# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 1 Introduction  
The angle is a fundamental element for defining the objects in the manufacturing process [1], and it is crucial in industry to evaluate the angular displacement using a fast and accurate method. In these years, optical measurement techniques have gained widespread attention and played an essential role in the measurement field because of their inherited noncontact, high-efficiency, and ultraprecision advantages. The optical encoder is one of the most well-used sensors for the angle measurement of the rotational shaft [2]. The encoder consists of a stationary read head and a scale disk mounted on the shaft. The read head detects the change of optical signal and records the rotational angle displacement as the shaft rotates, and the measurement range can generally cover the full 360\({}^{\circ}\). The precise measurement of a small angle is also significant, such as the tilt angle of the spindle in the machine tool [3]. Many optical measurement techniques can be engaged for small-angle measurements, including autocollimation and interferometry-based angular measurement [4, 5, 6, 7]. The autocollimation utilizes the internal-reflection effect for the angle measurement [8], and the typical interferometer for angular measurement is based on the effect of small angles on interference light intensity [9]. Differing from the encoder, they do not require a disk, and only a reflector needs to be mounted on the measured object. Accordingly, the measurement setup is more flexible, and thus they are also used for the flatness measurement [10], straightness measurement [11, 12], and the position measurement for a multiaxis system [13, 14, 15, 16]. In recent decades, angle measurement techniques have generally used the monochromatic laser due to its high intensity and excellent directionality compared with the lamp or light emitting diode (LED). With the progress of laser techniques, the pulse laser source was conceived and constructed, and the pulse duration developed from nanosecond to femtosecond. The femtosecond laser, also referred to as optical frequency combs, has both the advantages of the monochromatic laser and two significant features. One isthat the ultra-short pulse contains a number of equally-spaced stable discrete modes in the frequency domain, and the other is the high peak intensity. Because of the multiple frequency modes, many research works have been made so far in length and distance measurement [17; 18; 19; 20]. Recently, the femtosecond laser was also used in the optical angle measurement field, and some optical angle sensors were proposed and realized, such as the optical level. The optical level is based on the diffraction dispersion effects of optical frequency combs to produce an \"angle scale comb\" [21]. Specifically, after passing through the Fabry-Perot etalon, the femtosecond laser beam is projected into the refractive-type diffraction grating, generating a series of first-order diffracted beams where each optical mode corresponds to a single diffracted direction. The grating is mounted on the rotary stage, and the detector is stationary. The observed mode is different as the stage rotates, and the detected intensity changes accordingly. Meanwhile, a similar concept, the mode-locked femtosecond laser collimator, has been proposed to expand the measurement range of conventional monochromatic laser collimator [22; 23]. Compared with the optical level, the measurement sensitivity of the femtosecond laser collimator is not affected by the distance between the detector and the grating. Therefore, it can realize a more compact setup to conserve space. To further improve the sensitivity and visibility, the angle measurement in the frequency domain was proposed by employing the collimator objective, single-mode fiber, and spectrometer [24]. In this case, the mode can be observed clearly, and the visibility can almost arrive at 100%.