A New Optical Angle Measurement Method Based on Second Harmonic Generation with a Mode-Locked Femtosecond Laser
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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 generation1 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
It should be noted that P_{2}, which is the sum of the second harmonic waves generated in the nonlinear crystal, is also proportional to the square of P_{1}, 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 (n_{o}(λ_{1}) ≠ n_{e}(θ_{m}, λ_{2})), 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 (n_{o}(λ_{1}) = n_{e}(θ_{m}, λ_{2})), as shown in Fig. 2c, all the second harmonic waves are combined totally constructively, and P_{2} 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.
Parameters for calculation of the empirical equation of refractive indices
Crystal | A (−) | B (−) | C (µm^{2}) | D (µm^{−2}) |
---|---|---|---|---|
BBO | ||||
for n_{o} (θ = 0°) | 2.7539 | 0.01878 | 0.01822 | 0.01354 |
for N_{e} (θ = 90°) | 2.3753 | 0.01224 | 0.01667 | 0.01516 |
LiIO_{3} | ||||
for n_{o} (θ = 0°) | 3.4132 | 0.0476 | 0.0338 | 0.0077 |
for N_{e} (θ = 90°) | 2.9211 | 0.0346 | 0.0320 | 0.0042 |
LiNbO_{3} | ||||
for n_{o} (θ = 0°) | 4.91296 | 0.116275 | 0.048398 | 0.0273 |
for N_{e} (θ = 90°) | 4.54528 | 0.091649 | 0.046079 | 0.0303 |
Relative sensitivity of simulation results
Crystal | Relative sensitivity arbitrary units |
---|---|
BBO | 1.0 |
LiIO_{3} | 0.16 |
LiNbO_{3} | 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.
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.
Parameters used in the simulation of SHG
K (W^{−1}) | P_{1} (W) | λ_{1} (nm) | L (mm) | ϕ (mm) | f (mm) |
---|---|---|---|---|---|
3.66 × 10^{−9} | 1000 | 1560 | 2.0 | 3.6 | 40, 75, or 150 |
4 Experiments and Results
For the proposed angle measurement method, the power of the second harmonic wave P_{2} is preferred to be as high as possible. Regarding Eq. (8), a higher power P_{1} of the fundamental wave contributes to increasing P_{2}. 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.
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 |
Specification of the lens and the position of the BBO
Focal length of the focusing lens (mm) | R_{1} (mm) | R_{2} (mm) | R_{3} (mm) | t_{1} (mm) | t_{2} (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 |
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 |
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 |
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 L_{eff} 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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