The thermo-optic coefficient, a critical parameter dictating the temperature-dependent refractive index variation in birefringent crystals such as
Chinese Optics Letters, Volume. 23, Issue 8, 081201(2025)
Superresolution measurement of thermo-optic coefficient of KTP crystals based on phase amplification
In this work, we achieve a fourfold enhancement in thermo-optic coefficient measurement resolution for KTiOPO4 crystal using a self-stabilized birefringence interferometer integrated with cascaded second-harmonic generation. We observe the tunable interference beating phenomenon by rotating a birefringent crystal versus the temperature of the crystal. Furthermore, the fourth-harmonic interference fringes beat 4 times faster than the fundamental wave interference fringes. This beating effect is used to determine the thermo-optic coefficients of the two principal refractive axes with a single measurement. This work provides a feasible, real-time, and robust method for superresolution measurements based on birefringence interferometry.
1. Introduction
The thermo-optic coefficient, a critical parameter dictating the temperature-dependent refractive index variation in birefringent crystals such as
One well-known method that is used in quantum optics to realize phase amplification is based on the multiphoton number and path entangled state known as the NOON state[10–12], which utilizes all
In this work, we first give a theoretical description of a birefringent Mach–Zehnder interferometer (MZI) based on phase amplification and then achieve superresolution measurement of the thermo-optic coefficient of KTP crystal through a specially designed intrinsically stable birefringent polarization MZI and two polarization-independent SHG modules. After cascaded SHG processes, the oscillation period of the interference curve is reduced to 1/4 of the original, which means that the measurement resolution is improved by 4 times. In addition, the interference-beating phenomenon versus crystal temperature is observed for the fundamental wave (FW), second-harmonic (SH), and fourth-harmonic (FH). The beating intensity can be tuned by rotating the crystal, and the FH interference fringes beat 4 times faster than the FW interference fringes. This beating effect is used to determine the thermo-optic coefficients of the two principal refractive axes with a single measurement. This work overcomes the resolution limitations of traditional interferometry methods and provides a universal framework for characterizing complex thermo-optic behaviors in birefringent crystals.
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2. Methods
The general theoretical models is given first. Figure 1 shows a graphical summary of the main concept of this work. As shown in Fig. 1(a), an
Figure 1.Schematic diagram of superresolution interferometry principle. (a) Birefringence MZI. (b) Nonlinear birefringence MZI including SHG process. (c) Birefringence MZI with KTP crystal rotation angle δ. (d) Nonlinear birefringence MZI with KTP crystal rotation angle δ. The interference curve on the right side of the interferometer represents a simplified simulation schematic based on the KTP crystal parameters from Ref. [13] at the wavelength of 1550 nm.
In wave optics, the light field at the output port of the MZI can be expressed as
Therefore, by continuously tuning the temperature of the KTP crystal, the thermo-optic coefficient of the
Next, we rotate the KTP crystal in the birefringence MZI around the
The two SH superposition light fields can be given by
Comparison of Eq. (4) with Eqs. (5) and (6) demonstrate that the interference curve of the SH also exhibits beating behavior, and the beating curve oscillates twice as fast as that of the FW. Based on the same principle, the complex amplitude of the FH is given by
After the above theoretical derivation, a complete mathematical model of a birefringence interferometer based on the SHG process to achieve phase superresolution measurement is given. Next, we will further experimentally verify the predictions described in the above theoretical model.
3. Experimental Setup
A schematic of the experimental setup is shown in Fig. 2. The light source involved in the experiment is a homemade mode-locked fiber laser with a central wavelength of 1560 nm, a pulse duration of 212 ps, and a repetition frequency of 21.6 MHz. After being amplified by a homemade erbium-doped fiber amplifier, the power of the pulsed light can reach 2 W. The linearly polarized pulsed light is first transformed into a 45°-polarized beam using a half-wave plate (HWP), and then injected into a self-stabilized polarization MZI, which contains two KTP crystals; one KTP crystal is used for the measurements, while the other compensates for the optical path length differences between the two arms of the MZI.
Figure 2.Experimental setup for superresolution measurement. FC, fiber collimator; HWP, half-wave plate; DHWP, dichroic HWP; PBS, polarizing beam splitter; DPBS, dichroic PBS; KTP, potassium titanyl phosphate crystal; PPLN, periodically poled lithium niobate crystal; BBO, β-barium borate crystal; DM, dichroic mirror; BPF, 390-10 nm bandpass filter; OPM, optical power meter.
The self-stable MZI is based on a tilted Sagnac loop, where the clockwise and counterclockwise beams have a traverse distance of 10 mm. Since light beams in the two arms of the MZI are slightly tilted and in counterpropagating configurations, both light beams have nearly the same sensitivity to environmental turbulence, such as temperature fluctuation and vibrations. Both crystals are
The orthogonally polarized FW from the two arms of the self-stable MZI enters the first Sagnac-type polarization-independent SHG module, which consists of a dichroic PBS (DPBS), a dichroic half-wave plate (DHWP), and a periodically poled lithium niobate (PPLN, CTL Photonics Inc.) crystal with a length of 25 mm and a poling period of 19.62 µm. The operation temperature of the PPLN crystal is set to 39.4°C to fulfill the quasi-phase-matching condition. The Sagnac loop with a DHWP inserted is used to realize the SHG for both vertical and horizontal polarizations, which was demonstrated in our previous work[19].
Then, the generated SH is separated from the FW propagation path by a dichroic mirror (DM). The second polarization-independent SHG module is composed of two orthogonally glued β-barium borate (BBO, CASTECH Inc) crystals that satisfy type-I birefringence phase-matching conditions. Each BBO crystal has a thickness of 0.5 mm and a phase-matching angle of
4. Results and Discussion
Figure 3 shows the interference fringes of the FW, SH, and FH measured experimentally at different rotation angles of the KTP1 crystal. The
Figure 3.Interference beating versus temperature for the FW, SH, and FH. The panels on the left (from top to bottom) represent the FW cases at rotation angles of δ = 0, π/6, π/3, π/2. The panels on the middle and right represent the corresponding interference results of the SH and FH, respectively. Different offsets in each of the interference fringes come from different initial phases between the two arms of the interferometer.
When the polarization direction of the FW does not coincide with the principal axis of the birefringent crystal, the interference fringes exhibit beating behavior of the optical properties along the two axes, and we can determine the optical properties along both axes from any single measurement of this type of beating curve. For example, when
Next, we characterize the conversion efficiency of the two polarization-independent SHG modules. In our experiment, the polarization-independent SHG modules exhibit nearly identical conversion efficiency characteristics for both horizontally and vertically polarized input lights. Therefore, we show the power conversion efficiency of the two SHG modules in the case of vertically polarized input. The results are shown in Fig. 4, where the dashed lines are the theoretical fitting curves. The nonlinear efficiency of the first SHG process in the experiment exceeds 30%, while that of the second SHG process is less than
Figure 4.Power conversion efficiency of the (a) first and (b) second polarization-independent SHG modules.
5. Conclusion
In summary, we have achieved superresolution measurement of the thermo-optic coefficient of KTP crystal using birefringence interferometry based on the phase amplification method. Through the carefully designed self-stabilized birefringent MZI and polarization-independent SHG modules, the physical mechanism of superresolution interferometric measurements is revealed in detail. After phase amplification, the FH interference fringes oscillate 4 times faster than those of the FW, which indicates a fourfold enhancement in resolution capability. In addition, the interference beating phenomenon versus the crystal temperature has been observed for the FW, SH, and FH. This beating feature is used to determine the optical properties along both crystal axes with a single measurement. Although we have only achieved a fourfold improvement in resolution here, as described in our previous work[19], further optimization of the pump light source and nonlinear crystal parameters, along with the application of optical parametric amplification techniques, can enable even higher measurement resolution. We should point out that the present system is not limited to the determination of the thermo-optic coefficient of a birefringent crystal and can also be used to determine the wavelength dispersion[21] and the electro-optical coefficient of the birefringent crystal. More importantly, compared to methods for achieving superresolution measurements based on the NOON state, our scheme boasts real-time responsiveness and is compatible with existing mature interferometric measurement systems. This work will thus be of great importance for understanding superresolution measurements based on phase amplification.
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Wuzhen Li, Zhiyuan Zhou, Guangcan Guo, Baosen Shi, "Superresolution measurement of thermo-optic coefficient of KTP crystals based on phase amplification," Chin. Opt. Lett. 23, 081201 (2025)
Category: Instrumentation, Measurement, and Optical Sensing
Received: Feb. 28, 2025
Accepted: Apr. 9, 2025
Published Online: Jun. 24, 2025
The Author Email: Zhiyuan Zhou (zyzhouphy@ustc.edu.cn), Baosen Shi (drshi@ustc.edu.cn)