We report a strictly single-mode optical fiber with a record core diameter of 84µm and an effective mode area of ~3600µm2 at 1µm. We also demonstrate fundamental mode operation in an optical fiber with a record core diameter of 252µm and a measured mode field diameter (MFD) of 149µm at 1.03µm, i.e. an effective mode area (Aeff) of ~17,400µm2 at 1.03µm, an Aeff of 31,600µm2 at 1.5µm. All these fibers have near parabolic index profiles with a peak refractive index difference ΔN≈~6×10-5, i.e. a record low numerical aperture (NA) of ~0.013 in an optical fiber. This low refractive index difference was achieved by frozen-in thermal stress as a result of two different types of glass in the fibers. When the fundamental mode was excited in the 252µm core fiber using a 1.03µm ASE source, the output beam was measured to have M2x=1.04 and M2y=1.18.
©2009 Optical Society of America
Large mode area (LMA) optical fibers have been intensely studied in recent years for scaling high peak powers in optical fibers to MWs and beyond to break the bottleneck caused by detrimental optical nonlinearities in fibers. A few approaches are pursued. The first approach is photonic crystal fibers (PCF) using a large number of periodic very small air holes to achieve effectively very low NA fibers [1,2]. Unlike in conventional optical fibers, where a low fiber NA is usually achieved by carefully controlling dopant levels in the core and surrounding cladding during fabrication, effective NA in PCFs is controlled by the relative dimension of air holes in the cladding, i.e. d/Λ where d is hole diameter and Λ is center-to-center hole spacing. This allows much lower effective NA than what is possible in conventional optical fibers in a controllable way. Ytterbium-doped rod-type PCFs have been demonstrated with core diameters as large as 100µm, albeit somewhat multimode . The ytterbium-doped core had essentially uniform refractive index and was well index-matched to the surrounding silica glass in this case. The extremely weak guidance in these large core PCFs does not allow them to be bent due to excessive losses. Recently, leakage channel fibers with built-in mode filtering have been demonstrated to be an effective alternative for an improved bend performance . Passive all-glass leakage channel fibers have been demonstrated with core diameters as large as 170µm recently . The third approach to achieve large effective mode areas is to propagate a higher order mode in highly multi-mode fibers . However, this approach needs complex mode excitation and mode converters since fundamental mode output is preferred in practical applications. Conventional optical fibers, on the other hand, are constrained to NA>0.06, limited by a lack of reliable fine dopant control in the vapor deposition process, and can only be strictly single-mode for core diameters as large as ~13µm at ~1µm and ~20µm at 1.55µm. It has, however, been shown that reasonable fundamental mode propagation can be achieved in multi-mode fibers by carefully managed single-mode launch [7,8]. It has been further shown that coiling can be appropriately applied to further discriminate against higher order mode propagation . With these techniques, single-mode operation in core diameters of ~30µm at ~1µm is regularly achieved. Single-mode operation in highly multimode fibers with core diameters of 80µm was also recently demonstrated at ~1µm with very small coil diameters of 4.25cm . At such small coil diameter, the Aeff of the fiber is substantially reduced from its maximum value in a straight fiber .
It has been long speculated that conventional single-mode optical fibers with much larger core diameters and much lower NAs are not achievable due to the aforementioned challenges in controlling the very small index change in the vapor deposition process. In particular, it is difficult to control the NA of LMA fibers with an accuracy of ±0.005 in the fabrication process. For fibers with core size beyond 50µm, they are usually multi-moded and it is very challenging to get single-mode excitation. In this work, we demonstrate a different type of optical fibers by taking advantage of the photo-elastic effect and thermal stress, which have been previously studied in [11–14]. Using such method, we are able to control the NAs of such LMA fibers to be as small as 0.013, which are also reproducible. In addition, we demonstrate strictly single-mode index-guiding optical fibers with a record core diameter of 84µm and Aeff of ~3600µm2 at 1µm. We further report fundamental mode propagation at 1.03µm in a fiber with a core diameter of 133µm and Aeff of ~5000 µm2 at 1µm and ~9100µm2 at 1.55µm. Based on the fact that the main guiding mechanism is the formation of low-index trench introduced by frozen-in thermal stress during fabrication, we finally demonstrate fundamental mode operation in an extremely large core fiber with a diameter up to 252µm using a simpler design. When a 1µm ASE source was coupled into such fiber, it was measured to have M2x=1.04 and M2y=1.18. These fibers have near parabolic refractive index profiles and a record low peak index-difference between core and cladding ΔN~6×10-5, i.e. NA=0.013.
This record low NA was achieved through photoelastic effect by frozen-in thermal stress due to the existence of at least two different types of glass in the fiber. Efficient single mode operation is easily achieved by matching the MFD at the beam focus at input in all fibers and excellent beam quality at the output is confirmed by our M 2 measurement (M 2<1.1). Although such fibers cannot be significantly coiled due to their weak guidance, they can still find significant applications in high peak power pulse delivery, mode filters , and pulse compression via nonlinear spectral broadening, where a short length of fiber is sufficient and often desirable.
2.1 Stress-induced LMA fibers with three rings of fluorine-doped silica rods
We first fabricated fibers with a three-ring structure of two different cladding sizes, 0.835mm and 1.2mm respectively using the standard stack-and-draw technique . These fibers contain a hexagonal array of slightly fluorine-doped silica rods in a silica background. A single fluorine-doped rod is missing in the centre. The fluorine-doped rods have a refractive index ~1.2×10-3 lower than that of the silica. The cross section of a fiber is shown in Fig. 1 (a). In addition to the dark fluorine-doped silica rods and light silica background, a grey low index trench of hexagonal shape is clearly seen at the original surface of the rods used to build the preform. This low index trench arises due to thermal stress when the surface layer of the glass have slightly higher thermal expansion coefficient. The difference between surface layer and interior of the glass rods arises from diffusion, influence of atmosphere and other surface-related effects during the fabrication process. A core is formed at the center of the fiber by the low index trenches with a diameter of 133µm, defined as the distance between two parallel low index trenches shown in Fig. 1. The fiber structure is very similar to the fiber used in our recent all-glass, endless single-mode PCF of 50µm core diameter demonstrations . In that case, the stress-induced refractive index of similar magnitude is also expected within the PCF core. The dimension of the waveguide formed by the low index trench is, however, too small to support significant waveguide mode in that case. PCF guidance from the periodic cladding dominates in the 50µm core, endless single mode fiber.
Two additional fibers were drawn with smaller cladding diameters of 167µm and 262µm respectively under the same drawing temperature 1970°C just for refractive index measurements since we cannot measure refractive index of the large diameter fibers. Both sets of measurements show a parabolic index profile of the core and a refractive index difference between the center of the core and the low index trench ΔN of 6(±1)×10-5, largely independent of Λ. Since the index profile is independent on the cladding size, the refractive index profile in Fig. 1(b) is based on the measurements of smaller cladding fibers.
The fluorine-doped glass has a slightly larger coefficient of thermal expansion than that of the silica glass. It would contract more than the surrounding silica glass during cooling after exiting the furnace on the fiber drawing tower. This contraction is, however, constrained by the surrounding silica glass and will consequently put the surrounding silica glass under compression and the adjacent fluorine-doped glass under tension. This stress causes a refractive index rise in the silica glass next to each fluorine-doped rod and a corresponding refractive index reduction in the adjacent fluorine-doped glass. Frozen-in drawing tension can also contribute additional refractive index change. It is expected to be small in our case due to the geometry. It is worth noting that the refractive index change from a mismatch of coefficients of thermal expansion cannot be eliminated by thermal annealing, while the index change from the frozen-in drawing tension can be annealed above glass transition temperature. Similarly when the surface layer of each of the rods in the preform stack has a higher thermal expansion coefficient, a low index trench is developed at the original rod surfaces. This lead to a hexagonal low index trench, shown as dark line in the fiber cross section in Fig. 1(a), developed half way between the low index rods. This effectively forms an index guiding core in the centre of the fiber with a near parabolic refractive index profile and a core diameter 2R (see the right Fig. 1(b)). The refractive index scan in Fig. 1(a) is right through the center of the fiber and the centers of six low index rods. The six low index rods can be clearly seen, along with the refractive index rise in the silica glass adjacent to each low index rod due to a compressive stress as well as the refractive index reduction in the fluorinedoped silica glass adjacent to the silica glass due to a tensile stress. The low index trench between the low index rods (grey hexagonal line in Fig. 1(a)) can also be seen clearly in the refractive index profile in the Fig. 1(b).
To confirm that such a waveguide provides sufficient guidance, we simulated the optical modes using experimental measured index profile of the core. The first fiber (fiber I) has a d/Λ=0.36, Λ=90µm, 2R=84µm, and a fiber diameter of 835µm. The second fiber (fiber II) has Λ=142µm, 2R=133µm, and a fiber diameter of 1.2mm. Our waveguide mode analysis with a parabolic index profile of ΔN=6×10-5and 2R=84µm gives a LP11 cut-off wavelength at 960nm for the Fiber I. This makes the first fiber a strictly single mode fiber at 1µm. The analysis also gives a MFD of ~68µm and an Aeff of ~3600µm2 at 1µm. Similar analysis with 2R=133µm gives a LP11 mode cut-off wavelength of 1.5µm for the second fiber. This would make the second fiber two-moded at 1µm, but strictly single mode at 1.55µm. The same analysis also gives the second fiber a MFD of ~80µm and Aeff of ~5000µm2 at 1µm and a MFD of ~108µm and Aeff of ~9100µm2 at 1.55µm.
To verify our analysis, we checked the mode behavior using a silicon CCD camera and an ytterbium ASE source at ~1µm. A straight 90 cm length of fiber I was used. A lens L1 was used to collimate the light from a Hi1060 fiber carrying the ytterbium ASE source. A second lens L2 (f=50 mm, in this case) was used to focus the collimated light into the first fiber. Focus lengths of the lenses were selected to match the diameter of the focused light to that of the fundamental mode in the fiber. The measured output mode on the CCD camera after collimation is shown in Fig. 3, along with x and y profiles shown on the right side.
Strictly single-mode operation in fiber I at ~1µm was tested when the output mode was continuously monitored while the launch positioning stage with lens L2 was moved away from the optimal transverse position and then moved back. The result is shown in Fig. 4. The ytterbium ASE source was used for this test. No higher order mode was observed while adjusting the launch position up to the point where the fundamental mode was entirely turned off. At optimal position, total transmission with reference to the power just before the fiber was measured to be 91.4% at ~1µm. This included reflection losses at the two fiber ends, indicating extremely high launch efficiency and low transmission loss of less than 0.05dB/m at ~1µm. The mode at various wavelengths was further characterized with the ASE source being replaced by a broad band super-continuum source and a band-pass filter. Stable single-mode operation was easily seen above 850nm (see Fig. 5). At shorter wavelengths, evidence of higher order mode was observed.
To confirm that the mode was guided in the high index region in the center of the fiber, not by PCF guidance at the interface between silica and fluorine-doped silica rods, the fiber output endface was illuminated by an additional white light source (see Fig. 2) while the output mode was monitored at ~1µm. It could be clearly seen that the mode only occupied the center part of fiber within the index guiding core and did not extend beyond the low index trench (see Fig. 6). White circles mark the location of the inner six low index rods in Fig. 6(a). A further test was done by monitoring the output mode while increasing the launched power until the center part of the mode was saturated (see Fig. 6(b)). In this way, faint structures away from the mode could be observed. The weak light in the high index ring around each fluorine-doped silica rods could be clearly seen. Also visible is that the output mode was confined within the low index trench around the core. It is clearly evident from the two tests shown in Fig. 6 that the mode was guided by the local high index core in the center of the fiber.
Robust single-mode propagation was also easily observed in fiber II at ~1µm using a similar experimental set-up, the ytterbium ASE source and ~87cm straight fiber. The captured mode at ~1µm is shown in Fig. 7(a) and (b). When the output endface was side-illuminated by the white light source, fine structure of trenches of low index could be seen in Fig. 7(b). Based on the microscope image, we know the geometrical dimensions. So the MFD can be calibrated with this physical dimension, illustrated as an inset in Fig. 7(b). MFD of 83.4µm was measured, very close to 80µm of simulated MFD. The LP11 mode at ~1µm could also be excited in fiber II, when launch lens was deliberately offset transversely from the optimal conditions for single mode excitation, as shown in Fig. 7(c). No other higher order mode was observed, however. Both the fundamental and LP11 mode are clearly seen being guided by the parabolic index core in the centre of the fiber when the endface of the fiber was illuminated by the additional white light source. When fundamental mode was excited in the 133µm core fiber, M2 was measured to be M2x=1.08 and M2y=1.05, which is shown in Fig. 8.
To check whether the fundamental mode is robust against stress, we also applied up to 440 grams (4 metal posts, 880 grams in total, half of the weight is on fiber II) on fiber II. We did not see any significant change in the mode, as depicted in Fig. 9. The posts are supported equally in weight by this fiber and a 1.2mm thick metal piece. In this case, we used a higher resolution CCD camera.
2.2 Stress-induced LMA fibers with one ring of fluorine-doped silica rods
Since the waveguide is expected to be formed by thermal stress arising from the lower thermal expansion coefficient at the surface of the silica rods. We used only one ring of 6 fluorinedoped silica rods and two outer rings of silica rods in the second preform. Fibers were drawn at 1950°C with core diameter of 2R=144µm (fiber III) and 252µm (fiber IV) respectively. In this case, an index guiding core is still formed at the fiber center by the thermal stress.
2.2.1 Fiber III
The microscope image of the fiber III is shown in Fig. 10(a). It has a core size of 144µm and an outer cladding size of 1.15mm. We use similar setup as shown in Fig. 2 except using different lens combination and without using the white source. The 1µm ASE source was collimated with a lens L1 with a focus length of 8mm, L2 was chosen to be have a focus length of 100mm. In this case, the launched signal into the fiber III had a small mode mismatch with the fundamental mode in the center core. The output beam profile from a 84cm fiber III was captured as shown in Fig. 10(b). The MFD is measured to be 84.3µm. Since the launched beam had a slightly smaller size of 75µm, part of the energy was coupled into the rings formed by the slightly raised high index surrounding the fluorine-doped silica rods. From Fig. 10(a), we could see that the silica cells in the 2nd ring also had similar low index trenches of hexagonal shape. This should not be surprising since the low index trench arises from the higher thermal expansion of the surface layer of silica rods used. The low index trenches in each cell can form a guiding core. If we selectively change the launching conditions along x or y axis, we can excite a fundamental mode at each silica cell, as depicted in Fig. 11. Although the off-centered pure cells still guide fundamental mode, we observed slightly (~5–10%) higher loss in these cells due to the even weaker guiding property. Among these insets, the Fig. 11(e) is the image of the fundamental mode of the center core. The MFD of the fundamental mode in an off-centered cell was measured to be ~81µm, which is close to that of the center core within measurement error. This definitely shows that the guiding core was not formed by any photonic crystal fiber effect. Please note that in order to see all the structures of cells, we deliberately inject the power of 1µm ASE signals close to the saturation (255) of the CCD camera.
2.2.2 Fiber IV
In order to examine mode properties of this fiber with the largest core of 252µm, two endfaces with an angle of 0.5 degrees of a 1 m length fiber IV were prepared. The microscope image of the fiber endface is shown in Fig. 12(a). A refractive index scan was performed along x axis on a fiber with a smaller cladding diameter (made just for this measurement) and shown in Fig. 12(b). The center core is again formed by slightly lower refractive index trenches. The refractive index difference between the core center and the low index trench is 7×10-5 within an error of ±2×10-5. This error is similar to measurement in fibers with three rings of fluorine-doped silica rods measured previously.
Lens L1 with focus length of 8mm and another lens L2 with a focus length of 200mm were used to characterize this fiber. The mode profile is shown in Fig. 12(b). The focused beam size was matched to the mode size of fiber IV in this case. The coupling efficiency into the fundamental mode was very high, therefore we did not observe any light in the surrounding layers. The MFD at 1µm was measured to be 149µm. In addition, 92% transmission was measured transmitted through the fiber. Since a total of 8% Fresnel reflection losses were expected from the two endfaces (the lenses used here were all AR-coated), very low fiber loss was confirmed. When the fundamental mode propagation was observed, M2 was characterized using Spiricon M2-200 to be M2x=1.04 and M2y=1.18 (see Fig. 13(b).
This fiber was found to support higher order mode at 1µm. When lens L2 was moved away from the optimum position, higher order mode LP11 was observed in the center core. Propagation of fundamental mode in the surrounding pure silica cells, similar to the case of fiber III in Fig. 11, was also observed. But in this case, we could observe higher order mode.
Stress effects in optical fibers have been previously studied in the context of fiber birefringence , in general [11,12] and drawing dependent effect . The simple case of concentric cylinders has been well studied for its thermal stress [11,12]. Thermal stress arises due to the difference in the thermal expansion coefficients in component glass in the fiber. When fiber cools from elevated temperatures, the interface between the two component glass becomes fixed at certain lower temperature. Thermal stress develops as fiber is further cooled towards room temperature. The component glass with higher thermal expansion is put under tension while its contraction is constrained. This puts the component glass with lower thermal expansion under compression. Photo-elastic effect leads to a lower refractive index in the component glass with higher thermal expansion and a higher refractive index in the component glass with lower thermal expansion. Three stress component needs to be analyzed are radial, circumferential and axial. They lead to three corresponding refractive index components :
where Δn denotes refractive index change, σ stress, r radial coordinate, θ circumferential coordinate, and z axial coordinate respectively. B1 and B2 are stress-optical coefficients. For silica glass, B1 is near an order of magnitude larger than B2. In a weakly guided waveguide with dominating transverse field and negligible axial field, i.e. TEM modes, transverse refractive index determines the optical modes. Moreover, continuity of circumferential field components at the interfaces plays a significant role in determining the optical mode. This makes Δnθ the most relevant index component and, consequently, σr and σz the more relevant stress components.
In a simple concentric cylinder made with two homogenous glass components, first component in the center and the second surrounding the first one, all three stress components are expected to be constant over the first glass component in the center. σz is expected to be constant but at a different magnitude over the second glass component, while both σθ and σr are expected to decrease in a 1/r2 manner away from the interface. σr=0 while σθ≠0 at the free outer boundary. σr is continuous at the glass interface while σθ changes sign. One important result from the analysis in [11,12] is that the maximum stress happens at the glass interface and this maximum stress level is dependent only on the relative dimensions not absolute dimension of the fiber. In other words, fibers drawn to different diameters from the same preform are expected to have the same level of maximum thermal stress at the interface.
Another aspect of stress in a fiber is drawing-tension-dependent stress. This arises due to the fact that drawing tension exerted on a fiber is taken mainly by the glass component with high melting temperature at the point of neck-down region in the furnace where this glass is solidified while the other glass component is still liquid. The drawing tension is eventually removed and the contraction from the tension release will be constrained by the other glass component. This will put the glass component with high melting temperature under tension and the glass component with low melting temperature under compression. For silica and slightly doped silica glass, a low melting temperature is typically associated with high thermal expansion coefficient, the drawing-induced stress will be opposite in sign to that of thermal stress.
In our case, the dominating effect arises from the different glass on the surface of the silica rod and its interior. This difference arises from the fabrication process where the surface is strongly influenced by atmosphere, evaporation and diffusion, at elevated temperatures. This leads to a thin surface layer with slightly higher thermal expansion coefficient. Since we have not observed significant difference in the measured stress-induced refractive index at different drawing conditions in a large number of fibers, drawing-tension induced stress is expected to be small in our fibers. The observation that the measured stress-induced refractive index is nearly identical in fibers drawn to different outer diameters is also consistent with the theoretical prediction that the maximum stress is only dependent on the relative geometrical dimensions. The parabolic refractive index of waveguide is also consistent with the theoretical prediction that the stress following a 1/r2 dependence away from the interface. The theory also predicts a constant stress over the fluorine-doped silica rod if it is homogenous, while a structured stress-induced refractive index was observed (see Fig. 1(b)). The fluorine-doped silica rod was prepared by collapsing a tube. This would cause the center of the collapsed rod to be different from the rest of the rod due to evaporation and diffusion at the original inner surface at elevated temperatures. This inhomogeneous glass composition can explain the measured stress profile in fluorine-doped silica rods in the fiber.
The subtle difference between the surface and bulk of the silica rods and its effect on refractive index of a fiber made with stack-and-draw technique is reported for the first time according to our knowledge. The use of the small thermal stress-induced refractive index to make waveguide is also reported for the first time according to our knowledge. The small magnitude of the stress-induced refractive index is exploited to make large optical waveguides for single mode operation in this work. We demonstrated a strictly single mode fiber with a record core diameter of 84µm and an Aeff of 3600µm2 at 1µm and a two-moded fiber with a record core diameter of 252µm and an Aeff of ~17400µm2 at 1µm. The extremely low index-difference of ~6×10-5, i.e. NA=0.013, was achieved through the photo-elastic effect from frozen-in thermal stress caused by the mismatch in the coefficients of thermal expansion of the two types of the glass used in the fibers.
References and links
1. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14(7), 2715–2720 (2006). [CrossRef]
2. J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express 13(4), 1055–1058 (2005). [CrossRef]
3. C. D. Brooks and F. Di Teodoro, “Multi-megawatt peak-power, single-transverse-mode operation of a 100 µm core diameter, Yb-doped rod-like photonic crystal fiber amplifier,” Appl. Phys. Lett. 89(11), 111119–111121 (2006). [CrossRef]
4. L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007). [CrossRef]
5. L. Dong, J. Li, H. A. McKay, A. Marcinkevicius, B. T. Thomas, M. Moore, L. B. Fu, and M. E. Fermann, “Robust and practical optical fibers for single mode operation with core diameters up to 170µm,” Conference on Lasers and Electro-optics, post-deadline paper CPDB6, San Jose, CA, May 2008.
6. S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006). [CrossRef]
7. M. E. Fermann, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23(1), 52–54 (1998). [CrossRef]
8. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000). [CrossRef]
9. A. Galvanauskas, M. Y. Cheng, K. C. Hou, and K. H. Liao, “High peak power pulse amplification in large core Yb-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 559–566 (2007).
10. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006). [CrossRef]
11. K. Brugger, “Effects of thermal stress on refractive index in clad fibers,” Appl. Opt. 10(2), 437–438 (1971). [CrossRef]
12. U. C. Paek and C. R. Kurkjian, “Calculation of cooling rate and induced stress in drawing of optical fibers,” J. Am. Ceram. Soc. 58(7–8), 330–335 (1975). [CrossRef]
13. R. A. Sammut and P. L. Chu, “Axial stress and its effect on relative strength of polarization-maintaining fibers and preforms,” J. Lightwave Technol. LT-3(2), 283–287 (1985). [CrossRef]
14. Y. Hibino, F. Hanawa, T. Abe, and S. Shibata, “Residual stress effects on refractive indices in undoped silica-core single-mode fibers,” Appl. Phys. Lett. 50(22), 1565–1566 (1987). [CrossRef]
15. B. Ortaç, M. Baumgartl, O. Schmidt, and J. Limpert, “µJ-level femtosecond and picosecond fiber oscillators,” MB15,OSA, Advanced Solid-State Photonics 2009.
16. P. St. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]
17. L. Dong, H. A. Mckay, and L.B. Fu, “All-glass endless single mode photonic crystal fibers,” Opt.Lett. 33(21), 2440 (2008). [CrossRef]
18. M. Born and E. Wolf, “Principle of Optics,” Pergamon Press, 1991.