High Angular Tolerance Fiber-Optic Interferometric Probe Design

High Angular Tolerance Fiber-Optic Interferometric Probe: Optical
System Design and Analysis
Ou Chenlonga,b,c, Liu Fengweia,b,c, Wu Yongqiana,b,c*
aState Key Laboratory of Optical Field Manipulation Science and Technology, Chinese Academy
ofSciences,
Chengdu, Sichuan 610209, China
bInstitute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan, 610209,
China
cUniversity of Chinese Academy of Sciences, Beijing, 100049, China
ABSTRACT
Driven by the burgeoning demands of ultra-precision manufacturing for compact, high-precision displacement sensing, a
novel high-angular-tolerance fibre-optic differential interference probe is presented. Addressing limitations of
conventional single-mode fibre probes—such as small mode-field diameter and poor beam confinement—this Fizeau
interferometer-based design operates from 1510 to 1610 nm. It features a single-input/output optical path, enhancing
robustness and interference resistance, while simultaneously delivering superior beam focusing and a substantial
acceptance angle (~5°). Optical simulations confirm high coupling efficiency and stable interference contrast within a 200
μm working range around the focal point. Critically, the probe meets key specifications: maximum clear aperture <10 mm,
focused spot diameter ≤20 μm (at NA=0.12), and near-diffraction-limited imaging quality (MTF), providing a pivotal
foundation for the commercialization of embedded fibre-optic interferometric displacement sensors.
Keywords: High precision; Miniaturization; Angular tolerance; Optical system.
1. INTRODUCTION
In the field of modern industrial manufacturing, the rapid development of high-end manufacturing industries such as
semiconductor chips, aerospace precision components, and microelectromechanical systems (MEMS) has placed
unprecedentedly stringent demands on the precision, efficiency, and adaptability of measurement technology 1. Take the
semiconductor industry as an example: in the production of chips with a process node of 7 nanometers or below, line width
errors must be controlled within 0.15 nanometers. This means that traditional measurement methods are no longer
sufficient to meet the inspection requirements of ultra-precision machining 2, 3. Against this backdrop, non-contact optical
measurement technology, with its significant advantages of non-destructive testing, high precision, and high speed, has
gradually become a core technological pillar in the field of industrial inspection 4.
Traditional contact-based measurement methods, such as coordinate measuring machines, can achieve high precision but,
due to mechanical contact, may cause scratches or deformation on the measured surface, making them unsuitable for
measuring flexible materials or micro-sized parts 5. Additionally, their measurement efficiency is relatively low, making
them difficult to adapt to the real-time inspection requirements of modern production lines. In contrast, non-contact optical
measurement technology achieves measurement through the interaction of light signals with the object being measured,
fundamentally avoiding the negative effects of contact. It can also achieve measurement resolutions at the micron or even
nanometer level, making it an indispensable key technology for ultra-precision manufacturing.
Among various non-contact optical measurement technologies, interferometry stands out for its exceptional measurement
accuracy, which is based on the principle of light wave interference 6. Since the invention of the Michelson interferometer
in 1887, interferometry has undergone a transformative evolution from laboratory research to industrial applications 7.
While traditional block prism interferometric systems can achieve high-precision measurements, they are bulky, have
complex optical path adjustments, and are susceptible to environmental interference, making them unsuitable for the
compact spaces and complex conditions of industrial environments 8.

The emergence of fibre-optic interferometric measurement systems has provided a new approach to addressing these issues.
Compared to traditional block-based optical systems, fibre optic interferometric systems offer advantages such as compact
structure, strong resistance to electromagnetic interference, and the ability to perform remote measurements, enabling them
to carry out inspection tasks in confined spaces 9,10. However, existing fibre optic interferometric probes still face numerous
technical challenges: while the traditional design using single-mode fibre as the signal arm meets miniaturisation
requirements, the small mode field diameter results in rapid divergence of the output beam, limiting the measurement
range; Additionally, their tolerance angle is small (typically less than 1°), imposing stringent requirements on the
installation orientation of the measured object, and they are prone to failure under complex assembly errors in industrial
environments 11,12. Furthermore, single-wavelength interferometric systems are sensitive to environmental factors such as
temperature fluctuations and airflow disturbances, leading to phase drift and reduced measurement accuracy 13.
To overcome these limitations, the combination of multi-wavelength interferometry and high-precision optical design has
become a research hotspot. Multi-wavelength interferometry introduces multiple wavelength light sources to expand the
measurement range through wavelength combinations while suppressing phase noise caused by environmental interference,
significantly enhancing the system's stability and interference resistance 14, 15. Phase-shifting interferometry, which
achieves high-precision demodulation of interference fringes by precisely controlling phase changes, enabling nanometre-
level displacement measurements, remains a major challenge in current research 16, 17.
Based on the above background, this paper proposes an optical system design scheme for a high-tolerance angular
interferometer probe based on the Fizeau interferometer structure. This scheme employs a multi-wavelength light source
in the 1510–1610 nm band. By optimising the optical system parameters, it achieves a compact structure with single-beam
input and single-beam output, significantly increasing the measurement tolerance angle while enhancing interference
resistance. The optical system adopts a three-element objective lens design, ensuring imaging quality approaching the
diffraction limit while meeting the miniaturisation requirements of a maximum aperture diameter <10 mm and a spot
diameter ≤20 μm. Through optical transmission matrix simulation analysis, the system maintains stable interference signals
within a 200 μm measurement range at the focal point, with a mirror tolerance angle of approximately 5°, effectively
reducing alignment difficulties in industrial environments.
This research not only provides a high-precision, highly adaptable measurement tool for the ultra-precision manufacturing
field but also promotes the commercial application of fibre optic interferometry technology in embedded integration fields,
holding significant importance for enhancing China's self-reliance and control over high-end manufacturing equipment.
2. TECHNICAL FRAMEWORK AND METHODS
2.1 Design Specifications
The core optical specifications of this probe include the following five key parameters:
(1) This design selects four wavelengths of light signals in the 1510-1610 nm near-infrared band as the measurement light
source. Light in this band is not perceptible to the human eye, making it safe to use; Most industrial inspection materials
(such as semiconductors, optical glass, ceramics, etc.) exhibit good transmittance and reflectance in this band, making it
widely applicable; the four wavelengths are spaced reasonably, ensuring high-precision multi-wavelength interference
phase demodulation while minimizing wavelength interference and enhancing measurement stability.
(2) The numerical aperture (NA) of the fiber optic port is set to 0.12, with a beam waist diameter of 10 μm, ensuring
sufficient light flux while controlling aberrations;
(3) The optical working distance is ≥15 mm, meeting the safety distance requirements for non-contact measurement;
(4) The spot diameter is ≤20 μm, ensuring measurement resolution;
(5) The angular tolerance allows the measured surface to be tilted by ≥±5° while maintaining effective signal coupling,
demonstrating the system's robustness.
2.2 Design Process
The entire optical design process is shown in Figure 1. Based on the initial parameter analysis, key parameters such as
waist size and working distance are determined. Q parameters are then derived, and candidate structures are selected based
on compatibility and feasibility. If the requirements are not met, the initial structure of the objective lens is optimized. If
the requirements are met, aberration adjustment is performed, and software is used to correct spherical aberration,

chromatic aberration, etc. Following image quality evaluation using metrics such as MTF and spot patterns to validate
performance, the final system structure is determined. Through closed-loop design, precise implementation from
theoretical to actual performance is ensured, laying a solid optical foundation for high-precision measurement by the probe.
Figure 1. Optical design flowchart
2.3 Optical Structure
The high-angle-difference fibre optic interferometer system designed in this paper is based on the Fizeau interference
principle. The system adopts an amplitude-split interferometric structure, where the incident light wave is split into two
paths by a beam splitter: the reflected light path is focused onto a CCD detector by an imaging lens, while the transmitted
light path is collimated by a collimating lens to form parallel light, which then passes through a reference reference surface,
a converging lens, and an objective lens before reaching the measured surface. The two light beams are reflected from the
reference surface and the measured surface, respectively, and then recombine. The interference fringes are formed by the
path difference caused by the small air gap. Each interference fringe corresponds to a constant air gap thickness, and the
variation between adjacent fringes reflects thickness differences at the half-wavelength level. The interference patterns
captured by the CCD enable high-precision detection 18. This design ensures stable acquisition of interference signals and
high-resolution measurement capabilities through optimised collimation, convergence, and imaging optical path structures.
Although traditional Fizeau configurations are used for two-dimensional surface analysis in optical component inspection ,
we can also adopt a one-dimensional measurement scheme using a point probe system for linear surface scanning. This
method retains the basic principles of Fizeau interferometry while enabling displacement detection along a single axis.
Figure 2. Structure of the Fizeau interferometer
2.4 Gaussian ray tracing analysis based on the Jones matrix
This system employs a Gaussian beam transmission model for simulating the modal field output, with its optical field
characteristics determined by the modal field distribution at the fibre endface. As shown in Figure 3, the system probe
structure comprises a fibre, an air gap, and a three-element objective lens assembly. The Gaussian beam characteristics of
the fibre optic output are characterised by two key parameters: the waist radius  (determined by the fibre optic mode
field diameter MFD) and the air gap  between the fibre optic and the lens assembly. Using the ABCD transmission
matrix theory, the transmission transformation process of the Gaussian beam in the probe can be fully described using the
Q parameter 19, 20. After optical probe transformation, the waist position of the output beam is located at a distance L from
the probe, where the waist radius becomes . The working distance Z of the system is defined as the measured distance
between the mirror and the probe.

Figure 3. Gaussian beam transmission principle diagram
Based on the principle of Gaussian light matrix tracking, light transmission in a near-axis optical system can be described
by cascaded operations of the ABCD matrix, where:
Transmission matrix:
 , describes the propagation of light in a medium with refractive index n and thickness d;
Refraction matrix: 
, describes the refraction of light at an interface with radius of curvature r and refractive
indices and on either side.
3. EXPERIMENTS AND RESULTS
3.1 Optical design results
Based on the core technical indicators established in the design, this solution systematically designs the curvature radius,
thickness, and glass material parameters of the lens. The relevant parameter details are shown in Table 1 below. This
structure uses a three-piece objective lens composed of a double-laminated achromatic lens and a meniscus lens. This
design ensures optical performance while effectively correcting aberrations through reasonable lens combinations and
material selection.
Table 1. Lens Parameters
Surface (cali
ber)
Radius of cur
vature
Thickness
Materials (refractive in
dex)
surface
infinite
3.5
1（3）
8.120
10.530
H-ZF88（1.945950）
2（3）
5.440
3.100
H-ZPK7（1.569070）
3（3）
-5.440
0.100
4（3.5）
5.950
1.540
H-ZPK5（1.592800）
5（3.5）
16.870
16.890
noodles
infinite
From the lens parameters designed in Table 1, the designed focal point is 16.887 mm, which meets the requirement of
being greater than 15 mm.
As shown in Figure 4, the optical design results indicate that the system exhibits excellent focusing performance at both
1510 nm and 1610 nm wavelengths: the spot diameter at the fibre output is less than 9 μm (design requirement < 20 μm),
and the spot radius of the reflected light returning to the fibre end face is less than 10 μm of the fibre diameter, ensuring
high optical coupling efficiency; Additionally, the MTF curve indicates that the system's imaging performance approaches
the diffraction limit.

（a） (b)
(C)
Figure 4. Optical design results: (a) Focus position spot diagram, (b) Reflected light returning to the fiber end face spot
diagram, (c) Diffraction-limited MTF diagram
3.2 Optical System Performance Analysis
Based on the principles of the Jones matrix and the results of optical design in Part II, Figure 5 is obtained. Substituting
the transmission matrix and refraction matrix yields the total matrix M of the cascaded optical system designed.
Figure 7 Gaussian light matrix structure diagram
The total matrix M is as follows:.