make SHG. Experimental results with the developed measurement system have demonstrated feasibility of the proposed angular measurement. Meanwhile, the sensitivity observed in the experiments has been found to be lower than that predicted in the theoretical calculation. To clarify the reason for the discrepancy between the results of theoretical calculation and those of the experiments, investigations have been carried out based on ray tracing. It has been clarified that the chromatic aberration has been one of the main root causes of these problems, taking the results of ray tracing and observed second harmonic spectra into consideration. It has also been clarified that the shorter focal length of the focusing lens has made the influence of chromatic aberration stronger, and has made the effective crystal length Leff shorter, resulting in the degradation of the sensitivity.

It should be noted that, as the first step of research, this paper has been focused on the proposal of the angle measurement principle and on the verification of the feasibility of the proposed principle under the condition of a focused beam from a low power femtosecond laser source. Detailed investigation on the basic characteristics of the proposed method, design optimization of the optical setup for the achievement of higher resolution, as well as the measurement uncertainty analysis, will be carried out in future work. It should also be noted that the applications of angle measurement using a focused laser beam are limited and a high-power femtosecond laser source with a high-intensity collimated laser beam will also be employed in future work for expanding the applications of angle measurement based on SHG.

Notes

Acknowledgements

Grants-in-Aid for Scientific Research (KAKENHI), Japan Society for the Promotion of Science (JSPS).

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Copyright information

© International Society for Nanomanufacturing and Tianjin University and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Finemechanics, Graduate School of EngineeringTohoku UniversitySendaiJapan

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