A general theory of far-field optical microscopy image formation and resolution limit using double-sided Feynman diagrams

Optical resolution of far-field optical microscopy is limited by the diffraction of light, while diverse light-matter interactions are used to push the limit. The image resolution limit depends on the type of optical microscopy; however, the current theoretical frameworks provide oversimplified pictures of image formation and resolution that only address individual types of microscopy and light-matter interactions. To compare the fundamental optical resolutions of all types of microscopy and to codify a unified image-formation theory, a new method that describes the influence of light-matter interactions on the resolution limit is required. Here, we develop an intuitive technique using double-sided Feynman diagrams that depict light-matter interactions to provide a bird’s-eye view of microscopy classification. This diagrammatic methodology also allows for the optical resolution calculation of all types of microscopy. We show a guidepost for understanding the potential resolution and limitation of all optical microscopy. This principle opens the door to study unexplored theoretical questions and lead to new applications.


Supplementary Note 1: Category 1)
In category 1), a coherent system with the LO, we first consider laser microscopy with a finite-sized single-pixel photodetector as a typical example. For simplicity, we consider the easiest light-matter interaction: an (1) -derived interaction such as linear absorption. The diagram (including the LO) is shown in Fig. 1 (e)

Supplementary Note 2: Category 2)
In category 2), a coherent system without the LO, we again begin with laser microscopy. We first consider the third harmonic generation, which is a This means that although the information on the object frequency inside the 3-D aperture reaches the detector, the image changes to some extent based on the detector size. A smaller detector size seemingly allows for a better optical resolution, forming a sharper image of a single point object, but the largest object frequency acquired is independent of the detector size. In addition, the OTF cannot be defined in this case without the LO.
Next, we consider Kohler illumination microscopy with a (1) -derived interaction under the condition of partially coherent image formation. In this case, the interference between the diffracted lights is not negligible and the LO is absent. The 3-D aperture can be calculated by convolving ill (− ) with col ( ) in the relevant diagram (see Fig. 1

Supplementary Note 3: Category 3)
In category 3), an incoherent system with the LO, we take fluorescence as an example of the light-matter interaction. In fluorescence, which is one of the Next, we consider Kohler illumination fluorescence microscopy. The 3-D aperture can be computed in the same manner. By replacing ex ( ) with ill ( ) and using the relation ill (− ) ⊗ ill * ( ) = ( ), the 3-D aperture reduces to A( ) = col * (− ) ⊗ col ( ), assuming that the 2-D detector has a sufficiently small pixel size. The OTF of the imaginary part is also proportional to the 3-D aperture in this case, which is consistent with the well-known OTF of wide-field fluorescence microscopy 1 .

Supplementary Note 4: Category 4)
In category 4), an incoherent system without the LO, we use spontaneous parametric down-conversion (SPDC) as an example of the light-matter interaction. In SPDC, two fields referred to as the signal and idler are generated by the excitation filed 2 . The diagram of SPDC, which is a (2) -derived interaction, is described in Fig. 1 (b) of the main manuscript, where we presume that only the signal is observed; the idler is not detected. Again, we start with laser microscopy. By convolving all the functions corresponding to each arrow, the 3-D aperture is found to be The Feynman diagram expresses light-matter interactions, including information on the NAs and the wavelengths in terms of pupil functions. Thus, we propose a theorem concerning the resolution limit of a microscope system: for a given type of light-matter interaction, the Feynman diagram determines the upper limit of the 3-D aperture for far-field optical microscopy if a priori information on the sample does not exist. Even in the case of an enhanced high signal-to-noise ratio or a different type of excitation/signal-collection system from laser-scanning and Kohler illumination microscopy, the object frequency outside the upper limit determined by the Feynman diagram is never acquired. This holds true even for structured illumination microscopy 3 , image scanning microscopy 4 , and stimulated emission depletion microscopy 5 .

Supplementary Note 6: Real space representation
In order to address optical microscopy, we move from the frequency space to the real space. The plane wave of an excitation laser beam can be assumed to be in a coherent state with a frequency of = ( , , ): where α is a complex number and | ⟩ is the number state for the plane wave with frequency f (wavenumber k=2πf) 6 . Because the laser beam is focused on the sample by the excitation objective, the corresponding excitation state is represented by the direct product of all modes restricted by the NA and wavelength: where ex ( ) represents the 3-D pupil function for the excitation objective. In coherent light-matter interactions, we consider only the coherent states for laser beams. When two different laser beams (ex1 and ex2) are employed for the excitation, the state representing the excitation condition becomes | ⟩ ex1 | ⟩ ex2 .

Supplementary Note 7: The vacuum state in microscopy
For incoherent light-matter interactions, we incorporate the vacuum state |0⟩, which exists around the sample, into the formulation as one of the excitation sources. For this purpose, we consider the direct product of the coherent state and vacuum state | ⟩ ex |0⟩ as the excitation condition. The vacuum state |0⟩ contains all modes |0⟩ with frequencies f:

Supplementary Note 8: Operator in real space for microscopy
We now introduce the basic idea of the annihilation and creation operators in real space, â(x) and â + (x), using the 3-D pupil function P(f): where â(f) and â + (f) are the annihilation and creation operators in frequency space, respectively. In a similar way, we can define the annihilation operators in real space for the excitation laser field âex(x), the vacuum field around the sample âvac(x), the local oscillator field derived from the vacuum field âlo(v)(x), the local oscillator field due to the excitation laser field âlo(l)(x), the signal field emitted from the sample âsig(x), and the signal field collected by the detector âcol(x) as: Here, ex ( ) is the 3-D pupil function for the excitation system expressed by the product of the 2-D pupil function for the excitation system ex (2) ( , ) (which includes the laser beam profile) and the spherical shell truncated by NA ex (3) ( , , ), which has delta-function characteristics in the radial direction. V( ), describing the vacuum field, represents a complex random function whose modulus is unity, col ( ) is the 3-D pupil function for the signal-collection system, which is the partial sphere with a modulus of one, and fsig is the modulus of the wavenumber for the signal field. Note that fsig considers the refractive index of the sample. Information about the aberration is included in the pupil functions. Thus, we define the operators describing the excitation and signal fields in microscopy, which also allows for the calculation associated with the vacuum field.

Supplementary Note 9: Conversion of q-number into c-number
We convert q-number (operator) to c-number by acting on the state. To unify the framework for coherent and incoherent light-matter interactions, the classical field is replaced by the operator. Then, the operator acts on the bra or ket describing the excitation condition. Because the vacuum field inevitably exists around the sample, we always include both the coherent state for the laser and the vacuum state in the excitation condition, i.e. as | ⟩ ex |0⟩ . For example, using the relation ̂e x ( )| ⟩ = | ⟩ , the calculation is as follows: where ASFex(x) is the amplitude spread function (ASF) formed by the excitation laser beam on the sample through the excitation objective. The calculation related to the vacuum field is as follows: where hcol(xd) is the ASF formed by the signal field on the detector through the signal-collection

Supplementary Note 10: Commutation relation of operators
For the convenience of formula transformation and simplification, we calculate the commutation relation between the annihilation and creation operators in real space: using the random phase nature of * ( d ). Thus, we formulate the commutation relation of operators in real space by using the conventional operators in frequency space to address microscope system. The solid, dotted, and wavy arrows represent the excitation, vacuum, and signal fields, respectively.

Supplementary
For fluorescence, two diagrams exist that differ in the order of the electric field that first excites the ket side | ⟩ or bra side ⟨ |. | ⟩ and | ⟩ represent the ground and excited states, respectively.
Sequential excitation by two electric fields ← (̂e x + ) and → (̂e x ) correspond to a photon. (b) Interactions between molecular ensemble and light. The vacuum field is involved in incoherent interactions. (c) Examples of Feynman diagrams describing interactions such as sum frequency generation, third-order harmonic generation, coherent anti-Stokes Raman scattering, and spontaneous Raman scattering. Only one diagram is shown to represent an interaction, however in reality multiple simultaneous diagrams exist.