Abstract

Harmonic generation by tightly-focused Gaussian beams is finding important applications, primarily in nonlinear microscopy. It is often naively assumed that the nonlinear signal is generated predominantly in the focal region. However, the intensity of Gaussian-excited electromagnetic harmonic waves is sensitive to the excitation geometry and to the phase matching condition, and may depend on quite an extended region of the material away from the focal plane. Here we solve analytically the amplitude integral for second harmonic and third harmonic waves and study the generated harmonic intensities vs. focal-plane position within the material. We find that maximum intensity for positive wave-vector mismatch values, for both second harmonic and third harmonic waves, is achieved when the fundamental Gaussian is focused few Rayleigh lengths beyond the front surface. Harmonic-generation theory predicts strong intensity oscillations with thickness if the material is very thin. We reproduced these intensity oscillations in glass slabs pumped at 1550nm. From the oscillations of the 517nm third-harmonic waves with slab thickness we estimate the wave-vector mismatch in a Soda-lime glass as ΔkH= -0.249μm-1.

© 2015 Optical Society of America

1. Introduction

In harmonic-generation microscopies [1-3], and in particular second harmonic (SH) [4-6] and third harmonic (TH) [7-9] microscopies, the harmonic wave is excited by a tightly focused Gaussian laser beam, and the image information is carried by the intensity variations as this beam is scanned through the sample. Harmonic generation by focused beams is also common in short-pulse measurements [10], and of course, in various wavelength conversion schemes. Several researchers extended the focused beam theory to include vector field excitation, either in a general context [11,12], or in relations to microscopy [13].

Although it is often assumed that the signal is generated near the focal region ([11] talks about inverse of the wave-vector mismatch as a measure for the near-focus interaction length), it is well known that harmonic wave generation is the result of coherent interaction over an extended volume, and depends on material properties (susceptibility and chromatic dispersion) and on excitation geometry. The integral predicting the amplitude of the harmonic wave, which is found in textbooks (e.g. [14], Eq. (2).10.11b), was studied already in the late 1960’s in a classic paper by Boyd and Kleinman [15] and later by Ward and New, who studied TH generation in various gases by a focused laser beam [16]. Ward and New introduced analytic solutions of the amplitude integrals for Bulk excitation configuration [16].

In this paper we study intensity oscillations of SH and TH as predicted by the amplitude integral in a Gaussian excitation geometry. We derived new analytic expressions for the amplitude integral for each of the two harmonic waves, and present harmonics intensity maps vs. integration limits using these expressions. The maps show how harmonics intensities depend on the integration limits in a finite medium. Our calculated maps show that generally, for both SH and TH waves and a typical negative wave-vector mismatch, maximum harmonic intensity is achieved at Bulk excitation (Gaussian beam focused midway between surfaces) if the window is thin (one Rayleigh length) and by Surface or near-surface excitation if the window is thick (few Rayleigh lengths). For these negative wave-vector materials at some anti-resonant thicknesses, high harmonics are practically not generated, regardless of focal plane position within the material. The theoretical expressions also show that the harmonic intensity is affected quite strongly by regions a few Rayleigh-lengths away from the focal plane, and contribute to strong oscillations in the intensity. Note that the intensity oscillations discussed here differ from the sub-wavelength oscillations of back-propagated TH waves predicted by Olivier and Beaurepaire [13].

We have compared our results with experiments, where we have reproduced TH intensity oscillations quite close to those predicted by the amplitude integral for relatively thin windows (window thickness on the order of one Rayleigh length). The period of these oscillations offers a good estimate to the wave-vector mismatch characteristic to the material used. The estimate can be refined by a more careful fit of the theoretical curve to the measured curve.

We start by evaluating the amplitude integrals for SH waves in section 2 and for TH waves in section 3 where we analyze two geometrical situations, whether the pump beam waist is at the surface of a slab of nonlinear material or at its center. We present experimental results in section 4 and then summarize.

2. Amplitude integral for Gaussian-excited second harmonic waves

The amplitude of a Gaussian-excited second harmonic wave at an axial plane z2 is proportional to an amplitude integral J2(ΔkH,zR,z1,z2) which is the result of the coherent buildup of the harmonic wave from the input facet z1 up to z2 [14,16]. The amplitude integral is given as ([14], Eq. (2).10.11b with small notation changes):

J2(ΔkH,zR,z1,z2)=z1z2eiΔkHzdz1+iz/zR;ΔkH2kH(ωf)kH(2ωf)k0(ωf)=ωfc;kH(ωf)=k0(ωf)n(ωf);kH(2ωf)=2k0(ωf)n(2ωf)
where ωf is the angular frequency of the fundamental harmonic, c is the vacuum speed of light, n(ωf),n(2ωf) are the material’s refractive indices at the fundamental and doubled angular frequencies respectively. The coordinatesziin Eq. (1) are axial coordinates, measured from the plane of focus – the plane of minimum waist of the harmonic-exciting Gaussian beam (cf. Figure 1). The parameter zR in Eq. (1) is the Rayleigh length (equals half of the Confocal Parameter denoted by b in [14,16]), that relates to the beam waist radius w0 by:
zR(ωf,w0)12k0(ωf)n(ωf)w02
Now, define dimensionless parameters:
ζzzR;ζ1z1zR;ζ2z2zR;ΔkHzRu
and write the integral (1) as
J2(zR,u,ζ1,ζ2)=zRζ1ζ2eiuζ1+iζdζ
As it turns out, the amplitude integral of the second harmonic wave of Eq. (4), after splitting into purely real and purely imaginary integrals, has an analytic solution, in terms of i(u,ζ) (“CosIntegral”), and Si(u,ζ) (“SinIntegral”) functions (these functions are defined in Ref. [17]). For u>0 (positive wave-vector mismatch, associated with anomalous material dispersion), the indefinite integral of Eq. (4) can also be expressed by the Exponential Integral function [18].

 

Fig. 1 Harmonics generation geometry. a. Gaussian beam parameters. b. General position of the focal plane of the harmonics exciting Gaussian beam (with respect to the surfaces of the harmonics-generating window). c. Surface excitation geometry. d. Bulk excitation geometry. Note that we define the waist location as z=0, a distance measured to the left of the waist gets a minus sign.

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Two well defined and interesting cases of excitation geometry are described by a “Bulk” amplitude integral and by a “Surface” amplitude integral. In [12], these two geometries are termed “centered” and “offset”, respectively. For these two cases, the full thickness of the material, assumed to be shaped as a flat window, is 2ζ2zR. In the case of a Bulk amplitude integral, the Gaussian waist is located midway between the two window’s surfaces, and in the case of a Surface amplitude integral, the Gaussian waist is located at one of the window’s surfaces:

J2bulk(zR,u,ζ2)J2(zR,u,ζ2,ζ2)J2surf(zR,u,ζ2)J2(zR,u,0,2ζ2)

For infinite limits (ζ2=) the Bulk amplitude integral (Eq. (5)) evaluates as [12,16]:

J2bulk(zR,u,)={0u<0πzR2πzReuu=0u>0

Mathematically, we see two “steps” in J2bulk(zR,u,) at Δk=0 (or u=0). A πzR step at 00, and another πzRstep at 00+. For positive wave-vector mismatch (Δkor the dimensionless u - associated with anomalous dispersion), the Bulk SH amplitude integral exponentially decays with increasing mismatch (cf. Figure 2).

 

Fig. 2 Theoretical SH generated amplitude (absolute value of in units of μm) vs. wave-vector mismatch, for Gaussian excitation. a. Window of infinite thickness. Mathematically, the Bulk amplitude, for window of infinite thickness, is discontinuous at perfect wave-vector match (u=0, cf. Equation (12)). And mathematically, for window thickness increased to infinity, the Surface amplitude goes (very “slowly”) to infinity at perfect wave-vector match. b. Window of finite thickness. The parameter LwNis a normalized window thickness (cf. Equation (7)). Note that we show here amplitudes (absolute value of), not intensities.

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The amplitude integral depends explicitly on four variables – the Rayleigh length (zR), the wave-vector mismatch (given here by the dimensionless parameter u for a fixedzR), and the two integration limits (z1,z2) (Eq. (1)). In the following we keep the Rayleigh length fixed atzR=50μm. We are left with three variables. Bulk amplitudes and Surface amplitudes, shown by Fig. 2, are special cases of only two variables as the lower integration limit z1 is either zero or is symmetrical to the upper integration limit with respect to the focal plane. Similar to [19], we define a normalized window thickness (LwN) as

LwN(z2z1)/zR=ζ2ζ1
Note that when holding the window’s thickness fixed (for Bulk-Surface comparison), as z1 shrinks z2 grows until atz1=0z2(Surf)=2z2(Bulk). In the case of Bulk amplitude, the fundamental exciting Gaussian is focused at the center of the window (Eq. (5)). The curves of Fig. 2(a), plotted for an infinitely thick window, show two peculiarities of the theoretical SH amplitude atu=0: discontinuities of the Bulk amplitude and infinitely high Surface amplitude (absolute value). The curves of Fig. 2(b) are plotted for a finite-thickness window as designated in the figure. Note the non-vanishing Bulk amplitude for a finite thickness window even with negative wave-vector mismatch (u<0), as is the case with normal-dispersion materials. Note also that in Fig. 2 we plot absolute value of the amplitudes, not absolute value squared.

The curves of Fig. 3 display absolute value squared of Bulk amplitudes and Surface amplitudes. This time the wave-vector mismatch (u) is fixed and the variable is the normalized window thickness (given in these cases byζ2, cf. Equation (5) and Eq. (7)).

 

Fig. 3 Theoretical SH generated intensity (in units of μm2) vs. integration range (“window thickness”), for Gaussian excitation. a. u=4. The curves show that even for negative and relatively large wave-vector mismatch, SH waves are still generated at Surface excitation and even at Bulk excitation if the window is “thin” (say on the order of one Rayleigh length). b. u=1. At near perfect wave-vector match, high intensities are generated for both Surface and Bulk configurations if the window is thin. c. u=+1. Note the domination of Bulk over Surface for small positive wave-vector mismatch (compare with the curves of Fig. 2).

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The curves in Fig. 3(a) clearly show strong intensity oscillations for “thin” windows (windows’ thickness on the order of one Rayleigh length). Observe the curves in “c” showing higher intensities generated in the Bulk configuration (vs. the Surface configuration) for a small positive wave-vector mismatch (u=+1).

The last illustrative figure for SH generated intensities in a Gaussian excitation configuration is Fig. 4. The figure presents three SH intensity maps with the integration limits as the horizontal and vertical axes. Note that the maps are symmetric with respect to the diagonals and one redundant half is preserved in the shown maps only for convenience and aesthetic reasons. The maps clearly show the geometrical location of maximum intensity. By adjusting the Rayleigh length to a given window thickness (or vice-versa) and by adjustingthe position of the focal plane (relative to, say, the front surface of the window), these maximum intensities may be reached. Looking at the top-left quarter of Fig. 4(a), for example, sitting at the first peak of Bulk excitation (i.e. along the diagonal red arrow) and moving the focal plane away from the window’s center (moving along the other diagonal, in the direction of the dashed line, keeping the total window thickness constant), the map indicates quick reduction in generated SH power. Such power reduction was indeed observed by Bjorkholm during his 1965 SH experiments in ADP crystals [20].

 

Fig. 4 Theoretical SH generated intensity (in units of μm2) vs. integration limits (“window thickness” and position of the plane of focus), for Gaussian excitation. The parameter ζ1 is the distance from the focal plane to the front surface of the window (including sign) in units of zR. Similarly for ζ2 and the back surface. Wave-vector mismatch is a:u=4. b:u=0.1. c:u=+1. These maps can assist in designing experimental set-ups for max SH intensity generation. For example, the map in a indicates that for negative wave-vector mismatch, if the window is thick (say 4zR or more), then Surface excitation is far preferred over Bulk excitation (to get high SH intensity). Or, from c, if u=+1 (positive wave-vector mismatch), focusing beyond the front surface (e.g. 3 Rayleigh lengths deep) will yield (relatively) high SH amplitude (see the dark circles). The dashed line in a marks one of several anti-resonant window thicknesses at which SH waves are practically not generated. For all maps - note the differences in the color scales.

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3. Amplitude integral for Gaussian-excited third harmonic waves

The third harmonic amplitude integral J3(ΔkH,zR,z1,z2) ([14], Eq. (2).10.11b with small notation changes) is given as:

J3(ΔkH,zR,z1,z2)=z1z2eiΔkHzdz(1+iz/zR)2;ΔkH3kH(ωf)kH(3ωf)
In a dimensionless form, the amplitude integral of Eq. (8) is written as –

J3(zR,u,ζ1,ζ2)=zRζ1ζ2eiuζ(1+iζ)2dζ

As it turns out, divided into purely real and purely imaginary integrals, the amplitude integral of the third harmonic wave of Eq. (9) also has an analytic solution in terms of the i(u,ζ) and Si(u,ζ) functions.

Bulk TH amplitude and Surface TH amplitude are defined as in Eq. (5) with J3 replacingJ2. For infinite integration interval (ζ2=), the TH Bulk integral evaluates as [12,16]:

J3bulk(zR,u,)={0u02πzRueuu>0

As for the Surface integral, given an infinite medium, we find for perfect wave-vector match (u=0) (cf. (Fig. 5)):

 

Fig. 5 Theoretical TH generated amplitude (absolute value of in units of μm) vs. wave-vector mismatch, for Gaussian excitation. a. Window of infinite thickness. The TH Bulk amplitude peaks at u=1 (not at u=0) with a value of (2π/e)zR2.3zR (Eq. (10)). b. Window of finite thickness. The parameter LwN is a normalized window thickness (cf. Equation (7)). Note that we show here amplitudes (absolute value of), not intensities.

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J3surf(zR,u=0,)=zR

For positive wave-vector mismatch ΔkH (associated with anomalous dispersion), the Bulk TH amplitude integral peaks at u=1 giving a max value of (2π/e)zR2.3zR (Fig. 5). Compare this max of 2.3zR for the TH amplitude integral with the 2πzR for the max SH amplitude integral (at u=0+).

Graphical illustrations for TH intensities are given by Figs. 5-7. An important difference between SH intensities and TH intensities is the location of maximum intensity with respect to the wave-vector mismatch. While for SH waves and thick windows, maximum intensity occurs at perfect wave-vector match (u=0, Fig. 2(a)), TH wave intensity peaks at small positive wave-vector mismatch (u=1,[14] - Fig. 2.10.2, and Fig. 5(a)).

 

Fig. 6 Theoretical TH generated intensity (in units of μm2) vs. integration range (“window thickness”), for Gaussian excitation. a. u=4. The curves show here too that even for negative and relatively large wave-vector mismatch, TH waves are still generated at Surface excitation and even at Bulk excitation if the window is “thin” (say on the order of one Rayleigh length). b.u=0.5. At near-perfect wave-vector match, high intensities are generated for both Surface and Bulk configurations if the window is thin. c.u=+1. Optimum wave-vector mismatch for maximum TH intensity. Note the domination of Bulk over Surface for a “small” positive wave-vector mismatch (compare with the curves of Fig. 5).

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Fig. 7 Theoretical TH generated intensity (in units of μm2) vs. integration limits (“window thickness” and position of the plane of focus, cf. Figure 6), for Gaussian excitation. Wave-vector mismatch is a:u=4. b:u=0.1. c:u=+1. These maps can assist in designing experimental set-ups for maximizing TH intensity generation. For example, if u=0.1 (negative wave-vector mismatch, associated with normal dispersion), and the window is thin (one Rayleigh length) then focusing midway between surfaces (Bulk) will yield (relatively) high intensity with less sensitivity to exact focal-plane position (see the black dot in b). The map in c is calculated for the optimum TH wave-vector mismatch (u=1) (to maximize the intensity of generated TH). In this optimal wave-vector mismatch case, focusing the exciting fundamental Gaussian beam around four Rayleigh lengths away from the front window surface yields highest TH intensities with little sensitivity to the total thickness of the window (dashed vertical line). Along the dashed line in a (constant anti-resonant window thickness), TH waves are practically not generated. For all maps - note the difference in the color scales.

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4. Measured intensity oscillations of third harmonic waves

Third harmonic generation was studied in Soda-lime glass by focusing 1550nm short pulses of about 1picosecond in duration and around one µJ each at a repetition rate of 500kHz. The glass target was shaped as a wedged window, about 80 µm at its thinnest end, to allow easytesting of sample thickness effects (Fig. 8(a)). The linearly polarized laser beam was focused by a X10 objective and generated a beam with an estimated Rayleigh range (from measured input beam diameter and objective’s focal length) to be about 100µm. Generated green-color TH pulses followed the pump polarization and were conveniently visible at a wavelength of 517nm. TH intensity was measured using a standard power meter (Ophir LaserStar) after collimating the output beam and filtering it by a low-pass filter and a prism that completely eliminated the pump beam.

 

Fig. 8 a. Soda-lime wedge. Thickness of the thin edge is about 80μm. Translation of the wedge 100μm adds 4μm to its thickness (slope of 0.04). b. TH power vs. pump power. Dashed curve is (pump-power)3, adjusted to match a single (highest) point. Recorded TH generation efficiency at the highest point is ~2.10-6.

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In Fig. 8(b) we show the power dependence of the TH that, as expected, follows a cubic power-law (dashed curve adjusted only at a single data point). The highest conversion efficiency (PTH/Ppump) was measured to be ~2.10-6.

Our main experimental result is displayed by the curves of Fig. 9(a). To generate these curves we manually translated the wedge twice in the x-direction and recorded each time the detected TH power vs. the x position. One time with the exciting beam focused midway between the two wedge surfaces (Bulk), and a second time with the exciting beam focused at the front surface of the wedge (Surface). The measured curves of Fig. 9(a) are juxtaposed with the calculated curves of Fig. 9(b) (Eq. (9)). While generally showing good measured-calculated agreement, two differences are noted. First – low relative intensity of the measured Surface curve as compared with the Bulk wave. We have no explanation to this difference. Second – the Bulk curve does not go down all the way to zero at the minima. We attribute this deviation from theory to the asymmetry of the focused (aberrated) fundamental beam (axial asymmetry with respect to the focal plane – cf. Figure 10).

 

Fig. 9 Intensity oscillations of generated TH waves with increasing window thickness. a. Measured. b. Calculated (Eq. (9)).

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Fig. 10 Cross-sectional images of generated TH beams. a. Typical image of a TH beam generated with the fundamental Gaussian beam focused nearly anywhere in the xz plane of the thin wedge. b. Low intensity image of the TH beam consistently appearing at the Bulk minima. Theoretically should be all black (zero). We attribute the incomplete ring-shaped constructive-destructive interference to the z-asymmetry of the fundamental focused-Gaussian beam (with respect to the focal plane). In thick windows (thicker than about 10 Rayleigh lengths), the situation is reversed: weak ring-shaped images are generated with the Gaussian beam focused practically anywhere except near the surfaces.

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For the Soda-lime wedge, given the measured period of intensity oscillations (first few periods), we estimate the wave-vector mismatch to beΔkH=0.249μm1. This estimate is close to a calculated value of 0.243μm1 based on tabulated data [21], particularly in view of the fact that Soda-lime refractive index varies greatly with its chemical composition [22].

Regarding the relations between the period of intensity oscillations and the wave-vector mismatch, our simulations show that with very thin windows (about one Rayleigh length), the relations are nearly (but not exactly)|ΔkH|=2π/period. In fact, we find for thin windows and glass-typical negative wave-vector mismatch:

|ΔkH|<˜2πperiod;LwN1

Figure 10 displays two images of the generated TH beam cross-section. Figure 10(a) shows the cross-section of beams that are generated with the fundamental Gaussian beam focused nearly everywhere within the glass wedge (with large intensity variations). Figure 10(b) shows the (rather low-intensity) beam cross-section generated with the fundamental Gaussian beam focused mid-way between the wedge surfaces, at the Bulk minima. The ring-shaped image is an indication for an incomplete destructive interference. We attribute the incomplete interference (predicted to be complete by the theory - Fig. 9(b)) to the axial asymmetry of the (aberrated) fundamental Gaussian beam.

5. Summary

Geometrical parameters affecting intensities of Gaussian-excited harmonic waves are included in the theory by the amplitude integral (Eq. (1) and Eq. (8) for SH and TH, respectively). We derived analytic solutions to the SH and TH amplitude integrals. With these solutions in place we generated curves and maps enabling a closer look at the influence of geometry parameters on generated harmonics intensities. Such closer look is particularly important for harmonics generation in thin windows. We looked specifically at intensity oscillations in thin normal-dispersion medium characterized by a negative wave-vector mismatch. We showed that even with such negative wave-vector mismatch, high intensity harmonic waves can be generated by thin windows. In fact, looking at the integration limits maps (Fig. 4 and Fig. 7), one can select the excitation geometry to maximize the generated harmonic intensity. Essentially, the maps show that for both SH and TH, focusing the exciting Gaussian somewhere beyond the front surface will maximize the generated intensity and show less intensity-sensitivity to window thickness.

On the experimental front, we faithfully reproduced TH intensity oscillations, as predicted by the amplitude integral. From the period of the first few oscillations (with increasing window thickness) the characteristics wave-vector mismatch can be accurately estimated. For Soda-lime material we estimated a wave-vector mismatch ofΔkH=0.249μm1. From tabulated data for the refractive index of Soda-lime at 1551nm and at 517nm we calculatedΔkH=0.243μm1. Our estimation from the experimental data is close to the number calculated based on the tabulated data, and seems to be the correct one in view of the “cleanliness” of the experiment and in view of the uncertainty in chemical composition of Soda-lime glasses (and hence uncertainty in refractive index).

As intensities of Gaussian-excited SH and TH waves are of interest in a number of applications, a detailed account for the influence of the geometrical parameters on the generated intensities could assist in optimizing the harmonic-excitation configuration, for example, in short pulse measurement applications. In microscopy, on the other hand, it is important to properly evaluate the effective depth which is sampled, which could be significantly larger than one confocal parameter that is often naively assumed.

Acknowledgments

This work was supported by Icore (Israeli centers of research excellence of the ISF), the Crown Photonics Center and the European ICT project FAMOS.

References and links

1. K. Isobe, W. Watanabe, and K. Itoh, “Functional Imaging by Controlled Nonlinear Optical Phenomena, 1st ed. (Wiley, 2013).

2. F. S. Pavone and P. J. Campagnola, Second Harmonic Generation Imaging (CRC Press, 2013).

3. P. Friedl, K. Wolf, U. H. von Andrian, and G. Harms, “Biological Second and Third Harmonic Generation Microscopy,” Curr. Protoc. Cell Biol. 2007 (Chapter 4, Unit 4.15).

4. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337(6207), 519–525 (1989). [CrossRef]  

5. F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012). [CrossRef]   [PubMed]  

6. N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013). [CrossRef]  

7. T. Y. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995). [CrossRef]   [PubMed]  

8. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997). [CrossRef]  

9. D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express 5(8), 169–175 (1999). [CrossRef]   [PubMed]  

10. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, 2000).

11. B. Ya. Zel’dovich, “Theory of Second-Harmonic Generation of Light in Focused Beams,” Sov. Phys. JETP 23, 451 (1966).

12. S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23(10), 2134–2141 (2006).

13. N. Olivier and E. Beaurepaire, “Third-harmonic generation microscopy with focus-engineered beams: a numerical study,” Opt. Express 16(19), 14703–14715 (2008). [CrossRef]   [PubMed]  

14. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Ch. 2.

15. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968). [CrossRef]  

16. J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185(1), 57–72 (1969). [CrossRef]  

17. Wolfram Alpha”:http://mathworld.wolfram.com/CosineIntegral.html http://mathworld.wolfram.com/SineIntegral.html

18. Wolfram Alpha”:http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/introductions/ExpIntegrals/ShowAll.html

19. R. Asby, “Theory of resonant optical second-harmonic generation from a focused gaussian beam,” Phys. Rev. 187(3), 1070–1076 (1969). [CrossRef]  

20. J. E. Bjorkholm, “Optical second-harmonic generation using a focused gaussian laser beam,” Phys. Rev. 142(1), 126–136 (1966). [CrossRef]  

21. Optical constants of Soda lime glass”:http://refractiveindex.info/?shelf=glass&book=soda-lime&page=Rubin-clear

22. C. A. Faick and A. N. Finn, “The index of refraction of some soda-lime-silica glasses as a function of the composition,” J. Am. Ceram. Soc. 14(7), 518–528 (1931). [CrossRef]  

References

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  1. K. Isobe, W. Watanabe, and K. Itoh, “Functional Imaging by Controlled Nonlinear Optical Phenomena, 1st ed. (Wiley, 2013).
  2. F. S. Pavone and P. J. Campagnola, Second Harmonic Generation Imaging (CRC Press, 2013).
  3. P. Friedl, K. Wolf, U. H. von Andrian, and G. Harms, “Biological Second and Third Harmonic Generation Microscopy,” Curr. Protoc. Cell Biol. 2007 (Chapter 4, Unit 4.15).
  4. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337(6207), 519–525 (1989).
    [Crossref]
  5. F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012).
    [Crossref] [PubMed]
  6. N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
    [Crossref]
  7. T. Y. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995).
    [Crossref] [PubMed]
  8. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
    [Crossref]
  9. D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express 5(8), 169–175 (1999).
    [Crossref] [PubMed]
  10. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, 2000).
  11. B. Ya. Zel’dovich, “Theory of Second-Harmonic Generation of Light in Focused Beams,” Sov. Phys. JETP 23, 451 (1966).
  12. S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23(10), 2134–2141 (2006).
  13. N. Olivier and E. Beaurepaire, “Third-harmonic generation microscopy with focus-engineered beams: a numerical study,” Opt. Express 16(19), 14703–14715 (2008).
    [Crossref] [PubMed]
  14. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Ch. 2.
  15. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
    [Crossref]
  16. J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185(1), 57–72 (1969).
    [Crossref]
  17. “Wolfram Alpha”: http://mathworld.wolfram.com/CosineIntegral.html http://mathworld.wolfram.com/SineIntegral.html
  18. “Wolfram Alpha”: http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/introductions/ExpIntegrals/ShowAll.html
  19. R. Asby, “Theory of resonant optical second-harmonic generation from a focused gaussian beam,” Phys. Rev. 187(3), 1070–1076 (1969).
    [Crossref]
  20. J. E. Bjorkholm, “Optical second-harmonic generation using a focused gaussian laser beam,” Phys. Rev. 142(1), 126–136 (1966).
    [Crossref]
  21. “Optical constants of Soda lime glass”: http://refractiveindex.info/?shelf=glass&book=soda-lime&page=Rubin-clear
  22. C. A. Faick and A. N. Finn, “The index of refraction of some soda-lime-silica glasses as a function of the composition,” J. Am. Ceram. Soc. 14(7), 518–528 (1931).
    [Crossref]

2013 (1)

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

2012 (1)

F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012).
[Crossref] [PubMed]

2008 (1)

2006 (1)

1999 (1)

1997 (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

1995 (1)

T. Y. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995).
[Crossref] [PubMed]

1989 (1)

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337(6207), 519–525 (1989).
[Crossref]

1969 (2)

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185(1), 57–72 (1969).
[Crossref]

R. Asby, “Theory of resonant optical second-harmonic generation from a focused gaussian beam,” Phys. Rev. 187(3), 1070–1076 (1969).
[Crossref]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
[Crossref]

1966 (2)

J. E. Bjorkholm, “Optical second-harmonic generation using a focused gaussian laser beam,” Phys. Rev. 142(1), 126–136 (1966).
[Crossref]

B. Ya. Zel’dovich, “Theory of Second-Harmonic Generation of Light in Focused Beams,” Sov. Phys. JETP 23, 451 (1966).

1931 (1)

C. A. Faick and A. N. Finn, “The index of refraction of some soda-lime-silica glasses as a function of the composition,” J. Am. Ceram. Soc. 14(7), 518–528 (1931).
[Crossref]

Ajayan, P. M.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

Asby, R.

R. Asby, “Theory of resonant optical second-harmonic generation from a focused gaussian beam,” Phys. Rev. 187(3), 1070–1076 (1969).
[Crossref]

Barad, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Beaurepaire, E.

Bjorkholm, J. E.

J. E. Bjorkholm, “Optical second-harmonic generation using a focused gaussian laser beam,” Phys. Rev. 142(1), 126–136 (1966).
[Crossref]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
[Crossref]

Carrasco, S.

Ceballos, F.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

Cicchi, R.

F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012).
[Crossref] [PubMed]

Cui, Q.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

Eisenberg, H.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Faick, C. A.

C. A. Faick and A. N. Finn, “The index of refraction of some soda-lime-silica glasses as a function of the composition,” J. Am. Ceram. Soc. 14(7), 518–528 (1931).
[Crossref]

Finn, A. N.

C. A. Faick and A. N. Finn, “The index of refraction of some soda-lime-silica glasses as a function of the composition,” J. Am. Ceram. Soc. 14(7), 518–528 (1931).
[Crossref]

Fourkas, J. T.

Horowitz, M.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
[Crossref]

Kumar, N.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

Lou, J.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

Najmaei, S.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

New, G. H. C.

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185(1), 57–72 (1969).
[Crossref]

Olivier, N.

Pavone, F. S.

F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012).
[Crossref] [PubMed]

Sacconi, L.

F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012).
[Crossref] [PubMed]

Saleh, B. E. A.

Shen, Y. R.

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337(6207), 519–525 (1989).
[Crossref]

Silberberg, Y.

D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express 5(8), 169–175 (1999).
[Crossref] [PubMed]

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

Teich, M. C.

Tsang, T. Y.

T. Y. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995).
[Crossref] [PubMed]

Vanzi, F.

F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012).
[Crossref] [PubMed]

Ward, J. F.

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185(1), 57–72 (1969).
[Crossref]

Yelin, D.

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, “Theory of Second-Harmonic Generation of Light in Focused Beams,” Sov. Phys. JETP 23, 451 (1966).

Zhao, H.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

Appl. Phys. Lett. (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70(8), 922–924 (1997).
[Crossref]

J. Am. Ceram. Soc. (1)

C. A. Faick and A. N. Finn, “The index of refraction of some soda-lime-silica glasses as a function of the composition,” J. Am. Ceram. Soc. 14(7), 518–528 (1931).
[Crossref]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
[Crossref]

J. Biomed. Opt. (1)

F. Vanzi, L. Sacconi, R. Cicchi, and F. S. Pavone, “Protein conformation and molecular order probed by second-harmonic-generation microscopy,” J. Biomed. Opt. 17(6), 060901 (2012).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

Nature (1)

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337(6207), 519–525 (1989).
[Crossref]

Opt. Express (2)

Phys. Rev. (3)

R. Asby, “Theory of resonant optical second-harmonic generation from a focused gaussian beam,” Phys. Rev. 187(3), 1070–1076 (1969).
[Crossref]

J. E. Bjorkholm, “Optical second-harmonic generation using a focused gaussian laser beam,” Phys. Rev. 142(1), 126–136 (1966).
[Crossref]

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185(1), 57–72 (1969).
[Crossref]

Phys. Rev. A (1)

T. Y. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995).
[Crossref] [PubMed]

Phys. Rev. B (1)

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS2,” Phys. Rev. B 87(16), 161403 (2013).
[Crossref]

Sov. Phys. JETP (1)

B. Ya. Zel’dovich, “Theory of Second-Harmonic Generation of Light in Focused Beams,” Sov. Phys. JETP 23, 451 (1966).

Other (8)

K. Isobe, W. Watanabe, and K. Itoh, “Functional Imaging by Controlled Nonlinear Optical Phenomena, 1st ed. (Wiley, 2013).

F. S. Pavone and P. J. Campagnola, Second Harmonic Generation Imaging (CRC Press, 2013).

P. Friedl, K. Wolf, U. H. von Andrian, and G. Harms, “Biological Second and Third Harmonic Generation Microscopy,” Curr. Protoc. Cell Biol. 2007 (Chapter 4, Unit 4.15).

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, 2000).

“Wolfram Alpha”: http://mathworld.wolfram.com/CosineIntegral.html http://mathworld.wolfram.com/SineIntegral.html

“Wolfram Alpha”: http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/introductions/ExpIntegrals/ShowAll.html

“Optical constants of Soda lime glass”: http://refractiveindex.info/?shelf=glass&book=soda-lime&page=Rubin-clear

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Ch. 2.

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Figures (10)

Fig. 1
Fig. 1 Harmonics generation geometry. a. Gaussian beam parameters. b. General position of the focal plane of the harmonics exciting Gaussian beam (with respect to the surfaces of the harmonics-generating window). c. Surface excitation geometry. d. Bulk excitation geometry. Note that we define the waist location as z=0 , a distance measured to the left of the waist gets a minus sign.
Fig. 2
Fig. 2 Theoretical SH generated amplitude (absolute value of in units of μm) vs. wave-vector mismatch, for Gaussian excitation. a. Window of infinite thickness. Mathematically, the Bulk amplitude, for window of infinite thickness, is discontinuous at perfect wave-vector match ( u=0 , cf. Equation (12)). And mathematically, for window thickness increased to infinity, the Surface amplitude goes (very “slowly”) to infinity at perfect wave-vector match. b. Window of finite thickness. The parameter L wN is a normalized window thickness (cf. Equation (7)). Note that we show here amplitudes (absolute value of), not intensities.
Fig. 3
Fig. 3 Theoretical SH generated intensity (in units of μm2) vs. integration range (“window thickness”), for Gaussian excitation. a. u=4 . The curves show that even for negative and relatively large wave-vector mismatch, SH waves are still generated at Surface excitation and even at Bulk excitation if the window is “thin” (say on the order of one Rayleigh length). b. u=1 . At near perfect wave-vector match, high intensities are generated for both Surface and Bulk configurations if the window is thin. c. u=+1 . Note the domination of Bulk over Surface for small positive wave-vector mismatch (compare with the curves of Fig. 2).
Fig. 4
Fig. 4 Theoretical SH generated intensity (in units of μm2) vs. integration limits (“window thickness” and position of the plane of focus), for Gaussian excitation. The parameter ζ 1 is the distance from the focal plane to the front surface of the window (including sign) in units of z R . Similarly for ζ 2 and the back surface. Wave-vector mismatch is a: u=4 . b: u=0.1 . c: u=+1 . These maps can assist in designing experimental set-ups for max SH intensity generation. For example, the map in a indicates that for negative wave-vector mismatch, if the window is thick (say 4 z R or more), then Surface excitation is far preferred over Bulk excitation (to get high SH intensity). Or, from c, if u=+1 (positive wave-vector mismatch), focusing beyond the front surface (e.g. 3 Rayleigh lengths deep) will yield (relatively) high SH amplitude (see the dark circles). The dashed line in a marks one of several anti-resonant window thicknesses at which SH waves are practically not generated. For all maps - note the differences in the color scales.
Fig. 5
Fig. 5 Theoretical TH generated amplitude (absolute value of in units of μm) vs. wave-vector mismatch, for Gaussian excitation. a. Window of infinite thickness. The TH Bulk amplitude peaks at u=1 (not at u=0 ) with a value of (2π/e) z R 2.3 z R (Eq. (10)). b. Window of finite thickness. The parameter L wN is a normalized window thickness (cf. Equation (7)). Note that we show here amplitudes (absolute value of), not intensities.
Fig. 6
Fig. 6 Theoretical TH generated intensity (in units of μm2) vs. integration range (“window thickness”), for Gaussian excitation. a. u=4 . The curves show here too that even for negative and relatively large wave-vector mismatch, TH waves are still generated at Surface excitation and even at Bulk excitation if the window is “thin” (say on the order of one Rayleigh length). b. u=0.5 . At near-perfect wave-vector match, high intensities are generated for both Surface and Bulk configurations if the window is thin. c. u=+1 . Optimum wave-vector mismatch for maximum TH intensity. Note the domination of Bulk over Surface for a “small” positive wave-vector mismatch (compare with the curves of Fig. 5).
Fig. 7
Fig. 7 Theoretical TH generated intensity (in units of μm2) vs. integration limits (“window thickness” and position of the plane of focus, cf. Figure 6), for Gaussian excitation. Wave-vector mismatch is a: u=4 . b: u=0.1 . c: u=+1 . These maps can assist in designing experimental set-ups for maximizing TH intensity generation. For example, if u=0.1 (negative wave-vector mismatch, associated with normal dispersion), and the window is thin (one Rayleigh length) then focusing midway between surfaces (Bulk) will yield (relatively) high intensity with less sensitivity to exact focal-plane position (see the black dot in b). The map in c is calculated for the optimum TH wave-vector mismatch ( u=1 ) (to maximize the intensity of generated TH). In this optimal wave-vector mismatch case, focusing the exciting fundamental Gaussian beam around four Rayleigh lengths away from the front window surface yields highest TH intensities with little sensitivity to the total thickness of the window (dashed vertical line). Along the dashed line in a (constant anti-resonant window thickness), TH waves are practically not generated. For all maps - note the difference in the color scales.
Fig. 8
Fig. 8 a. Soda-lime wedge. Thickness of the thin edge is about 80μm. Translation of the wedge 100μm adds 4μm to its thickness (slope of 0.04). b. TH power vs. pump power. Dashed curve is (pump-power)3, adjusted to match a single (highest) point. Recorded TH generation efficiency at the highest point is ~2.10-6.
Fig. 9
Fig. 9 Intensity oscillations of generated TH waves with increasing window thickness. a. Measured. b. Calculated (Eq. (9)).
Fig. 10
Fig. 10 Cross-sectional images of generated TH beams. a. Typical image of a TH beam generated with the fundamental Gaussian beam focused nearly anywhere in the xz plane of the thin wedge. b. Low intensity image of the TH beam consistently appearing at the Bulk minima. Theoretically should be all black (zero). We attribute the incomplete ring-shaped constructive-destructive interference to the z-asymmetry of the fundamental focused-Gaussian beam (with respect to the focal plane). In thick windows (thicker than about 10 Rayleigh lengths), the situation is reversed: weak ring-shaped images are generated with the Gaussian beam focused practically anywhere except near the surfaces.

Equations (12)

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J 2 (Δ k H , z R , z 1 , z 2 )= z 1 z 2 e iΔ k H z dz 1+iz/ z R ;Δ k H 2 k H ( ω f ) k H (2 ω f ) k 0 ( ω f )= ω f c ; k H ( ω f )= k 0 ( ω f )n( ω f ); k H (2 ω f )=2 k 0 ( ω f )n(2 ω f )
z R ( ω f , w 0 ) 1 2 k 0 ( ω f )n( ω f ) w 0 2
ζ z z R ; ζ 1 z 1 z R ; ζ 2 z 2 z R ;Δ k H z R u
J 2 ( z R ,u, ζ 1 , ζ 2 )= z R ζ 1 ζ 2 e iuζ 1+iζ dζ
J 2bulk ( z R ,u, ζ 2 ) J 2 ( z R ,u, ζ 2 , ζ 2 ) J 2surf ( z R ,u, ζ 2 ) J 2 ( z R ,u,0,2 ζ 2 )
J 2bulk ( z R ,u,)={ 0 u<0 π z R 2π z R e u u=0 u>0
L wN ( z 2 z 1 )/ z R = ζ 2 ζ 1
J 3 (Δ k H , z R , z 1 , z 2 )= z 1 z 2 e iΔ k H z dz (1+iz/ z R ) 2 ;Δ k H 3 k H ( ω f ) k H (3 ω f )
J 3 ( z R ,u, ζ 1 , ζ 2 )= z R ζ 1 ζ 2 e iuζ ( 1+iζ ) 2 dζ
J 3bulk ( z R ,u,)={ 0 u0 2π z R u e u u>0
J 3surf ( z R ,u=0,)= z R
| Δ k H | < ˜ 2π period ; L wN 1

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