Second Harmonic Femtosecond Pulse Generation in Nonlinear

Chapter 35

Downloaded by NORTH CAROLINA STATE UNIV on October 14, 2012 | http://pubs.acs.org Publication Date: August 11, 1995 | doi: 10.1021/bk-1995-0601.ch035

Second Harmonic Femtosecond Pulse Generation in Nonlinear Polymer Thin-Film Structures Andrew Dienes, Erkin Sidick, Richard A. Hill, and AndréKnoesen Department of Electrical and Computer Engineering, University of California, Davis, CA 95616

Ultrashort-pulse second harmonic applications of nonlinear poled polymeric films (NPPF) and their optimization in terms of nonlinear chromophore concentration and thin film structure are discussed. The objective of such optimization is to maximize the conversion efficiency and, more importantly, minimize the pulse distortions in ultrashort -pulsesecond harmonic generation. It is shown that each NPPF has a specific optimum chromophore concentration density. It is also shown that the quasi-phase-matched structures of NPPFs not only enhance the efficiency of ultrashort-pulse second harmonic generation but also minimize pulse distortions.

During the last decade the potential of polymeric thin films for applications in nonlinear optics has received considerable attention. In particular, second-order nonlinear processes, which include second harmonic generation (SHG), two wave mixing and electro-optic modulation, have been extensively investigated. It has been demonstrated that polymeric thin films with large second-order nonlinearities can be created by permanently orienting molecular components with a large hyperpolarizability within a polymer host. The magnitudes of the nonlinearities can exceed those of inorganic second-order materials. In addition, the refractive index of organic polymers is low, a great advantage for electro-optic applications. Nonlinear poled polymeric films (NPPFs) have other attractive properties for SHG. Perhaps the most important of these is the flexibility which exists through both chemical synthesis and through the creation of multilayer structures for optimization of various physical properties for specific device requirements. One broad class of applications for NPPFs is guided wave nonlinear optics. Hie creation of nonlinear guided wave structures from these films for use in continuous-wave (cw) or quasi-cw SHG of optical waves has been explored recently (i). For these applications the main challenge lies in reducing the losses which severely limit the usable waveguide length. NPPFs are uniquely suited for ultrashort-pulse nonlinear optical applications (2). Ultrashort optical pulses as short as 10 fs can now be generated directly from modelocked lasers (5) and generation of new frequencies by nonlinear optical means is of great interest. In these applications the optical waves are not guided by the film but

0097-6156/95/0601-0484$12.00/0 © 1995 American Chemical Society In Polymers for Second-Order Nonlinear Optics; Lindsay, G., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995.


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485 Second Harmonic Femtosecond Pulse Generation

must instead propagate through the film and the interaction length is on the order of microns. The short interaction length has the important advantage of reducing the pulse distortions caused by the frequency dispersion of linear refractive index. To avoid these distortions in inorganic nonlinear crystals, very thin materials must be used. This is particularly true if the center wavelength of the second harmonic (SH) pulse is in the UV. For pulses in the few tens of femtosecond range the required thickness is so small (on the order of 50 pm or less) that the crystals are difficult if not impossible to fabricate and polish (4). In contrast, single and multilayer NPPFs with excellent optical quality can be made in thicknesses ranging from 1 Jim to 30 pm with relative ease, making NPPF very attractive for ultrashort pulse second-order nonlinear optics in general and for ultrashort pulse second harmonic generation (USP-SHG) in particular. Although a short interaction length limits the efficiency, the second order nonlinear coefficient (^-coefficient) of NPPFs can be large enough that this is not a severe restriction. In this chapter we will first review and explain in simple terms the important issues in USP-SHG using nonlinear polymeric films, focusing on the effects that influence the generated pulse width and the SHG efficiency. Following that, experimental results using a single layer NPPF will be briefly reviewed. We will then present some recent theoretical results on the optimization of the pulse shape and the efficiency in single-layer and multilayer quasi-phase-matched (QPM) structures of NPPF. Limitations due to optical damage will also be briefly discussed. Although we will focus on SHG, it is obvious that the discussion and results are also relevant to the more general cases of sum and difference frequency generation as well. Factors influencing pulse width and efficiency in USP-SHG Although a complete theory of USP-SHG in NPPF is analytically complex, all of the important issues can be understood through simple physical arguments. The source term "driving" the generation of the electric field at the second harmonic (SH) frequency 2 co is the polarization P2©. This polarization is proportional to an effective nonlinear ^-coefficient and to the square of die fundamental electric field. Thus the SH intensity (power density) generated is proportional to the square of the effective dcoefficienL The basic geometry is illustrated in Figure 1. The ^-coefficient is actually a tensor, but its dominant component coincides with the poling direction, which is in the present case perpendicular to the film plane. For SHG, the fundamental field must have component along these nonlinear dipoles and therefore the beam must be incident at an angle to the normal as shown and p polarized. Conversion efficiency is maximized near Brewster angle, which minimizes the Fresnel losses. The nonzero angle of propagation in the medium adds analytical complexity to the problem but does not alter the basic effects and their physical understanding. Therefore, in the subsequent discussions the nonzero propagation angle will be ignored. Since the SH intensity is proportional to the fourth power of the fundamental field (i.e., the square of the fundamental power density), strong focusing is needed to obtain practical efficiency with short interaction lengths. One important simplification is that, since the interaction length is short, beam diffraction effects can be ignored. Clearly, SHG efficiency is maximized by achieving the largest possible ^-coefficient and by using the tightest possible focusing. The first involves both "engineering" the active chromophore for highest hyperpolarizability and optimizing the alignment of the nonlinear dipoles (poling). Practical limitations to these processes exist but will not be discussed here. The minimum focused beam diameter is limited by optical fundamentals to the order of a few wavelengths but damage to the nonlinear material may pose a more severe limit. This issue will be addressed later in this chapter. The various effects caused by the frequency dispersion of refractive index, as well as those caused by linear absorption losses, must be considered. These effects interact in a complicated fashion in a nonlinear material, but before trying to account for their

In Polymers for Second-Order Nonlinear Optics; Lindsay, G., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

486

POLYMERS FOR SECOND-ORDER NONLINEAR OPTICS

complex interplay, it is important to obtain a basic physical picture for each of them separately. All the dispersive effects are due to the propagation of the optical electromagnetic fields in the nonlinear material. This propagation is described by exp[y(Ctyf-£|Z)], where the propagation constant at frequency CD/ is k=(Oin((Oi)/c, the phase velocity is Vi=c/n((Oi) and i"=7 is for the fundamental and 2 for the SH. Because of material dispersion, normally n(0)7)^0)2). A fundamental effect of this refractive index dispersion is the phase-mismatch which is important in both cw- and USP-SHG. Linear absorption results in a complex propagation constant, and its effects are also present in both cw- and USP-SHG.


9

Phase mismatch. The polarization of the nonlinear medium, is proportional to the square of the fundamental field E((oi) so it propagates with the propagation constant 2kj. But the SH field generated by this polarization propagates with the propagation constant fe. Normally 2ki*ki because flfcty^flfGfc), and thus a phase difference of Mz={k2-2ki)z exists between the SH fields generated at planes separated by a distance z. If this phase difference equals rc, exact cancellation occurs. The corresponding distance L =7tf M l is called the nonlinear coherence length. Integration over z of the exp(/A/z) phase mismatch gives the dependence of the total SH intensity on the nonlinear interaction length t

c

F(L)

=

I?

sinAfcL/2

(1)

AkL/2

If A k is non-zero, only one coherence length effectively contributes to the SHG, and the SH intensity oscillates with length as shown with a solid-curve in Figure 2. If Ak can be made zero, a condition called phase matching, then the SH intensity builds up as L . Some inorganic crystals can be phase-matched by exploiting the anisotropy of their refractive indices. Unfortunately, NPPFs cannot be phase-matched, due to their particular symmetry properties, and phase mismatch will always be present in a single layer film. Typically, coherence lengths are on the order of a few microns, which are also the typical thicknesses of spin-coated NPPFs. Such short interaction lengths preclude the use of NPPFs for cw-SHG in non-guided configurations. Ultrashort pulses, however, have extremely high power densities and even such thin films can give practical efficiency for some applications. In fact, such a short nonlinear interaction length is another important advantage of NPPFs because ultra-thin films minimize pulse distortions in ultrashort-pulse nonlinear optical applications. 2

Absorption loss. Nonlinear organic materials often have absorption in the visible, near UV or near IR wavelengths. Such resonances have several effects on USP-SHG. One of these is the presence of linear absorption losses at either the fundamental or at the SH wavelength. Another is the modification of the refractive index through the well known Kramers-Kronig relations. (The real and imaginary parts of a resonant susceptibility contribute to the linear refractive index and to the extinction coefficient, respectively). This resonant index contribution can be exploited for a quasi-phasematching effect by varying the active chromophore concentration. Changing the concentration, however, also changes the ^-coefficient, the absorption loss, and other parameters. Here, we consider the effects of the linear loss. Because of the high fundamental powers involved, even a very small absorption at the fundamental wavelength inevitably results in damage to an NPPF. Therefore, only absorption at the SH wavelength can be tolerated, and the NPPF is useful only if this absorption is small. Analysis of SHG in the presence of a linear loss at the SH wavelength introduces the physically obvious modification of replacing the real propagation constant ki by the complex equivalent &2-./CC2, 2 is the absorption coefficient, assumed w

n

e

r

e

a


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constant over the SH bandwidth. This ja.2 term then appears in the phase mismatch function given in equation 1, which is modified to F(L) = Z , V 2


a

+jAk)L/2 jAk)L/2

t

L

r-

(2)

The effect is to limit the efficiency and to damp out the oscillatory dependence with length as shown with a dashed-curve in Figure 2 . If the value of (X2 is sufficiently small, the efficiency is not greatly reduced in a single layer of - 1 pm thickness. If (X2 is not constant over the SH bandwidth (loss dispersion), the effects are more complicated and cannot be fully described by equation 2 . Increasing the absorption at the edges of the SH spectrum will act to broaden the pulses by decreasing the bandwidth. Clearly, a broad low absorption "window" at the SH wavelength is necessary for USP-SHG. Group velocity effects. In addition to phase mismatch, the frequency dispersion of refractive index causes other more important effects specific to USP-SHG. To understand these we must consider the optical pulses in both the time domain and the frequency domain. It is well known from Fourier transform theory that a pulse modulated sinusoidal wave with a pulse width x has a finite frequency bandwidth Aa> about the carrier co , with Ac0pT «7C (5). While the sinusoidal carrier propagates with the phase velocity, the pulse envelope travels with a group velocity v . The group delay per unit length is l/v =3#3co. In a dispersive material the group velocity is different from the phase velocity, and moreover, v» itself is not constant with frequency. This is known as group velocity dispersion (oVD). p

0

p

P

g

g

Intra-pulse GVD. It is well known that GVD within the frequency bandwidth of a pulse causes pulse broadening in a linear medium, due to the different group delays experienced by the different frequency components within the bandwidth. The physically obvious measure of the broadening, over the total propagation distance L , is the group delay difference between the opposite edges of the frequency band,

This broadening is clearly present in any nonlinear medium as well, affecting the fundamental pulse as it propagates, and through it the SH (source) polarization / 2 Additionally, the SH is similarly affected during its propagation. Owing to the short interaction length in USP-SHG, however, this intra-pulse GVD (IGVD) is negligible for all but the shortest pulses ( ~ 1 0 fs) that can presently be generated. >

W

Group velocity mismatch. The frequency dependence of also results in another more important effect, i.e., inter-pulse GVD or group velocity mismatch (GVM) between the fundamental and the SH pulses. Their envelopes propagate with different velocities. Since the P2a> envelope also travels at the fundamental group velocity, a time delay of

exists between the second harmonic pulses generated at points separated by distance z, as illustrated in Figure 3. Integration of the contributions over all z gives a stretched



488


Figure 1. Schematic of a substrate-nonlinear polymer thin-film structure used in ultrashort-pulse second harmonic generation.

2

3

Thickness L/L

c

Figure 2. Normalized SH intensity versus film thickness L (in units of nonlinear coherence length Lc) calculated using equation 2. Nonlinear Medium L

Fundamental

A*

A~ A* JX ...ii 12

123

1 2

1 2 3

S H (Small G V M )

S H (Large G V M )

4~

7

V

Figure 3. Broadening of SH pulses due to the group velocity mismatch. The SH pulses numbered 1, 2 and 3 represent those generated in the input side, middle, and output side of a nonlinear medium, respectively.


35. DIENES ET AL.


out, flat topped S H pulse if Aig(L), the group delay difference over the interaction length, is large compared to the input fundamental pulsewidth as illustrated. On the other hand, if Aig(L) is negligible, the generated S H pulse is shorter than the fundamental, due to the square law dependence of the S H power on the fundamental power. The value of the shortening factor is V2 for Gaussian shape pulses. Further understanding of the effect of G V M may be gleaned from considering the process in the frequency domain. G V M arises from the different phase mismatches at different (corresponding) frequencies within the fundamental and the S H frequency bands. If we assume, for simplicity, that the center frequencies of the bands (o>oi, CO02) are phase-matched, then it is obvious that phase mismatches must exist at other frequencies. Using the Taylor expansion %(0)=k((Q W(fr(Oo)dk/dG) , where dk/dcois evaluated at C0=C0Q, at both the fundamental and the S H center frequencies we find that the phase mismatch at an arbitrary frequency within the fundamental frequency band is equal to M(CO)=Z(9*:2/3CO-9A:I/3CO)(CD-COOI) between points separated by distance z. Referring to the effects of a phase mismatch described above, it is not surprising that integration over z results in imposing a spectral filter over the S H band generated without GVM. The filter is


0

r

G(co)

=

sin[AT (L)((o-co )] \

2

g

ol

ATg(LXoo-G)oi)

(5)

The larger the value of Ax (L), the narrower the filter, and the larger the spreading of the pulse in time. Quantitatively, multiplication by the filter function in the frequency domain corresponds to a convolution of the SH pulse shape with a rectangular pulse in the time domain. This is the obvious equivalent of the result derived above by the group velocity mismatch argument. G V M is the most important limitation to USP-SHG with inorganic crystals, since typical values of the parameter Aig(L) are on the order of 1 fs per |im. Thus, even the thinnest (50-100 pin) crystals result in broadening of the SH pulses when the fundamental pulses are shorter than -50 fs. If the material is phase-matched at the carrier frequencies, then the energy conversion efficiency is not adversely affected since the carrier of the SH pulses originating from different points adds up coherently. However, the peak power conversion efficiency is obviously reduced by the same factor that the SH pulse is broadened. Thus, it is the peak power conversion efficiency that is a good measure of the performance of any medium for USP-SHG. Values of the G V M parameter, ATo(L), in NPPFs are comparable to those in inorganic crystals. Therefore, in single layer films of a few microns thicknesses all effects of G V M are negligible, and even for fundamental pulses of around 10 fs duration the generated SH pulses are shorter than the fundamental (6). SHG efficiency of a single layer NPPF, however, is low even for materials with the highest nonlinear d-coefficients. As will be shown later, the use of multilayer quasi-phase-matched (QPM) structures can eliminate the limitations of the phase mismatch in NPPF and considerably increase the conversion efficiency. But G V M is still one of the important limiting effects in these structures. If both phase mismatch and G V M are present in a long interaction length, the combination of the destructive interference and the G V M time "slippage" can result in more severe pulse distortions, such as double peaks in the SH pulse profile (7). It will be shown later that in QPM structures of NPPFs such combined effects of phase mismatch and G V M are eliminated. In the above, a simple physical description of the main factors influencing USPSHG in a single layer NPPF has been presented, with emphasis on the two most important issues, namely preservation of the ultrashort pulse width and obtaining the best possible efficiency. A comprehensive analysis, in die form of coupled nonlinear differential equations, has been developed (7) that is able to account quantitatively for g


490


the sometimes rather complex interplay of the various factors. Additionally, it describes other phenomena such as the effects of non-transform limited fundamental pulses (i.e., pulses which containfrequencydomain nonlinear phase, or equivalently have nonlinearly chirped carrierfrequency),shaping of the generated SH pulses, etc. The analysis is also applicable to USP-SHG in various multilayer QPM structures and has been used to obtain the numerical results discussed later in this chapter.


Generation of 312 nm femtosecond pulses using a single layer NPPF Before summarizing some of the results and predictions of the analysis, a brief summary of recent experimental results on the generation of ultrashort UV pulses using a single layer NPPF will be presented. The nonlinear polymer used in these experiments was a coumaromethacrylate copolymer [P(MMA-CMA)-11] (8). The spin-cast film had a thickness of 2.5 pm and a chromophore density of 9.8xl0 mol/cm . The poling conditions were optimized, and the linear and nonlinear properties of the polymer were characterized (9). The absorption spectrum shown in Figure 4 has a low absorption window around X=318 nm in fortunate coincidence with the second harmonic frequency of the well known R6G CPM dye laser. This window can pass the spectrum of pulses as short as ~5 fs (AA,~30 nm). Shorter pulses will be broadened due to attenuation at their extreme wavelengths. Additionally, the absence of any absorption at 625 nm permits the doubling of pulses with large peak powers (8). The fundamental pulses were derived from a R6G CPM dye laser, amplified by an 8 KHz repetition rate Cu-vapor laser (70) to ~13 fs duration and ~50 nJ energy at 625 nm. The configuration was the same as shown in Figure 1, with the fundamental focused to a beam diameter of 26 pm in the film. The results are shown in Figure 4. This figure shows the measured SH pulse spectrum superimposed on the absorption spectrum of P(MMA-CMA). The full width half maximum is ~19 nm, which is consistent with a pulse length of ~11 fs. The GVM broadening parameter, Aig(L), for this film was estimated to be around 1 fs. It was found that the interaction length was only about seven optical cycles which is approximately the FWHM of the fundamental pulse. These observations indicate that no broadening of the SH pulse occurred and that the SH pulsewidth is ~11 fs. The inset in Figure 4 shows the SHG efficiency vs average fundamental power (8). The dependence is linear, as expected. The highest efficiency observed was -0.13% at an input energy of 51 nJ (peak power density of 740 GW/cm ). This efficiency is high for such a thin film. However, at this power density the film was damaged and the SH signal gradually decayed. At the approximate damage threshold of 164 GW/cm (11.4 nJ) the efficiency was 0.025% . By scaling the focal spot size and using the full 51 nJ fundamental, 13pJ UV pulses were obtained. The film used in this experiment was not optimized for maximum efficiency. From the measured index at 625 nm, and a Kramers-Kronig prediction of the refractive index at 312 nm (see the next section), L was determined to be 1.7 pm. Using this and the measured (X2=2.66xl0 nr in equation 4, and accounting for the off-normal propagation direction, it was calculated that the 2.5 pm thickness of the film is longer than would be needed for maximum efficiency. The dashed-curve of SH intensity vs thickness shown on Figure 2 corresponds to the parameters of this film and the actual thickness is indicated. A factor of 2 improvement could be obtained by using a thickness of -1.7 pm for this material at this particular chromophore concentration. The next issue that will be examined is how both efficiency and pulse duration can be optimized for a particular material by changing the chromophore concentration. 20

3

2

2

c

5

1


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491

Second Harmonic Femtosecond Pulse Generation


Optimization of a NPPF for the concentration of its active chromophore In the experiments summarized in the preceding section, the fundamental and SH wavelengths were located at opposite sides of the main absorption band of the active chromophore of the NPPF. In such a case, the contribution from the real part of the resonant susceptibility causes anomalous dispersion and the index at the SH wavelength may be lower than that at the fundamental. The possibility of exploiting a trend toward anomalous-dispersion phase-matching (11,12) thus exists. By changing the concentration of the active chromophore, the coherence length can be increased and the absorption loss at the SH lowered as well. However, decreasing the concentration also lowers the ^-coefficient and, unlike in a cw guided wave configuration, the increased interaction length may not be useful due to G V M and IGVD effects. An optimum concentration exists for a particular polymeric material which maximizes the SHG efficiency of a single layer film while still preserving the ultrashort pulse width. The optimum chromophore concentration has been calculated (7) for a representative NPPF, which is P(MMA-CMA) discussed in the previous section. In the method of Ref. 7, first the refractive index and the extinction coefficient were found from the measured absorption spectrum (Figure 4) at the original concentration using a Kramers-Kronig calculation. Next, an analytical fit to the complex permittivity of the material was obtained and related to the concentration through a scaling factor x . Finally, the analytical expression of the complex permittivity was used to calculate the coherence length L , the G V M and the IGVD parameters, and (X2 vs fundamental wavelength for various values of the JC . The results for the calculation of L are shown in Figure 5(a). Parameters a ^ c and d ffL (dtffis the effective ^-coefficient) remained relatively constant for a given fundamental wavelength and are not shown. Following that, the comprehensive analysis was used to calculate the SH efficiency and pulse width, taking proper account of a number of factors not covered in this chapter, such as the optimization of the non-normal incidence on the film, etc. The results, which are given in Figure 5(b), show that there exists an optimum concentration (x =0.2) at which the efficiency is near maximum and the SH pulse is not broadened. It should be noted that this concentration is much higher than what would be needed for achieving exact anomalous-dispersion phasematching. For the particular NPPF used in these calculations, the efficiency maximum is very broad, and there is approximately a factor of 10 range of nearly optimum concentrations. Nevertheless, the molar concentration of the chromophore cannot be arbitrarily increased. For other nonlinear polymers the results may be somewhat different, but the method used for optimization is very general. c

c

c

c

Q

c

c

USP-SHG in quasi-phase-matched structures For NPPFs, which cannot be phase-matched by the conventional anisotropy method used in nonlinear inorganic crystals, the method of quasi-phase-matching (77) offers an effective way to increase the interaction length and thus the efficiency. In this method, the cancellation of the SH signal due to the n phase mismatch after one coherence length is eliminated by using multilayer structures as shown in Figure 6. Figure 6(a) is a so called Bipolar-QPM structure (B-QPM), in which the orientation of the nonlinear dipoles is reversed after each coherence length. This cancels the accumulated n phase mismatch after Lc, and the generated SHfieldsadd in phase. The efficiency of each length, L , is still reduced by the phase mismatch, but the SH generated in successive layers now adds, enhancing the conversion efficiency. Such a structure using NPPFs can be made by mechanical assembly of multiple poled layers (75). Figure 6(b) shows a Unipolar-QPM structure (U-QPM) in which every second layer of length L 2 is a linear dielectric film, thus eliminating the out of phase contributions of these layers. U-QPM is considerably easier to make. Mechanical assembly is not required, therefore some unnecessary damage to the material can be c

C



492


Figure 4. The measured spectrum of the SH pulse and the absorption spectrum of P(MMA-CMA). The inset shows the conversion efficiency as a function of the average power of the fundamental pulse. (Adapted from ref. 8).

Figure 5. (a) Nonlinear coherence length L versus the fundamental wavelength for different chromophore concentrations, (b) Conversion efficiency and normalized SH pulsewidth Tp2/tpi versus Xc for an initial fundamental peak intensity of 100 GW/cm^, where the efficiency is magnified by the indicated factor. c

(a) Blpolar-QPM

(b) Unlpolar-QPM

L

c2

L

c2

Figure 6. Schematic diagrams of Bipolar- and Unipolar-QPM structures. Each period of the Bipolar-QPM structure consists of two nonlinear layers oriented in the opposite directions. While each period of the Unipolar-QPM structure consists of one nonlinear layer and one linear layer, and all the nonlinear layers are oriented in the same direction.


35.

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493

prevented. In this case the multilayer structure can be fabricated by successively spincoating the films and poling all the nonlinear layers together. Another advantage of this structure over B-QPM is that in situ poling (14) can be used to eliminate the orientational decay of the material. It is obvious, however, that a B-QPM structure is more efficient than a U-QPM structure consisting of same number of layers. Beside optical damage, the effects of phase mismatch and absorption at the SH wavelength are the only practical limiting factors in single layer films, since IGVD is negligible and the effects of G V M can be avoided. In QPM structures, however, GVM, and even IGVD also become significant. Numerical analyses have been performed to clarify the effects of G V M and loss on both types of QPM structures, assuming L i=La=L in the U-QPM structure. The conversion efficiency and pulse distortion effects in the two types of structures were examined for a hyperbolic secant fundamental pulse shape and for some typical parameters, which correspond approximately to those of P(MMA-CMA) described previously with a concentration range 03£x

Second Harmonic Femtosecond Pulse Generation in Nonlinear

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