## Iterative processing of second-order optical nonlinearity depth profiles

Optics Express, Vol. 12, Issue 15, pp. 3367-3376 (2004)

http://dx.doi.org/10.1364/OPEX.12.003367

Acrobat PDF (374 KB)

### Abstract

We show through numerical simulations and experimental data that a fast and simple iterative loop known as the Fienup algorithm can be used to process the measured Maker-fringe curve of a nonlinear sample to retrieve the sample’s nonlinearity profile. This algorithm is extremely accurate for any profile that exhibits one or two dominant peaks, which covers a wide range of practical profiles, including any nonlinear film of crystalline or organic material (rectangular profiles) and poled silica, for which an excellent experimental demonstration is provided. This algorithm can also be applied to improve the accuracy of the nonlinearity profile obtained by an inverse Fourier transform technique.

© 2004 Optical Society of America

## 1. Introduction

1. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. **8**, 21–22 (1962). [CrossRef]

*d*(

*z*). However, since the phase of this Fourier transform is not known the Fourier transform cannot be inverted uniquely to obtain

*d*(

*z*). To overcome this difficulty, we have recently demonstrated a number of techniques where two nonlinear samples, either identical or different, are pressed against each other and the MF curve of this sandwich is measured [2

2. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. **82**, 1362–1364 (2003). [CrossRef]

3. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. **84**, 681–683 (2004). [CrossRef]

*d*(

*z*) of the two samples unambiguously. The concept of forming a sandwich structure has also been used to estimate the spatial width of the nonlinearity profile of poled silica samples [4

4. C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. **39**, 197–198 (2003). [CrossRef]

*single*sample instead of sandwich structures. This novel technique makes use of an iterative loop known as the Fienup algorithm [5

5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. **3**, 27–29 (1978). [CrossRef] [PubMed]

2. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. **82**, 1362–1364 (2003). [CrossRef]

3. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. **84**, 681–683 (2004). [CrossRef]

## 2. Iterative processing of d(z) using the Fienup algorithm

5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. **3**, 27–29 (1978). [CrossRef] [PubMed]

6. T. F. Quatieri Jr. and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing **29**, 1187–1193 (1981). [CrossRef]

*f*(

*t*), together with the known properties of this function, such as being real or causal, and iteratively corrects an initial guess for

*f*(

*t*). In the context of the present work, we use it to recover the nonlinearity spatial profile

*d*(

*z*) of a thin (up to hundreds of microns) nonlinear sample from the measured MF curve of this sample. Note that without loss of generality we can assume that

*d*(

*z*) is a real and causal function, i.e.,

*d*(

*z*)=0 for

*z*<0, where

*z*=0 defines the edge of the nonlinear wafer. The MF measurement of a single sample provides the magnitude of the spectrum |

*D*

_{M}(

*f*)| of the FT of

*d*(

*z*), where

*f*is the spatial frequency, [3

3. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. **84**, 681–683 (2004). [CrossRef]

*ϕ*(

*f*) of this FT. The role of the Fienup algorithm is to recover this missing FT phase. To start, one makes an initial guess,

*ϕ*

_{0}(

*f*), for the unknown FT phase. We have found empirically that this initial guess does not strongly impact the convergence of the algorithm, so for convenience it can simply be

*ϕ*

_{0}(

*f*)=0. As illustrated in Fig. 1, the first step in the algorithm is to calculate numerically the IFT of the complex quantity |

*D*

_{M}(

*f*)exp(

*jϕ*

_{0}(

*f*)). The second step is to take the real part of this IFT and retain only the

*z*≥0 region, which gives a function

*d*

_{1}(

*z*) that constitutes a first estimate of the nonlinearity profile. The third and final step is to compute the FT of this profile, |

*D*

_{1}(

*f*)|exp(

*jϕ*

_{1}(

*f*)). The phase

*ϕ*

_{1}(

*f*) of this FT provides a new estimate for the missing FT phase

*ϕ*(

*f*). At this point the FT of

*d*(

*z*) has a known (measured) amplitude |

*D*

_{M}(

*f*)| and a best-estimate (calculated) phase

*ϕ*

_{1}(

*f*). The previous three-step cycle is then repeated using

*ϕ*

_{1}(

*f*) instead of

*ϕ*

_{0}(

*f*) as the new FT phase, which yields a second estimate

*d*

_{2}(

*z*) for the profile and a second estimate of the FT phase

*ϕ*

_{2}(

*f*). This process is iterated

*n*times until convergence is achieved, i.e., until the average difference between the profiles

*d*

_{n-1}(

*z*) and

*d*

_{n}(

*z*) obtained during two consecutive cycles is less than a preset value, for example 0.1%. What the algorithm has done is reconstruct a more accurate spectrum

*ϕ*

_{n}(

*f*) than the initial guess for the originally unknown FT phase.

6. T. F. Quatieri Jr. and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing **29**, 1187–1193 (1981). [CrossRef]

7. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing **28**, 672–680 (1980). [CrossRef]

*z*-transform with all its poles and zeros on or inside the unit circle. As a result of this property, the FT phase and the logarithm of the FT magnitude of an MPF are the Hilbert transform of one another. Consequently, the FT phase of an MPF can always be recovered from its FT amplitude, and an MPF can always be reconstructed from its FT magnitude alone. This reconstruction can of course be done by taking the Hilbert transform of the logarithm of the function’s FT magnitude to obtain the FT phase, then inverting the full (complex) FT. However, this direct approach is not the preferred solution because of difficulties in its implementation, in particular phase unwrapping.[6

6. T. F. Quatieri Jr. and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing **29**, 1187–1193 (1981). [CrossRef]

**29**, 1187–1193 (1981). [CrossRef]

*the*minimum-phase function whose FT magnitude is |

*D*

_{M}(

*f*)|. Since this solution is unique, if we knew

*a priori*that the profile to be reconstructed was an MPF, then we would be certain that the solution provided by the Fienup algorithm is the correct profile. Conversely, if the original profile is

*not*a minimum-phase function, then strictly speaking the algorithm will not converge to the correct profile. Since in general we do not know whether the original profile is an MPF, it follows that we also do not know whether the recovered profile is the correct one.

*d*

_{min}(

*n*), where

*n*is an integer that numbers sampled values of the function variable, e.g., distance

*z*into the sample in the present case of a nonlinearity profile. Because all physical MPFs must be causal (although not every causal function is an MPF),

*d*

_{min}(

*n*) must be equal to zero in at least half of the space variation, for e.g., for

*n*<0, as is the case of nonlinearity profiles. Second, the energy of a minimum-phase function, which is defined as

*d*

_{min}(

*n*)|

^{2}for

*m*samples of the function

*d*

_{min}(

*n*), must satisfy the inequality [8];

*m*>0. In Eq. 1,

*d*(

*n*) represents any of the functions that have the same FT magnitude as

*d*

_{min}(

*n*). This property suggests that most of the energy of

*d*

_{min}(

*n*) is concentrated around

*n*=0 [8]. Stated differently, any profile with either a single peak or a dominant peak will be either a minimum-phase function or close to one and thus work extremely well with the Fienup algorithm. There could be other function besides functions with a dominant peak that are MPFs, but so many nonlinearity profiles, including the most common ones, satisfy this criterion, i.e., exhibit a dominant peak, that this sub-class of profiles is well worth investigating by itself. This sub-class of MPFs is at the origin of the broad success of the Fienup algorithm in this application. This criterion is obviously satisfied by the functions illustrated in Fig. 2, which again are all MPFs except for the rectangular profile. Although a rectangular profile is not truly an MPF, because it has a single peak it is expected to be close to an MPF, which is indeed the case (almost all the poles and zeros of its

*z*-transform are on or inside the unit circle, and the remaining few are just outside). This explains the success of the Fienup algorithm with rectangular profiles (see Fig. 2): even though a rectangular profile is not a true MPF, it is close enough to one that the recovery is still amazingly good, with an average error of only 0.008%. The two-peak profile in Fig. 2 has a dominant peak (the secondary peak is only ~1/3 as high as the main one) and it is, as expected, a minimum-phase function. As a result, for this function the average error in the recovery is also extremely low (under 10

^{-5}%). The nonlinearity profile of poled silica, which typically exhibits a sharp dominant peak just below the sample’s anodic surface [3

**84**, 681–683 (2004). [CrossRef]

*d*(

*z*=0) is increased to a value much larger than the rest of the profile, for example

*d*(0)=5·max{

*d*(

*z*)}, the convergence of the algorithm improves substantially in terms of both accuracy and speed. This improvement is certainly expected because Eq. (1) is now satisfied, even if it were not satisfied prior to this change, which means that the function becomes an MPF. As an example, for the three-peaked nonlinearity profile of Fig. 4, if

*d*(0) is increased to

*d*(0)=10·max{

*d*(

*z*)}, the new recovered profile (red dashed curve in Fig. 4) is significantly closer to the original profile. This powerful result opens the possibility of recovering absolutely

*any*nonlinearity profile by depositing on it a stronger and very thin nonlinear material, such as LiNbO

_{3}. The film should be preferably thin, not because its thickness affects convergence (it does not) but because it is easier both to deposit and to measure its MF curve.

*d*(

*z*) is buried below the surface of sample. Second, it cannot determine unequivocally the sign of the nonlinearity profile. Consequently, if

*d*

_{n}(

*z*) is the solution provided by the Fienup algorithm for a given nonlinear sample, then all ±

*d*

_{n}(

*z*-

*z*

_{0}) functions are also solutions. However, in many cases these limitations are fairly inconsequential because being able to determine the shape of a nonlinearity profile is much more important than determining its sign or exact location. Furthermore, if need be both the sign and location of

*d*(

*z*) can be determined by other means, for example by using the two-sample IFT technique [3

**84**, 681–683 (2004). [CrossRef]

## 3. Application to experimental poled silica samples

9. R. A. Myers, N. Mukerjkee, and S. R. J. Brueck, “Large second-order nonlinearity in poled fused silica,” Opt. Lett. **16**, 1732–1734 (1991). [CrossRef] [PubMed]

10. Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science **7**, 89–95 (2003) [CrossRef]

**84**, 681–683 (2004). [CrossRef]

**84**, 681–683 (2004). [CrossRef]

11. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. **39**, 1834–1836 (2003). [CrossRef]

*D*

_{M}(

*f*) |of

*d*(

*z*). The maximum achieved internal propagation angle (

*θ*

_{max}) in this measurement is ~87°. However a large

*θ*

_{max}of this sort is not a necessity, e.g., even

*θ*

_{max}~78° is shown to supply adequate information for a fairly accurate recovery of the nonlinearity profile.[12] The FT phase and the profile recovered from this spectrum using the Fienup algorithm are plotted in Fig. 6 (green curve) and Fig. 7 (dotted curve), respectively. For comparison, on the same two figures we also plotted the FT phase (blue curve) and the profile (solid curve), respectively, recovered by the two-sample IFT technique. The profiles retrieved by the Fienup algorithm and by the IFT technique are in excellent agreement: both exhibit a sharp nonlinearity coefficient peak of magnitude

*d*

_{33}≈-1.05 pm/V just below the surface of the sample, a sign reversal at a depth of about 12 µm, and a wider positive nonlinear region extending to a depth of about 45 µm. These observations are in keeping with other profiles obtained by other IFT techniques in similar poled samples (for further discussions about the physics of these observed nonlinearity profiles, please see Refs. 3 and 13–15). The FT phase spectra recovered by these two very different techniques (Fig. 6) are also in very good agreement with each other. These excellent agreements lend support to both our processing technique and the two-sample technique.

*ϕ*

_{0}(

*f*) when using our processing algorithm. Now that we have access to the FT phase recovered by the two-sample technique (blue curve in Fig. 6), we can run our processing algorithm using this spectrum instead of zero as a better initial guess for this FT phase. This operation is equivalent to using our algorithm to post-process the FT phase recovered by the IFT technique, with the hope to obtain an even more accurate nonlinearity profile. The profile obtained by this post-processing technique after 100 iterations is shown as the dot-dashed curve in Fig. 7. Comparison to the profile obtained with the two-sample IFT technique (Fig. 7, solid curve) shows that post-processing did not modify the overall profile shape, but that it significantly smoothed out the artificial oscillations introduced by the IFT technique. The post-processed profile of the two-sample technique (dot-dashed curve) and the profile obtained with the Fienup algorithm assuming zero initial phase (dotted curve) are very close to each other: the average difference between them is only 0.14%, which clearly demonstrates the validity of both approaches. The similarity between the profiles before and after post-processing confirms that the IFT technique came very close to recovering the actual profile. It also demonstrates the usefulness of our processing algorithm in another application, namely post-processing a profile obtained by an IFT technique. To better illustrate the beauty of this new post-processing technique, we show in Fig. 5 the MF spectrum of the post-processed nonlinearity profile, calculated numerically (solid curve). The agreement between this theoretical curve and the measured MF data (open circles) is very good.

2. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. **82**, 1362–1364 (2003). [CrossRef]

**84**, 681–683 (2004). [CrossRef]

*W*of the nonlinear region is known. Knowing this thickness means that at the outset

*d*(

*z*) can be set to 0 not only over the

*z*<0 space but also for

*z*>

*W*, which restricts the range of

*z*values over which

*d*(

*z*) is unknown. Thus the iterative loop does not need to recover as many discrete values of

*d*(

*z*) and it converges faster.

## 4. Conclusions

_{3}) or nonlinear organic materials. This is also true of profiles with two comparable peaks (average error of 0.1%) and even three-peak functions (error of a few %). Furthermore, the same algorithm can be applied to the nonlinear profile obtained by an IFT technique to improve the profile accuracy, in particular to remove unphysical oscillations introduced by the IFT technique data processing algorithm. The validity of these conclusions was verified experimentally with a nonlinear sample of poled silica. The profile recovered from the measured MF curve of this sample using the Fienup algorithm was found to be in excellent agreement with the profile of the same sample measured using an IFT technique.

## Acknowledgments

## References and links

1. | P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. |

2. | A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. |

3. | A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. |

4. | C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. |

5. | J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. |

6. | T. F. Quatieri Jr. and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing |

7. | M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing |

8. | V. Oppenheim and R. W. Schafer, |

9. | R. A. Myers, N. Mukerjkee, and S. R. J. Brueck, “Large second-order nonlinearity in poled fused silica,” Opt. Lett. |

10. | Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science |

11. | A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. |

12. | A. Ozcan, M. J. F. Digonnet, G. S. Kino, and Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called “Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films.” |

13. | M. Mukherjee, R. A. Myers, and S. R. J. Brueck, “Dynamics of second-harmonic generation in fused silica,” J. Opt. Soc. Am. B |

14. | T. G. Alley, S. R. J. Brueck, and R. A. Myers, “Space charge dynamics in thermally poled fused silica,” J. Non-Cryst. Solids. |

15. | D. Faccio, V. Pruneri, and P. G. Kazansky, “Dynamics of the second-order nonlinearity in thermally poled silica glass,” Appl. Phys. Lett. |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 2, 2004

Revised Manuscript: July 9, 2004

Published: July 26, 2004

**Citation**

A. Ozcan, M. Digonnet, and G. Kino, "Iterative processing of second-order optical nonlinearity depth profiles," Opt. Express **12**, 3367-3376 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3367

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### References

- P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, "Effects of dispersion and focusing on production of optical harmonics," Phys. Rev. Lett. 8, 21-22 (1962). [CrossRef]
- A. Ozcan, M. J. F. Digonnet, and G. S. Kino, "Inverse Fourier transform technique to determine secondorder optical nonlinearity spatial profiles," Appl. Phys. Lett. 82, 1362-1364 (2003). [CrossRef]
- A. Ozcan, M. J. F. Digonnet, and G. S. Kino, "Improved technique to determine second-order optical nonlinearity profiles using two different samples," Appl. Phys. Lett. 84, 681-683 (2004). [CrossRef]
- C. Corbari, O. Deparis, B. G. Klappauf and P. G. Kazansky, "Practical technique for measurement of second-order nonlinearity for poled glass," Electron. Lett. 39, 197-198 (2003). [CrossRef]
- J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27-29 (1978). [CrossRef] [PubMed]
- T. F. Quatieri, Jr., and A. V. Oppenheim, "Iterative techniques for minimum phase signal reconstruction from phase or magnitude," IEEE Trans. Acoust., Speech, Signal Processing 29, 1187-1193 (1981). [CrossRef]
- M. Hayes, J. S. Lim, and A. V. Oppenheim, "Signal reconstruction from phase or magnitude," IEEE Trans. Acoust., Speech, Signal Processing 28, 672-680 (1980). [CrossRef]
- V. Oppenheim and R. W. Schafer, Digital Signal Processing, (Prentice Hall, 2002), Chap. 7.
- R. A. Myers, N. Mukerjkee, and S. R. J. Brueck, "Large second-order nonlinearity in poled fused silica," Opt. Lett. 16, 1732-1734 (1991). [CrossRef] [PubMed]
- Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, " Advances in poling and permanently induced phenomena in silica-based glasses," Current Opinion in Solid State & Materials Science 7, 89-95 (2003) [CrossRef]
- A. Ozcan, M. J. F. Digonnet, and G. S. Kino, "Cylinder-assisted Maker-fringe technique," Electron. Lett. 39, 1834-1836 (2003). [CrossRef]
- A. Ozcan, M. J. F. Digonnet, and G. S. Kino, Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called "Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films."
- M. Mukherjee, R. A. Myers, and S. R. J. Brueck, "Dynamics of second-harmonic generation in fused silica," J. Opt. Soc. Am. B 11, 665-669 (1994). [CrossRef]
- T. G. Alley, S. R. J. Brueck, and R. A. Myers, "Space charge dynamics in thermally poled fused silica," J. Non-Cryst. Solids. 242, 165-176 (1998). [CrossRef]
- D. Faccio, V. Pruneri, and P. G. Kazansky, "Dynamics of the second-order nonlinearity in thermally poled silica glass," Appl. Phys. Lett. 79, 2687-2689 (2001) [CrossRef]

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