Magnetic versus Optical Solutions for Transport-of-Intensity Equation in Lorentz Phase Microscopy of Patterned Magnetic Nano-Arrays

Phase retrieval from simple intensity measurements based on the transport-of-intensity equation (TIE) opens a new dimension of scientific research along with the well developed electron holography for better understanding of non-trivial physics of various functional, especially magnetic, materials on a wide length scale. The TIE approach developed originally for light optics by Teaque [1] was found to be promising for studying magnetic structures [2] at nanoscale with electron optics. The direct relation of TIE-formalism to TEM image formation theory was proved by De Graef [2: Ch.2)] and examined by Barty et al. [2: Ch.5]. However, several questions remained unanswered. In current work we re-examine a key moment of the optical TIE solution, which has proved to work well in the light optics when the vector field I∇φ is replaced by the gradient of some function ∇ψ with I and ∇φ being measured the intensity and gradient of the retrieved object phase (i.e. vector field I∇φ must be “potential” since rot(I∇φ)= rot(∇ψ)= 0). Basically, this assumption may be a source of significant errors when TIE-formalism is applied to quantitative magnetic imaging, as follows from our independent derivation and analysis of the so-called magnetic transport-of-intensity equation (MTIE) [3] based on the Aharonov-Bohm (AB) solution for the phase shift derived from the Shrodinger equation. This equation makes a quantitative relation of defocused image contrast to micro-magnetic parameters of materials such as local magnetization and nanoscale currents. Recently, a practical fast Fourier solution for more general situation of non-potential field I∇φ was suggested in [4]. To illustrate the difference in optical (by TIE) and magnetic (by MTIE) solutions applied to general continuity equation (CE), we consider a simple example in Fig.1. The defocused Fresnel images (Fig.1 a-b) generated from known amplitude and phase (Fig.1 c) by using a standard Fresnel integral are further used to retrieve the phase by optical TIE [5] (Fig.1 e,g) and magnetic MTIE [3, 4] (Fig.1 f,h) solutions. It is clear that optical solution fails in accurate retrieval of the “phase pyramid” (Fig.1 c) associated with closure vortex structure of the squared Permalloy element of size 1.4 μm characterized by the product Bt = 1.2 T x 10 nm. A similar situation happens with the TIE solution when MTIE/TIE concepts are used for retrieval of the object phase information (Fig.2 a) from simple intensity measurements of the defocused Fresnel images of Permalloy elements. The subsequent procedure for separation of magnetostatic (Fig.2 b) and electrostatic (Fig.2 c) potentials is described in [4]. Fig. 2d shows the difference map of magnetic and optical solutions for experimental defocused Fresnel images (def = 174 μm). The most significant deviations (Fig.2 e) of the TIE-solution from magnetic one are observed at the sharp edges of permalloy elements of 400 nm in size and 5-10 nm thick. From direct MTIE-phase reconstruction we obtained the following results: (a) electrostatic phase shift of 0.8 ± 0.1 rad (in excellent agreement with independent holographic data [6]): V⋅t =122.6 V nm (24.5V⋅ 5 nm); (b) the phase gradients due to AB-magnetostatic potential suggest the product (B⋅t)max ≅ 11 ± 1 T⋅nm. This value is consistent with a requirement of independence of reconstructed phase from the experimental defocus [7].