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Abstract
In the myopia correction surgery by femtosecond laser, such as Laser in Situ Keratomileusis (LASIK) and Small Incision Lenticule Extraction (SMILE), the nonlinear refractive index of the cornea may cause the deviation of cutting depth. In order to improve the cornea cutting’s accuracy and reduce the possibility of undercorrection, the nonlinear refractive index coefficient n2 of the human cornea must be measured with high accuracy. The spectral domain Z-scan technique can measure n2 of the highly scattering biological tissues with much better signal to noise ratio and thus better accuracy than the conventional methods. In this paper, the n2 coefficient of one ex-vivo human corneal sample was measured by the spectral domain Z-scan technique. Experimental results show that as this corneal sample gradually dehydrates, its n2 coefficients are 1.1 ± 0.1×10−19 m2/W, 1.4 ± 0.2×10−19 m2/W and 1.6 ± 0.2 ×10−19 m2/W respectively for the corneal sample with water contents of 89%, 82%, and 78%. The increase of the water content reduces the value of n2, which is reasonable since the nonlinear refractive index coefficient of water is one order of magnitude smaller than that of the cornea.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, owing to the advantages of high precision and low damage, femtosecond laser-induced breakdown has been widely used in femtosecond laser-assisted myopia correction surgery, such as Laser in Situ Keratomileusis (LASIK) and Small Incision Lenticule Extraction (SMILE) [1–4]. The stromal tissue in the cornea is cut and reshaped to correct the diopter by femtosecond laser. Since the thickness of the corneal tissue is ∼500 μm and the range of treatment is 30 ∼200 μm, the cutting accuracy is very important to the cornea surgery. Therefore, persistent efforts are devoted to improve the accuracy of femtosecond laser-assisted myopia correction surgery. M. Miclea et al. [5] reported that the cutting depth devised by the surgical machine program is inconsistent with the actual cutting depth, and the deeper the depth is, the greater the error happens. For femtosecond laser with peak power of 4.3 MW, the maximal error even can reach ∼100 μm when a layer of physiological saline with a thickness of 300 μm is covered on the pig corneal. The error in cutting depth is mainly induced by the self-focusing effect of the high intensity femtosecond laser. Therefore, measuring the refractive index coefficient n2 of the cornea is critical for improving the cornea surgery’s accuracy.
Additionally, the measurement of the cornea’s n2 also benefits the cornea’s 3D characterization and the treatment of keratoconus which may cause blindness. Bradford et al. used a 760 nm femtosecond laser to induce a two-photon absorption effect in the rabbit cornea such that the corneal collagen is cross-linked (CXL) to treat keratoconus [6–9]. Moritz Winkler et al. used the second harmonic wave (SHG) generated by the cornea to obtain the 3D imaging of the cornea’s collagen fiber [10]. The performances in both scenarios are strongly affected by the refractive index coefficient n2 of the cornea. Therefore, the characterization of the cornea’s n2 attracts intensive attentions of scientists from the fields of both optics and cornea surgery.
In the past decades, in order to obtain the nonlinear refractive index n2 of medium, many methods, such as nonlinear interferometry [11], ellipsometry method [12], optical Kerr effect method [13], the third-harmonic method [14], Z-scan method [15], etc. have been developed. Among them, the Z-scan technique is the simplest and most widely used method. Its core idea is to transform the wavefront distortion induced by the nonlinear optical process into the easily detected far-field diffraction intensity distribution. However, the random wavefront distortion due to the surface roughness of the sample can severely interfere the measurement. Therefore, the conventional Z-scan technique has rigorous requirements on the surface roughness of the sample and it is challenging to measure highly scattering biological samples such as the ex-vivo cornea by the conventional Z-scan technique. M. Miclea et al. [5] measured n2 of pig cornea using the conventional Z-scan technique. The authors found that the data quality is strongly affected by the sample scattering and ∼100 repeated measurements are required to obtain accurate results.
In 2015, K. C. Jorge et al. [16] proposed the scattered light imaging (SLIM) method to measure the n2 coefficient of the scattering media. SLIM measured the change of the divergence angle caused by the self-focusing effect via recording the side image of the laser beam in the highly scattering medium. However, the finite imaging range leads to the poor sensitivity of SLIM, which cannot measure n2 less than 10−19 m2/W. In addition, SLIM is not suitable for measuring n2 of the inhomogeneous scattering media because the inhomogeneity of the sample will deflect the light beam and disrupt the measurement of the divergence angle. Samineni et. al. [17–19] proposed a new-type Z-scan technique based on the hole refilling principle to measure n2 of the highly scattering sample. This technique can detect the extremely weak change in the spectrum caused by the self-phase modulation (SPM). During the measurements, the peak region of the initial spectra was firstly suppressed with an acoustic optical modulator (AOM) and then the resulting hole was “filled” after passing through the nonlinear media due to the self-phase modulation. The spectral domain Z-scan method [20] proposed in our previous work is based on the similar principle to the hole refilling method, but employs a much more simple apparatus. The spectral domain Z-scan method is immune to the scattering and has been demonstrated to be suitable to measure n2 of the frost glass. In this paper, the n2 coefficient of the ex-vivo cornea sample was measured by the spectral domain Z-scan technique, which paves the way for accurately characterizing the nonlinear optical properties of the highly scattering biological samples.
2. Experimental setup and method
This study was approved by the Ethics Committee of Tianjin Eye Hospital (201922) and adhered to the tenets of the Declaration of Helsinki. The corneal sample was cut from the donor’s cornea during the myopia correction surgery. The donor of the corneal sample is a 22-years-old male. The operation was performed using the VisuMax femtosecond laser system made by Carl Zeiss Meditec AG. The corneal sample was extracted by femtosecond laser cutting. Due to the small heat affected zone of femtosecond laser processing, only the surface of the extracted sample losses the characteristics of the in vivo tissue and micro-/nano-holes appear on the surface of the corneal sample. The interior tissue of the corneal sample does not denature after extraction. The details of the extraction procedures of the sample can be found in Refs. [3,21]. After extraction, the sample was soaked in the 20% dextran solution prepared with the phosphate buffered saline (Solarbio Beijing Inc.) and dextran T-500 (Solarbio Beijing Inc.), and preserved in a refrigerator at 4°C. Within 48 hours the corneal sample preserved by the above method can maintain the characteristics of in vivo tissues in the term of clarity, structural integrity, and immunogenicity [22,23]. In order to maintain the characteristics of the in vivo cornea to the maximum extent, the n2 measurements were carried out within 24 hours after the corneal sample was cut from the donor. In addition, the time needed by each n2 measurement was controlled to be no more than 5 minutes. The corneal sample has a diameter of 6.6 mm, a center thickness of 98 μm when the water content is 78%. During the measurement, the sample were exposed in the constant natural environment (temperature: 22.4 ± 0.3°C; humidity: 20.0 ± 3%).
The measurement setup is shown in Fig. 1, which is the same as that in our previous work [20]. A Ti: sapphire femtosecond laser amplifier (Spitfire, Spectra-physics) was employed to generate 55 fs, 800 nm, 0.45 mJ, 1 kHz laser pulses with a beam waist radius of ∼ 4 mm. The total uncertainty of the pulse duration is determined to be 2.98 fs by the measurement results of the single shot autocorrelator (SSA, Positive light Inc.). The laser power used in experiments is 73 μW and the total uncertainty of the laser average power is 2.49 μW which is measured by the power meter (Nova II, Ophir Photonics Inc.).
Fig. 1. Schematic diagram of the spectral domain Z-scan technique for measuring n2 of the ex-vivo cornea sample.
Using a tweezer, the corneal sample was taken out from the dextran solution and placed on the 0.15 mm thick cover glass, then the corneal sample attached on the cover glass was placed in the beam path of the spectral domain Z-scan setup for n2 measurement. The cornea sample was mounted on an electronic controlled translation stage and placed horizontally near the focal point of Lens 1 (f = 10 cm). The electronic controlled translation stage moves at a step of 0.1 mm along z direction during the measurements. The beam intensity profile at the focal spot was measured by imaging the beam spot using a CCD camera with a magnification of 48 (2S036M-H2, Do3think Inc.). The beam waist radius at the focal point was measured to be 15.0 μm and the total uncertainty of the beam waist radius was determined to be 0.5 μm. Therefore, the peak intensity I0 at the focal point was calculated to be 397 GW/cm2 and its total uncertainty is 36 GW/cm2 which is determined by the uncertainties of the laser power, pulse duration and beam waist radius. After interacting with the sample, the femtosecond laser was focused into the integrating sphere (FOIS-1, Ocean Optics Inc.) by Lens 2. The light collected by the integrating sphere was fed into the spectrometer (HR-2000, Ocean Optics Inc.) via a multimode fiber. For the cornea sample at each z position, the spectrum was collected 3 times and the average spectral curve was obtained.
The spectral domain Z-scan technique measures the material's nonlinear refractive index n2 by detecting the spectral broadening caused by the self-phase modulation. In our previous work [20], it has been demonstrated that the spectral transmittance T in certain wavelength range from λ1 to λ2 can be used to quantify the amount of the spectrum broadening, which is defined as [20]:
Fig. 2. Measured transmitted spectral curves at z/zr = 0 and z/zr =−5 for the human corneal sample with ∼ 78% (a), ∼ 82% (b), and ∼ 89% (c) water contents and the corresponding z-dependent transmittances T(z) are presented in (d)-(f). The transmittance T(z) is calculated using the spectral window from 799 nm and 801 nm.
3. Experimental results and discussions
3.1 Effect of water content on n2 of cornea
The corneal tissue water content (CTWC) can be calculated by:
where mwet represents the weight of the wet cornea, mdry is the weight of the dried cornea which is measured after all the n2 measurements are done and dried for 3 hours at 90℃. According to the experimental results by our group [24], the water content of the cornea decreases linearly with time and the dehydration rate is ∼1%/min at 22.4 ℃. Since the transmission valley appears just at the middle time of each n2 measurement (see Figs. 2(d)–2(f)), the sample mass corresponding to the transmission valley is just the average mass (m1+m2)/2 where m1 and m2 are respectively the mass of the corneal sample measured before and after n2 measurement. It is worth noting that Ref. [25] demonstrates the variation of the water content in the corneal sample can result in the change of the cornea thickness L. In order to measure the n2 coefficient accurately, the water content’s effect on the sample thickness should be considered. The thickness of the cornea sample with 78% water content used in Ref. [25] is 0.59 mm which is six times that of the sample used in our experiments when the water content is 78%. Therefore, according to the relation between the corneal sample thickness and water content derived in Ref. [25], the thickness of the cornea sample used in our experiments can be estimated by the following equation: where l = 0.098 mm and l0 = 0.59 mm, which respectively represent the thicknesses of the sample with 78% water content used in our experiments and the corneal sample with identical water content used in Ref. [25]. The thicknesses of the cornea sample with 82% and 89% water contents used in experiments were calculated to be 131 μm, and 214 μm respectively. During the measurements, the average power of the femtosecond laser incident on the corneal sample is 73 μW, the corresponding pulse energy is 73 nJ, and the peak intensity I0 is 397 GW/cm2. To avoid the loss of the water content, the time needed for measuring the n2 coefficient of the cornea sample is less than 15 minutes. Since the scattering and absorption of the sample can reduce the light energy, the spectra collected at different z positions were all normalized to ensure the total light energy is 1.The normalized spectral curves collected at z/zr = 0 and z/zr = -5 for one corneal sample with different water contents are shown in Figs. 2(a) – 2(c), where zr is the Rayleigh length of the focused femtosecond laser. when the spectral window is chosen to be 799 nm ∼ 801 nm, i.e. log10S = −1.1, the z-dependent transmittance T(z) was calculated based on the measured spectral curves and shown in Figs. 2(d)–2(f). From Figs. 2(a)–2(c), it is found that the water content has a great influence on the spectral curve measured at z = 0. This indicates that the higher the water content of the cornea sample is, the deeper the transmittance valley can be obtained at z = 0, as shown in Figs. 2(d)–2(f). According to Ref. [20], the maximum nonlinear phase shift Δϕmax induced by the nonlinear refractive index of the corneal sample can be calculated by the empirical equation
3.2 Effect of cover glass on the n2 measurement of cornea
The optical nonlinearity of the cover glass will affect the measurement of n2 of the corneal sample. In order to reduce the influence of the cover glass as much as possible, a thin cover glass with a thickness of ∼0.15 mm was used in the experiments. Using the laser pulses with the same pulse duration and intensity as those used in measuring n2 of the corneal sample, the transmitted spectral curves when the cover glass placed at different z positions were measured. The spectral curves at z = 0 and z = -5zr are shown in Fig. 3(a). From Fig. 3(a), it is found that there is nearly no difference between the two spectral curves, indicating that the spectral broadening effect caused by the nonlinear phase shift can be neglected. The transmittance T(z) with a spectral window of 799 nm ∼ 801 nm are shown in Fig. 3(b). From Fig. 3(a), it is found that TV is 0.006 and the resultant maximal nonlinear phase shift Δϕmax is only 0.15 calculated by Eq. (5). Since the cornea has a certain degree of scattering and absorption in the experiment, the actual power on the cover glass should be less than the peak intensity I0 ∼ 397 GW/cm2 incident on the corneal sample. Therefore, the spectral change caused by the self-phase modulation effect of the cover glass is negligible.
Fig. 3. Measured results of the cover glass’s n2. (a) Normalized spectral curves at z/zr = 0 and z/zr =−5; (b) normalized transmittance T at different positions along z axis. The laser pulse duration and intensity are the same as those used in Fig. 2.
3.3 Effect of nonlinear absorption on the n2 measurement of cornea
In this section, the nonlinear absorption’s effect on measuring n2 of the corneal sample is investigated. In order to measure the nonlinear absorption of the corneal sample, the nonlinear absorption of the cover glass must be firstly investigated. Our previous work has shown that using the total spectral band as the spectral window is a simple way to measure the nonlinear absorption, i.e. in Fig. 1 the spectrometer was replaced by a power meter and the spectral domain Z-scan technique was converted into the conventional Z-scan technique [20]. Using the laser pulses with the same parameters as those used in Fig. 2, the total spectral band transmittance of the cover glass at different z positions is presented in Fig. 4(a). It can be seen that T(z) is randomly distributed around 1, and the standard deviation is less than 0.006. This indicates that the nonlinear absorption at the laser peak intensity of ∼397 GW/cm2 is negligible. In addition, the experimental results presented in Ref. [28] show that when 800 nm femtosecond laser is used, the threshold intensity of the nonlinear absorption for silica glass is 3×103 GW/cm2, which is one order of magnitude larger than the peak intensity used in the experiments. Therefore, the nonlinear absorption of the cover glass is negligible during the measurement of n2 in this paper.
Fig. 4. (a) Normalized transmittance T(z) of the cover glass when the total spectral band (200 nm - 1100 nm) is used as the spectral window; (b) normalized transmittance T(z) of the cornea with the water content of 89% when the total spectral band (200 nm - 1100 nm) is used as the spectral window.
It is worth noting that measuring the nonlinear absorption coefficient by employing the total spectral band as the spectral window is only reliable when the material properties remain constant in the illuminated area. However, the ex-vivo cornea is an optically non-uniform sample, especially for low water content ones. During the z scanning process, the beam diameter on the corneal sample inevitably changes. Therefore, the non-uniformity of the sample will bring random disturbance to the transmitted signal. The pores on the surface of the corneal sample with high water content can be filled by water, reduce the scattering and non-uniformity of the sample to some extent. Therefore, we use the experimental results of the cornea sample with the highest water content in Fig. 2(c) to calculate the nonlinear absorption coefficient. The calculated results are shown in Fig. 4(b). It is seen that when the peak intensity is ∼397 GW/cm2, no nonlinear absorption is found in the cornea sample.
4. Conclusion
The nonlinear refractive index of the biological samples has significant impact on the imaging and laser treatment of biological tissues. However, since the traditional Z-scan technique determines the n2 coefficient by measuring the change of the beam spot size in the far-field, the traditional Z-scan technique is not suitable for determining the n2 coefficient of the highly scattering biological samples, such as ex-vivo corneal samples. Fortunately, the newly proposed spectral domain Z-scan technique [20] is suitable for the highly scattering samples since it measures the intensity variation in certain spectral band to determine the n2 coefficient. In this paper, by using the spectral domain Z-scan technique, the n2 coefficient of the corneal sample cutting from the donor’s cornea during the femtosecond laser myopia correction surgery is successfully measured. The corneal sample may dehydrate gradually when it exposes in air. Successive n2 measurements can determine the dependence of the n2 coefficient on the corneal sample’s water contents. The n2 coefficients are measured to be 1.1 ± 0.1×10−19 m2/W, 1.4 ± 0.2×10−19 m2/W and 1.6 ± 0.2 ×10−19 m2/W respectively for the corneal sample whose water contents are 89%, 82%, and 78%. The measurement results show that the n2 coefficients of corneal sample decrease as the increase of the water content. It demonstrates that the spectral domain Z-scan technique is an effective method to characterize the n2 coefficient of highly scattering biological samples.
Funding
National Key Research and Development Program of China (2018YFB0504400).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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