## Retrieving the displacement of strained nanoobjects: the impact of bounds for the scattering magnitude in direct space |

Optics Express, Vol. 21, Issue 23, pp. 27734-27749 (2013)

http://dx.doi.org/10.1364/OE.21.027734

Acrobat PDF (3846 KB)

### Abstract

Coherent X-ray diffraction imaging (CXDI) of the displacement field and strain distribution of nanostructures in kinematic far-field conditions requires solving a set of non-linear and non-local equations. One approach to solving these equations, which utilizes only the object’s geometry and the intensity distribution in the vicinity of a Bragg peak as *a priori* knowledge, is the HIO+ER-algorithm. Despite its success for a number of applications, reconstruction in the case of highly strained nanostructures is likely to fail. To overcome the algorithm’s current limitations, we propose the
*a priori* knowledge of the local scattering magnitude and remedies HIO+ER’s stagnation by incorporation of randomized overrelaxation at the same time. This approach achieves significant improvements in CXDI data analysis at high strains and greatly reduces sensitivity to the reconstruction’s initial guess. These benefits are demonstrated in a systematic numerical study for a periodic array of strained silicon nanowires. Finally, appropriate treatment of reciprocal space points below noise level is investigated.

© 2013 OSA

**I**

_{QB}(

*q*) in the vicinity of one or more Bragg peaks

*Q*

_{B}, which is determined by the displacement field

*u*, the shape Ω of the nanocrystal and its chemical composition profile within a specific framework of approximations.

*q*denotes the distance in reciprocal space from the Bragg peak

*Q*

_{B}.

**I**

_{QB}(

*q*) is proportional to |𝔣(

*Q*

_{B}+

*q*)|

^{2}= |

**FT**

_{QB}

_{+}

_{q}_{←}

*{*

_{x}*ρ*

_{el}(

*x*)}|

^{2}where

*ρ*

_{el}(

*x*) is the electron density of the illuminated sample. However, a measurement of the intensity distribution

**I**

_{QB}(

*q*) does not reveal the

*q*-dependence of the phase information arg(𝔣(

*Q*

_{B}+

*q*)). Once this phase information is available, the displacement field

*u*is obtained by inverse Fourier transform. Consequently, considerable efforts are put in the development of robust algorithms for retrieving this lost phase information based on proper

*a priori*knowledge [1

1. M. Köhl, A. A. Minkevich, and T. Baumbach, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express **20**, 17093–17106 (2012). [CrossRef]

7. R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input-output method of phase retrieval,” Appl. Opt. **52**, 3031–3037 (2013). [CrossRef] [PubMed]

*u*from the coherently scattered intensity

**I**

_{QB}(

*q*) utilizes the object’s geometry Ω as

*a priori*knowledge combined with proper reconstruction algorithms. The shape Ω can be accessed by complementary techniques like GISAXS [8

8. G. Renaud, R. Lazzari, and F. Leroy, “Probing surface and interface morphology with grazing incidence small angle x-ray scattering,” Surf. Sci. Rep. **64**, 255–380 (2009). [CrossRef]

9. G. Binnig, C. F. Quate, and C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. **56**, 930–933 (1986). [CrossRef] [PubMed]

12. N. Jalili and K. Laxminarayana, “A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences,” Mechatronics **14**, 907–945 (2004). [CrossRef]

13. L. Reimer, *Scanning Electron Microscopy: Physics of Image Formation and Microanalysis*, vol. 45 (Springer Series in Optical Sciences, 1998), 2nd ed. [CrossRef]

2. J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. **46**, 312–318 (2013). [CrossRef] [PubMed]

5. A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B **76**, 104106 (2007). [CrossRef]

6. Y. Chushkin and F. Zontone, “Upsampling speckle patterns for coherent x-ray diffraction imaging,” J. Appl. Crystallogr. **46**, 319–323 (2013). [CrossRef]

14. A. Diaz, V. Chamard, C. Mocuta, R. Magalhães-Paniago, J. Stangl, D. Carbone, T. H. Metzger, and G. Bauer, “Imaging the displacement field within epitaxial nanostructures by coherent diffraction: a feasibility study,” New J. Phys. **12**, 035006 (2010). [CrossRef]

28. I. A. Vartanyants and I. K. Robinson, “Partial coherence effects on the imaging of small crystals using coherent x-ray diffraction,” J. Phys.: Condens. Matter **13**, 10593–10611 (2001). [CrossRef]

29. J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

31. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. **78**, 011301 (2007). [CrossRef]

1. M. Köhl, A. A. Minkevich, and T. Baumbach, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express **20**, 17093–17106 (2012). [CrossRef]

2. J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. **46**, 312–318 (2013). [CrossRef] [PubMed]

31. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. **78**, 011301 (2007). [CrossRef]

*a priori*knowledge in direct space beyond the object’s geometry proved valuable for retrieving the displacement field

*u*from a particular experimental data set of a highly inhomogeneously strained nanostructure [2

2. J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. **46**, 312–318 (2013). [CrossRef] [PubMed]

4. A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B **78**, 174110 (2008). [CrossRef]

5. A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B **76**, 104106 (2007). [CrossRef]

*a priori*knowledge does not eliminate the non-convex operations involved during reconstruction. These operations are claimed to cause stagnation or convergence to local minima and typically imply a strong dependence on the initial guess of the reconstruction.

*a priori*knowledge in comparison to the existing algorithms are developed [1

1. M. Köhl, A. A. Minkevich, and T. Baumbach, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express **20**, 17093–17106 (2012). [CrossRef]

3. D. E. Adams, L. S. Martin, M. D. Seaberg, D. F. Gardner, H. C. Kapteyn, and M. M. Murnane, “A generalization for optimized phase retrieval algorithms,” Opt. Express **20**, 24778–24790 (2012). [CrossRef] [PubMed]

7. R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input-output method of phase retrieval,” Appl. Opt. **52**, 3031–3037 (2013). [CrossRef] [PubMed]

31. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. **78**, 011301 (2007). [CrossRef]

32. S. Marchesini, “Phase retrieval and saddle-point optimization,” J. Opt. Soc. Am. A **24**, 3289–3296 (2007). [CrossRef]

_{OR}+ER-algorithm [1

**20**, 17093–17106 (2012). [CrossRef]

29. J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

30. J. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A **4**, 118–123 (1986). [CrossRef]

33. A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A **1**, 932–943 (1984). [CrossRef]

34. D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 - theory,” IEEE Trans. Med. Imaging **1**, 81–94 (1982). [CrossRef]

**20**, 17093–17106 (2012). [CrossRef]

**78**, 011301 (2007). [CrossRef]

*a priori*knowledge beyond shape Ω in direct space and intensity distribution

**I**

_{QB}(

*q*) in reciprocal space. In particular, the dependence on the initial guess of the reconstruction is reduced significantly. Therefore, the combination of randomized overrelaxation and additional

*a priori*knowledge is very promising.

_{OR}+ER-algorithm. We refer to the resulting algorithm as

## 1. Theoretical background

35. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. **444**, 101 (2007). [CrossRef]

40. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

**I**(

*q*) of the scattered incident radiation by

**I**(

*Q*) ∝ |𝔣(

*Q*)|

^{2}where

*Q*=

*k*

_{Out}−

*k*

_{In}is the scattering vector.

*k*

_{In}and

*k*

_{Out}are the incident and outgoing wave vector respectively.

*ρ*

_{el}(

*x*) is the electron density of the nanostructure. We restrict to the displacement in samples which can be considered as inhomogeneously strained crystalline structures with fully elastic strain. Heterogeneous material systems are included as long as their interface is coupled elastically. In the vicinity of a Bragg peak

*Q*

_{B}≠ 0⃗, the contribution of amorphous domains in the sample to the scattered intensity is typically negligible. So, amorphous domains may also be present in the nanostructure.

*ρ*

_{eff}(

*x*) given either only the shape Ω(

*x*) and the amplitudes

*ζ*

_{QB}(

*x*). If this additional knowledge is exploited in the reconstruction procedure, the range of applicability of phase retrieval is significantly extended to higher values of strain and the robustness of the method for equal strain distribution is increased. Some remarks on uniqueness of the solution and numerical discretization can be found in Sec. A in the appendix.

### 1.1. The
HIO O R M + ER M -algorithm

_{OR}+ER-algorithm [1

**20**, 17093–17106 (2012). [CrossRef]

29. J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

*ζ*

_{QB}. The HIO

_{OR}+ER-algorithm is an iterative procedure which is build up by two blocks, HIO

_{OR}and ER. Each iteration of the HIO

_{OR}+ER-algorithm

*N*

_{HIO}repetitions of the HIO

_{OR}-algorithm are performed, followed by

*N*

_{ER}repetitions of the ER-algorithm. We shortly summarize both building blocks:

**21**, 2758–2769 (1982). [CrossRef] [PubMed]

30. J. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A **4**, 118–123 (1986). [CrossRef]

**P**

_{Ω}is a linear projection operator on the interior of the object’s geometry Ω, i.e.,

**P**

_{Γ}is a non-linear and non-convex projection operator in reciprocal space. It enforces the amplitudes Γ

*in 𝔣*

_{q}^{(i)}(

*q*) without modifying the phases arg (𝔣

^{(i)}(

*q*)). Hence, its formal definition if The HIO

_{OR}-algorithm can be interpreted as a feedback based iterative mapping with feedback parameter

*β*. A single iterative step is defined as where the mapping incorporates all available constraints and information except finite direct space geometry Ω.

**Q**

_{Γ;λΓ}is the overrelaxed map of

**P**

_{Γ}, i.e., Whereas overrelaxation with a

*fixed*relaxation parameter

*λ*

_{Γ}typically even decreases the performance of the HIO-algorithm, randomization of the relaxation parameter turned out to be a key feature for overcoming stagnation in the iterative procedure [1

**20**, 17093–17106 (2012). [CrossRef]

*λ*

_{Γ}is drawn from a uniform random distribution in the interval [1−

*ν*, 1+

*ν*],

*ν*≥ 0. Unless stated otherwise, we choose

*ν*= 0.5. The success of the HIO

_{OR}+ER-algorithm neither depends sensitively on the value of parameter

*β*in the range [0.5, 1.0] nor on the particular choice of the internal parameters

*N*

_{HIO}and

*N*

_{ER}for

*ν*≈ 0.5 [1

**20**, 17093–17106 (2012). [CrossRef]

*ν*= 0 (i.e.,

*λ*

_{Γ}≡ 1).

*at each point*

_{q}*q*.

*q*) is found (and, therefore, the solution for

*ρ*

_{eff}(

*x*)), we can rely on Eq. (2) to extract the displacement field

*u*. Depending on the amount of strain in the sample and the value of

*Q*

_{B}, unwrapping [43

43. J.-F. Weng and Y.-L. Lo, “Novel rotation algorithm for phase unwrapping applications,” Opt. Express **20**, 16838–16860 (2012). [CrossRef]

*a priori*knowledge may significantly expand the range of applicability of CXDI to highly inhomogeneously strained samples [4

4. A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B **78**, 174110 (2008). [CrossRef]

5. A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B **76**, 104106 (2007). [CrossRef]

*ζ*(

*x*) (see Eq. (2)) in the HIO

_{OR}+ER-algorithm. The following property is true for most experimental samples: We can define sample domains Ω

*⊆ Ω in which the local scattering magnitude*

_{j}*ζ*(

*x*) deviates only slightly from its average value (e.g., substrate region or full sample in case of homogeneous chemical composition). The unification of the domains Ω

*does not need to coincide with the full sample Ω. Moreover, the domains may even overlap.*

_{j}*ζ*(

*x*) is constant as long as we neglect the changes originating in strain. Hence, the deviation of

*ζ*(

*x*) from its average is equal to zero. If the nanostructure is not fully chemically homogeneous, at least the substrate region can typically be considered chemically homogeneous and almost unstrained. Hence, the scattering magnitude in this subdomain of the full structure deviates only slightly from this subdomain’s average value – independent of the local scattering magnitude of additionally grown nanostructures on top. For the nanostructure on top of the substrate, we can constrain

*ζ*(

*x*) in addition with bounds representing a different average and a larger distance to this average.

*ζ*(

*x*) by where Ω

*are the domains to be constrained. M*

_{j}_{L,j}≤ 1 and M

_{H,j}≥ 1 are additional parameters which need to be known

*a priori*for every domain

*j*. Moreover, the lower and upper bounds M

_{L,j}and M

_{H,j}fulfill M

_{H,m}

*ζ̄*≥ M

_{m}_{L,n}

*ζ̄*and M

_{n}_{L,m}

*ζ̄*≤ M

_{m}_{H,n}

*ζ̄*, if the domains Ω

_{n}*and Ω*

_{m}*intersect.*

_{n}*i*) labels iterations and (

*j*) labels the domains Ω

*. Each iteration, the averages*

_{j}*ζ̄*are estimated during reconstruction by evaluating Note, that in general the mappings

_{j}_{L,j}= M

_{H,j}= 1. In this case, applying a mapping

_{OR}+ER-algorithm by modifying

**M**

_{C}(

*λ*

_{Γ}) (see Eq. (7b)) to At this point, we define the

*ρ*

_{eff}(

*x*) remains a fixed point of the iterative procedure for the proposed extension (11) due to the fact that

**M**

_{C}reduces to

**M**

_{C}≡ 1 if applied to

*ρ*

_{eff}(

*x*). For the ER

^{M}-algorithm, we employ instead of Eq. (4). Finally, we define the

*N*

_{HIO}iterations of

*N*

_{ER}iterations of the ER

^{M}-algorithm. The

_{OR}+ER and standard HIO+ER: All extensions can be implemented numerically as local, independent single pixel operations. Thus, they can be performed very efficiently and executed in parallel. Hence, computational efficiency is limited by the

*N*log(

*N*)-scaling of the FFT algorithm.

*same*set of given

*a priori*knowledge — either {Ω(

*x*), Γ

*} or {Ω(*

_{q}*x*), Γ

*, M*

_{q}_{L,}(

*x*), M

_{H,}(

*x*)} in our case — more successfully, the constraints

**M**

_{M}aim at regularization of the model (2) by adding the

*additional a priori*knowledge {M

_{L,}(

*x*), M

_{H,}(

*x*)}.

### 1.2. Generation of the input data for the reconstruction

*u*has been obtained by finite element modeling (FEM) of linear elasticity theory (LET) [44

44. T. Benabbas, Y. Androussi, and A. Lefebvre, “A finite-element study of strain fields in vertically aligned InAs islands in GaAs,” J. Appl. Phys. **86**, 1945–1950 (1999). [CrossRef]

49. P. Schroth, T. Slobodskyy, D. Grigoriev, A. Minkevich, M. Riotte, S. Lazarev, E. Fohtung, D. Hu, D. Schaadt, and T. Baumbach, “Investigation of buried quantum dots using grazing incidence x-ray diffraction,” Mater. Sci. Eng., B **177**, 721–724 (2012). [CrossRef]

50. M. Eberlein, S. Escoubas, M. Gailhanou, O. Thomas, J.-S. Micha, P. Rohr, and R. Coppard, “Investigation by high resolution x-ray diffraction of the local strains induced in Si by periodic arrays of oxide filled trenches,” Phys. Status Solidi A **204**, 2542–2547 (2007). [CrossRef]

51. M. Eberlein, S. Escoubas, M. Gailhanou, O. Thomas, P. Rohr, and R. Coppard, “Influence of crystallographic orientation on local strains in silicon: a combined high-resolution x-ray diffraction and finite element modelling investigation,” Thin Solid Films **516**, 8042–8048 (2008). [CrossRef]

_{2}(gray domain in Fig. 2(a)). Different thermal expansion coefficients of the crystalline and amorphous region result in non-vanishing strain in the nanostructure after cooling down to room temperature.

*u*. Translational symmetry along the wires allows performing the simulation in a planar cut perpendicular to the direction of the wires. Thus, the system can be treated in two dimensions. Due to the periodic arrangement of the wires, simulations are restricted to a single block of the periodic object.

*u*. The strain distribution is characterized by its maximum strain

*ε*

_{M}= max(∂

*) in the crystalline part on the wire’s central axis. The resulting phase field of the effective electron density for the values*

_{z}u_{z}*ε*

_{M}= {0.10%, 0.30%, 0.60%} is depicted in Figs. 2(b)–2(d). In addition, Figs. 2(e)–2(g) gives an impression of the scattering signal Γ

*for*

_{q}*ε*

_{M}= {0.10%, 0.20%, 0.28%}.

### 1.3. Measures for success in numerical simulations

*ρ*

_{eff}of the iterative reconstructions after (

*i*) iterations of the

*ρ*

_{eff}(

*x*) by evaluating This measure eliminates the undefined global phase in the reconstructed effective electron density (see Sec. A in the appendix).

*ρ*

_{eff}from iteration to iteration dropped below a certain value. This change has been monitored by the angle In [1

**20**, 17093–17106 (2012). [CrossRef]

### 1.4. Artifacts in reciprocal space

*exhibits a signal to noise ratio which is limited only by finite digit precision. In contrast to that, the signal to noise ratio of the experimental scattering data used by Minkevich et al. in [24*

_{q}24. A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett. **94**, 66001 (2011). [CrossRef]

25. A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B **84**, 054113 (2011). [CrossRef]

_{N}from above, i.e., Γ

*≤ Γ*

_{q}_{N}. As no beamstop is required in case of imaging strained nanocrystals at

*Q*

_{B}≠ 0⃗ (in contrast to performing diffractive imaging measurements in forward direction

*Q*

_{B}= 0⃗ [52

52. P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr., Sect. A: Found. Crystallogr. **62**, 248–261 (2006). [CrossRef]

_{N}is well defined and experimentally accessible.

*≤ Γ*

_{q}_{N}. We hope that our investigation will contribute to establishing CXDI of inhomogeneously strained nanostructures as a robust technique of experimental value. The investigation of other artifacts (like the “inconsistent” Bragg peak in experimental data, photon noise in the signal

**I**

*[7*

_{q}7. R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input-output method of phase retrieval,” Appl. Opt. **52**, 3031–3037 (2013). [CrossRef] [PubMed]

**I**

*by instrumental effects and corrections from higher order scattering beyond kinematic approximation) goes beyond the scope of this manuscript.*

_{q}**P**

_{Γ}as given in Eq. (6) is modified to The individual approaches differ in the definition of

*κ*and are defined in Table 1.

_{q}*A*) incorporates the noisy region as the weak signal limit, i.e., approximating all amplitudes below the lower cutoff Γ

_{N}as zero. However, small inconsistencies between direct space support Ω and Fourier space amplitudes

*κ*are inevitable in this approach. Therefore, our second, third and fourth approach allow the reciprocal space amplitude at every

_{q}*q*-point to evolve freely until it exceeds the given cutoff Γ

_{N}. The approaches differ in the behavior if the cutoff is exceeded during the iterative procedure: In the second approach (

*B*), the respective amplitude is reset to zero. In the third approach (

*C*), it is reset to a random value in the interval [0, Γ

_{N}] (uniform distribution). The fourth approach (

*D*) resets the amplitude only to its upper bound Γ

_{N}. Finally, the fifth approach (

*E*) contains a small damping

*c*

_{D}≲ 1 in regions below Γ

_{N}. This way, we regularize the corrupted region such that high frequency contributions to the direct space effective electron density

*ρ*

_{eff}(

*x*) are suppressed. Note, that the last approach (

*E*) reduces to the fourth approach (

*D*) in the limit

*c*

_{D}→ 1.

## 2. Results

*s*∈ [0%, 100%] of fully automated reconstructions (no user interaction like tuning of parameters during reconstruction). The success rate is a statistical quantity for all cases we consider, because random numbers influence the reconstruction at two stages: first, the initial guess itself is based on random phases for the given amplitudes in reciprocal space. Therefore, the success rate

*s*is a statistical quantity even for the traditional HIO+ER-algorithm. Second, for those reconstructions that exploit randomized overrelaxation (

*ν*> 0), the iterative approximation to the solution varies from one trial to another even for the same initial guess. Therefore, we estimate the success rate

*s*by evaluating a set of N

_{Real}= 100 trials. Every trial has its own random initial guess and its own random overrelaxation parameters

*λ*

_{Γ}. For a robust, automated reconstruction, this success rate should reach values close to 100% within a practical number of iterations

*N*

_{Iter}.

*s*may depend on the strain

*ε*

_{M}, the number of iterations (

*i*) which has been performed and our choice of the angle

*φ*

_{Max}which is used to distinguish successful reconstructions from failed reconstructions based on Eq. (13). We will investigate its behavior by two-dimensional cuts through this three dimensional parameter space and use a unified color-coding to encode the success rate

*s*.

_{Tot}= 185136). A single pixel corresponds to a distance of 0.862nm in x-direction and 1.575nm in z-direction. The substrate was truncated in such a way that an oversampling

*σ*of 3.6326 has been achieved. The parameters of the reconstructions were chosen as

*N*

_{ER}= 10,

*N*

_{HIO}= 130,

*β*= 0.8,

*c*

_{D}= 0.99 and either

*ν*= 0 (“without randomized overrelaxation”) or

*ν*= 0.5 (“with randomized overrelaxation”). The reconstruction procedure was terminated, if either

*N*

_{Iter}= 500 iterations have been performed or the change of the iterative approximation from the current to the previous approximation

*χ*

^{(i)}as defined in Eq. (14) dropped below 10

^{−6}rad.

*2.1. Full reciprocal space information within the framework defined by Eqs.*(2)*and*(3)

_{N}= 0 in Fig. 3. Figures 3(a)–3(d) are devoted to the classical HIO+ER-algorithm (

*ν*= 0 and no constraints on

*ζ*). The range

*ε*

_{M}from 0.02% to 0.40% was covered in steps

*δε*

_{M}= 0.02%. For a maximum strain up to

*ε*

_{M}= 0.10%, all trials converged to the solution

*ρ*

_{eff}within very few iterations (see Fig. 3(a) as well as Figs. 2(b) and 2(e)). For

*ε*

_{M}= 0.12% the success probability dropped down to

*s*= 38%, for

*ε*

_{M}= 0.16% to

*s*= 15% and for any

*ε*

_{M}> 0.20% (see Fig. 2(f)), success probability was essentially

*s*= 0%. In the range

*ε*

_{M}= 0.12% to

*ε*

_{M}= 0.20%, sometimes stagnation close to the solution

*ρ*

_{eff}can be observed (see Figs. 3(b) and 3(c)). However, in most cases stagnation is observed on a level

*φ*far from the true solution (

*φ*> 20°). Figures 3(c) and 3(d) demonstrate that once the classical HIO+ER-algorithm is stuck in a level of stagnation, the mean number of iterations that the iterative procedure remains there is very high, even if this level is far from the solution

*ρ*

_{eff}.

*ν*> 0) in Figs. 3(e)–3(h). In the range

*ε*

_{M}≤ 0.10%, no negative penalty of randomized overrelaxation has been discovered (see Fig. 3(e)). Within the range

*ε*

_{M}= 0.12% to

*ε*

_{M}= 0.28% (see Fig. 2(g)), the HIO

_{OR}+ER-algorithm is clearly superior to the classical HIO+ER-algorithm (see Fig. 3(e)). Successful reconstructions are possible independent of the random initial guess with a success probability

*s*close to 100% within

*i*= 500 iterations. However, the number of iterations which is required to achieve a success rate close to 100% is increasing with increasing strain (see Fig. 3(e)). Figures 3(f)–3(h) investigate this observation in more detail and are to be compared to Figs. 3(c)–3(d) for the traditional HIO+ER-algorithm. In contrast to the traditional HIO+ER-algorithm, the iterative reconstruction procedure with randomized overrelaxation manages to escape from levels of stagnation up to

*ε*

_{M}≥ 0.28% within a reasonable number of iterations (

*i*< 500). This behavior can be observed particularly well for

*ε*

_{M}≥ 0.26% in Fig. 3(h). Finally, for

*ε*

_{M}≥ 0.30% (see Fig. 2(c)), robust automatic reconstructions are not possible even with randomized overrelaxation within

*i*= 500 iterations.

*s*for the traditional HIO+ER-algorithm extended for bounds on the local scattering magnitude

*ζ*without randomized overrelaxation, Figs. 3(m)–3(p) demonstrate the benefits from the combination of randomized overrelaxation and our constraints on the local scattering magnitude.

*ζ*(

*x*) was constrained by M

_{L,}= M

_{H,}= 1.0 in the full domain Ω. Most importantly, we observe, that such strict lower and upper bounds on

*ζ*(

*x*) tremendously enhance the range of applicability: Without randomized over-relaxation, almost stagnation free reconstructions proved to be possible up to approximately

*ε*

_{M}≲ 0.56% (see Fig. 2(d)) — in contrast to

*ε*

_{M}≲ 0.10% without such bounds (see Fig. 3(a)). Moreover, even up to

*ε*

_{M}= 1.00%, some random initial trails have converged to the solution

*ρ*

_{eff}(compared to

*ε*

_{M}≲ 0.20% for traditional HIO+ER-algorithm). However, if a random initial trial stagnated, it typically stagnated on a level

*φ*≥ 20°, i.e., far from the solution.

*a priori*knowledge, but without randomized overrelaxation.

*ε*

_{M}= 1.0% we stopped our investigation because the number of pixels in discrete numerical grid representing the fastest 2

*π*-oscillation in the effective electron density

*ρ*

_{eff}dropped to approximately eight pixels.

*ε*

_{M}≥ 1.00% (see Fig. 3(m)): reconstructions typically reach a distance of

*φ*= 1.0° to the solution

*ρ*

_{eff}for iterations

*i*≪ 500.

_{L,}= M

_{H,}= 1.0 only in the substrate region Ω

_{Sub}(thickness in z-direction equal to 80 pixel, hatched domain in Fig. 2(a)), we can observe the behavior depicted in Fig. 3(k) for

*ν*= 0 (no overrelaxation) and in Fig. 3(o) for

*ν*= 0.5 (with overrelaxation). Again, the benefits of lower and upper bounds on the magnitude in direct space and randomized overrelaxation are clearly visible.

_{L,}and M

_{H,}. The results for M

_{L,}= 0.7 and M

_{H,}= 1.3 are depicted in Figs. 3(l) and 3(p). Without randomized overrelaxation (see Fig. 3(l)), stagnation is still very strong, but at least some random initial trials converged to a level close to the true solution. Interestingly, even randomized overrelaxation does not eliminate stagnation for such bounds (see Fig. 3(p)): nevertheless, the behavior becomes independent of the random initial trail and manages to come close to the true solution. The latter is not true for

*ε*

_{M}≥ 0.30% without bounds on the local scattering magnitude. Up to

*ε*

_{M}= 0.28% — the limit for successful reconstructions without bounds for the local scattering magnitude — no negative penalty is observed.

### 2.2. Low signal cutoff in reciprocal space

_{N}> 0 (see Sec. 1.4). To be specific, we define for every value of strain

*ε*

_{M}the low cutoff Γ

_{N}as Γ

_{N}=

*μ*max

*(Γ*

_{q}*) where*

_{q}*μ*= 0.005. If we exclude the Bragg peak, this value for

*μ*corresponds approximately to the experimental value for the signal to noise ratio which we mentioned in the beginning of the section 1.4. Figures 2(e)–2(g) show which part of the scattering signal for the particular values of strain

*ε*

_{M}= 0.10%,

*ε*

_{M}= 0.20% and

*ε*

_{M}= 0.28% exceeds this noise level Γ

_{N}. We define the effective oversampling ratio

*σ*

_{eff}as the ratio of the number of data points exceeding the noise level Γ

_{N}divided by the number of data points inside the direct space support. It is listed in Table 2 together with the number and percentage of data points in the scattering signal exceeding the cutoff Γ

_{N}. Moreover, this table contains the fraction of the

*ℒ*

_{1}-and

*ℒ*

_{2}-norm which is accumulated in the scattering signal below the cutoff. The effective oversampling ratio is in the range of

*σ*

_{eff}= 0.03 to

*σ*

_{eff}= 0.05 for the cases we present here.

_{N}is sufficient for a successful reconstruction. Of course, the precise values in Table 2 depend on the spacing of the discretization grid in direct space (or equivalent: the extension of the domain in reciprocal space): A grid with larger interpixel spacing in direct space will suffer from aliasing artifacts. On the contrary, a finer grid will reduce the effective oversampling even further and, therefore, relies even more on an efficient approach for low signal data points. However, the optimal choice of the direct space pixel grid goes beyond the scope of this manuscript.

*A*) to (

*E*) as defined in Table 1, we need to eliminate the amplitude information Γ

*from the initial guess (8) if Γ*

_{q}*≤ Γ*

_{q}_{N}. Therefore, we enforce Γ

*= 0 for Γ*

_{q}*≤ Γ*

_{q}_{N}in the initial guess. Moreover, we now need to take instabilities into account which have not occurred in our reconstructions for ideal data: For every reconstruction trial (each of the N

_{Real}= 100 trials with new random initial phases) we extract two numbers: First, we extract the smallest distance

*φ*

^{(i)}which was achieved at any iteration

*i*≤ 500 (left bars in Figs. 4(a) and 4(b)). Second, we extract the distance

*φ*

^{(500)}, i.e., the distance to the solution

*ρ*

_{eff}after

*i*= 500 iterations (right bars in Figs. 4(a) and 4(b)). Thus, the left bars in Figs. 4(a) and 4(b) are important if we find some robust criterion for selecting the best approximation

*i*∈ [0, 500], whereas the right bars are important as long as such a criterion is not available. Figure 4(a) compares the models (

*A*) to (

*E*) if no bounds on local scattering magnitude are applied. For the data depicted in Fig. 4(b) strict bounds on the local scattering amplitude (M

_{L,}= M

_{H,}= 1.0 in entire domain Ω) have been exploited. We observe that model (

*A*) (i.e., setting the low signal amplitudes simply to zero) results in very unstable behavior. Model (

*B*) is unstable without constraints on the local scattering magnitude, but partially regularized in presence of such constraints. Nevertheless, none of the models (

*B*) to (

*D*) succeeds in achieving a small angle

*φ*≤ 10.0° to the solution

*ρ*

_{eff}. The small damping underlying model (

*E*) provides the by far best results.

*E*), the success rate

*s*is depicted as a function of strain

*ε*

_{M}vs. iteration

*i*and as a function of angle

*φ*

_{Max}vs. strain

*ε*

_{M}in Figs. 4(c)–4(f), where the reciprocal space data described by the non-zero cutoff in Table 2 has been used. In both cases, the success rate is almost independent of the choice

*φ*

_{Max}for

*φ*

_{Max}∈ [3.0°, 20.0°].

_{N}= 0), the maximum strain

*ε*

_{M}which could be reconstructed in the framework of model (

*E*) (with low cutoff

*μ*= 0.005 resulting in effective oversampling ratios

*σ*

_{eff}≈ 0.03 to

*σ*

_{eff}≈ 0.05) dropped from

*ε*

_{M}≈ 0.28% to

*ε*

_{M}≈ 0.24% without constraints on the local scattering magnitude

*ζ*. If strict bounds on

*ζ*in the entire domain Ω are enforced in the framework of the

*ε*

_{M}which could be reconstructed drops from

*ε*

_{M}≥ 1.0% to

*ε*

_{M}≈ 0.68%. This corresponds approximately to the phase field illustrated in Fig. 2(d). Keep in mind that many other important artifacts in experimental data need to be taken care of which we did not discuss in this manuscript, but will be subject of future research.

## 3. Conclusion

*a priori*knowledge of the local scattering magnitude to the

*σ*

_{eff}≪ 1 (see Table 2). In our comparison the combination of a small damping and limiting the reciprocal space amplitude by noise level Γ

_{N}from above turned out to be the most efficient model by far.

**M**

_{M}separately or combined can be found. From this figure, our three major results can be read off easily: First, the additional bounds on the local scattering magnitude — incorporated by the mapping

**M**

_{M}— enhance the range of applicability up to a factor of five, but the result still depends on the random initial guess. Second, randomized overrelaxation manages to eliminate the sensitivity to the random initial guess to a large degree — with and without the constraints

**M**

_{M}. Third, randomized overrelaxation manages to increase the range of applicability further (without any additional

*a priori*knowledge) in all cases we presented. Therefore, we are confident that the combination of randomized overrelaxation

**Q**

_{Γ;λΓ}and the local scattering magnitude constraints

**M**

_{M}enhances current possibilities to reconstruct the atomic displacement field from CXDI measurements.

*a priori*known correlations of the effective electron density are added to the reconstruction process.

## A. Uniqueness and discretization

*δx*and

_{m}*δq*of the grid in direction

_{m}*m*is connected to the domain boundaries ranging from (−

*X*,

_{m}*X*) and (−Ω

_{m}*, Ω*

_{m}*) via*

_{m}*and*

_{m}*δq*, we implicitly choose values for

_{m}*N*,

_{m}*δx*and

_{m}*X*. In theory, knowledge of the amplitudes Γ

_{m}*and the shape Ω(*

_{q}*x*) is sufficient, if the shape Ω is finite, the dimensionality of the structure is at least equal to two and the distance

*δq*of the measured points in reciprocal space is small enough to guarantee a sufficient oversampling ratio

*σ*on the grid (

*σ*≥ 2 is a lower bound) [1

**20**, 17093–17106 (2012). [CrossRef]

26. R. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A **7**, 394–411 (1990). [CrossRef]

54. R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A **13**, 725–734 (1996). [CrossRef]

## Acknowledgments

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**OCIS Codes**

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

(290.3200) Scattering : Inverse scattering

(100.3200) Image processing : Inverse scattering

(110.3200) Imaging systems : Inverse scattering

**ToC Category:**

Image Processing

**History**

Original Manuscript: June 21, 2013

Revised Manuscript: August 30, 2013

Manuscript Accepted: September 2, 2013

Published: November 5, 2013

**Citation**

Martin Köhl, Philipp Schroth, A. A. Minkevich, and Tilo Baumbach, "Retrieving the displacement of strained nanoobjects: the impact of bounds for the scattering magnitude in direct space," Opt. Express **21**, 27734-27749 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27734

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### References

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