Optical nano-imaging via microsphere compound lenses working in non-contact mode

Microsphere lens for nano-imaging has been widely studied because of its superior resolving power, real-time imaging characteristic, and wide applicability on diverse samples. However, the further development of the microsphere microscope has been restricted by its limited magnification and small field-of-view. In this paper, the microsphere compound lenses (MCL) which allow enlarged magnification and field-of-view simultaneously in non-contact imaging mode have been demonstrated. A theoretical model involving wave-optics effects is established to guide the design of MCL for different magnifications and imaging configurations, which is more precise compared with common geometric optics theory. Experimentally, using MCL to image the specimen with a tunable magnification from 2.8× to 10.3× is realized. Due to the enlarged magnification, a high-resolution target with 137 nm line width can be resolved by a 10× objective. Besides, the field-of-view of MCL is larger than that of a single microsphere and can be further increased through scanning working manner, which has been demonstrated by imaging a sample with ∼76 nm minimum feature size in a large area. Prospectively, the well-designed MCL will become irreplaceable components to improve the imaging performances of microsphere microscope just like the compound lens in the conventional macroscopic imaging system.


und lenses working in non-contact mode: supplemental document
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The
imaging schematics of MCL working in different imaging configurationsFig. S1.The imaging schematics of MCL working in (a) virtual-virtual (VV), (b) real-real (RR) and (c) real-virtual (RV) imaging configurations.

Derivation of image distance, agnification and conditions for virtual-real imaging configuration

Fig. S2.MCL nano-imaging schematic.P1 and P2 are the pole of bottom spherical surface of bottom and upper microsphere, respectively.n0, n1 and n2 are the refractive index of environment, bottom microsphere and upper microsphere.r1 and r2 are the radius of bottom microsphere and upper microsphere, respectively.l and g represent the object distance and gap between microspheres, respectively.

The schematic of MCL for nano-imaging is presented in Fig. S2.According to the geometric optics theory [1,2], the imaging distance (position) and magnification of a single microsphere can be calculated by Eq. (S1) and Eq.(S2), respectively.

( ) ( )
2 0 1 0 1 0 0 1 2 - 2 +2 2 - 2
n n rl n r l r n n l n r n r
− ′ = + − ,(S1)
( ) ( )
1 1 0 0 1 = 2 - 2 n r l n n r n n β + − . (S2)
The derivation of Eq. (S1) and Eq.(S2) are based on the relationship between object and image distance for a single spherical surface [1].Where l and  ‫׳‬ are object distance and image distance, respectively.The origin point used for measuring the object distance and image distance is same, i.e. the pole of the bottom spherical surface of the microsphere.If object or image locate at the bottom side of the origin point, l or  ‫׳‬ takes a negative value.Otherwise, they take a positive value.So, in equation (S1) and (S2), the object distance l should take a negative value for calculation. is the magnification coefficient.r is the radius of microsphere.n 0 and n 1 are the refractive index of environment and microsphere, respectively.

In the MCL, the image distance and magnification of the bottom microsphere can be calculated with Eq. (S1) and Eq.(S2).The upper microsphere t ct distance and image distance of the upper microsphere is the pole alculations for bottom and upper microspheres, the object distance for the upper microsphere is the image distance of the bottom microsphere plus the gap between two microspheres and minus diame er of the bottom microsphere.The calculation of image distance (position) and magnification for upper microsphere can also reuse Eq. (S1) and Eq.(S2) by replacing the value of object dis ance accordingly.It is an iterative process.So, the image distance  ‫׳‬ and magnification  MCL of a MCL can be calculated by Eq. (S3) and Eq.(S4) as following:
( ) ( ) ( ) ( ) ( )( ) 2 0 1 1 0 1 1 1 1 0 0 1 1 1 2 0 2 2 1 1 0 2 MCL 2 2 0 1 1 0 2 2 n r l r n n l n r n r n n n l g r n r n r
 − ′ =  + −   ′ − −  ′ =  ′ − + −  , (S3) ( ) ( ) ( )( ) (
)
1 1 1 1 0 1 0 1 2 2 2 2 0 1 1 2 0 2 MCL 1 2 = 2 2 = 2 + 2 2 = n r l n n r n n n r n n l g r r n n β β β β β   − + −     ′ − − + −   ×    , (S4)
To note, if a distance locates at the bottom side of P2, it is a negative value.Otherwise, it is a posit r calculation.

Besides, if we obtained a positive  ‫׳‬ , it means the image produced by the MCL te at the bottom side of P2.

When the special case that a MCL is put near the sample surface ( ≈ 0) and the two microspher r (g=0) is considered, based on the above iterative processes, we can obtain a simplified magnification coefficient  MCL of the MCL:
( )( ) ( ) 1 2 2 MCL 0 2 0 1 2 0 2 0 1 = 2 2 4 n n r n n n n r n n n r β − − − − . (S5)
Where r 1 and r 2 are the radiu

of bottom and upper microsphere, respectively.n 0 , n 1 and n 2 are the refractive ind
x of environment, bottom microsphere, and upper microsphere, respectively.When the upper mi rosphere works near the boundary between real and virtual imaging manner,  → ∞.The radius ratio of two microspheres should satisfy:
( ) (
)( )
0 2 0 2 1 1 0 1 0 2 4 = 2 2 n n n r k r n n n n − = − − , (S6)
When the magnification  MCL of the MCL is 1, the radius ratio of two microsphere should satisfy:
( ) (
)( )
0 2 0 1 2 2 2 0 0 1 0 2 2 = n n n r k r n n n n n − = + − − ,(S7)
Hence, the condition for the MCL working in virtual-real (VR) imaging configurations and remaining magnifying function is:
1 2 1 2 k d d k > > .
(S8) Where k 1 and k 2 are the two boundaries.Significantly, if we substitute the refractive indices of material with effective refractive index of microsphere in all the above equations, we can calculate the related parameters with higher precision.


The differences in the focal lengths of microsphere simulated by 3D and 2D FDTD model

The focal length of microsphere simulated by 3D and 2D FDTD models are slightly different.

To check the difference, we have simulated focal lengths of five silica microspheres (diameter 10 μm -18 μm) at 405 nm wavelength using both 3D and 2D FDTD models.The results are shown in the Fig. S3.It can be found that the relative ratio ∆ (∆ = | −  |/ ) of change in the focal length obtained by 2D and 3D FDTD models is very small (<0.044) within the simulation range.Besides, the overall variation trend is that ∆ reduces as the increase of microsphere diameter.From this study, we predict that the difference between 2D and 3D FDTD models should have limited effects on our conclusion.In addition, the 3D model always requires a significantly larger running memory.For example, the RAM of the working station used for simulation in this work is 36 GB, which ultimately can support the full-wave simulation for a single ~18 μm silica microsphere with 3D FDTD model (meshing accuracy is set as default level 2 in FDTD software).Although the simulation for microsphere by 3D model should be more precise, limited by the simulation resources and for consistency, the 2D FDTD models are employed to simulate the focal length of all 10 μm -120 μm microspheres in this work.Actually, simulating the focal length of microsphere with 2D model is widely adopted in literatures.Although the focal lengths of microsphere obtained by 2D model are not very accurate, without loss of generality, it can still reflect the fundamental trend of changes.In this work, the simulations for silica microsphere and MCL are mainly utilized as an example to demonstrate the effectiveness of the concept "effective refractive index".Since both the calculation of effective refractive index and full-wave imaging simulation are based on 2D FDTD model, considering the slight difference between 2D and 3D models, their results presented in Fig. 2 will not have significant changes if 3D model simulation is used.In comparison with the ray tracing results in the same figure, such small discrepancies do not affect our analysis and conclusion.


Predictio

of imaging configurations with effective refractive index and refractive index of material


The SEM i
age of Blu-ray disc

The minimum feature size and period of the Blu-ray disc used in experiments are100 nm and 320 nm, respectively.The SEM image of Blu-ray disc is presented in Fig. S5.


Experimental procedures to control and validate the gap distance between the two microspheres

The schematic of experimental procedures to control and validate the gap distance between the two microspheres are shown in Fig. S6.The two microspheres in MCL are moun

d on two holders which are con
ected to and controlled by two separate nano-stages.The two nanostages together with the sample are fixed on a microscope sample stage which possesses a high movement resolution (~100 nm).The real-time z coordinate of the sample stage surface is displayed in the control software.In the first

ep, the two microspheres are kissed with each other.The equator of the upper microsphere is mo
ed up to the imaging plane of the objective lens together with the sample stage.Then, the z coordinate of the sample stage is recorded.In the second step, an isolated upward movement of the upper microsphere to a certain distance is induced by the connected nano-stage.In the third step, the upper microsphere is moved down with the sample stage so that its equator is relocated at the imaging plane of the objective lens.

The current z coordinate of the sample stage is recorded, and it minus the z coordinate recorded in step 1 is equal to the gap distance g1 between the microspheres after the first movement of nano-stage connected to the upper microsphere.Steps 4 and 5 are just repeated procedures described in steps 2 and 3, and then we can obtain the final gap distance after the series of movements between microspheres.Since the movement resolution of the sample stage is ~100 nm and the depth of field of the microscope with 100× objective lens is ~800 nm, the measurement accuracy for the gap distance is ~800 nm.According to experimental results shown in Fig 4(c) in the manuscript, the magnification does not change significantly with a ~800 nm variation of gap distance.So, the ~800 nm measurement accuracy for gap distance is good enough to verify the relationship between magnification and gap distance of microspheres.From the experimental procedures, the feedback mechanism for controlling and validating the gap distance is using a close-loop sample stage with accurate and quantified positioning to measure the equivalent gap distance with the assistance of optical microscopic imaging when the upper microsphere settles down after each movement.The gap distance measured by the feedback mechanism can be compared with the given movement distance of the nano-stage for further validation.


The effect of the working distance of bottom microsphere on magnification of microsphere compound lens

The working distance of the bottom microsphere in experiment is very short and difficult to be accurately measured by simple methods, including the one described in section 6 for gap distance measurement.So, in the experiment, we estimated that the microsphere compound lens working in non-contact mode through some indirect evidences, like the microsphere can smoothly move above sample for scanning mode imaging.Since the working distance of the bottom microsphere in experiment is quite short, th

effect of working distance on magnification calculation can be almost neglected.As an example, the magn
fication of an MCL consisting of 23 μm and 110 μm silica microspheres when its working distance varies from 0 to 500 nm is calculated based on the modified theory and shown in Fig. S7.The magnification decreases as the increase of working distance.However, the variation within this range is very small.Therefore, considering the working distance is quite short, its effect on calculations of magnification of the compound lens system is insignificant.

Fig. S3 :
S3
Fig. S3: (a) Focal lengths of silica microsphere obtained by 2D (blue line) and 3D (red line) FDTD models respectively.The black line indicates the absolute value ∆ (( ∆ = | −  |)), which is the deviation of focal length obtained by 2D and 3D FDTD models.(b) The relative ratio ∆ (∆ = | −  |/ ) of change in the focal lengths obtained by 2D and 3D FDTD models.


Fig. S4 .
S4
Fig. S4.The images of three-point sources simulated by ray tracing with (a) refractive index of silica, (b) effective refractive index of silica microspheres, and by (c) FDTD full-wave simulation with refractive index of s

ica.The MCL used in this case consists of 10 μm and 68 μm silica microsphere.P represents optical power.To demo
strate the effectiveness of the effective refractive index on predicting imaging configurations, an MCL consisting of 10 μm and 68 μm silica microspheres is designed to image three dipole sources by simulation.The results of 2D ray tracing simulation with refractive index of silica and effective refractive index are presented in Fig. S4.(a) and (b) respectively.The results with refractive index of silica and effective refractive index predict that the MCL should works in VV imaging configuration and VR imaging configuration, respectively.The 2D full-wave simulation