Supplementary Information for - PNAS

Supplementary Information for How to better focus waves by considering symmetry and information loss Kai Lou, Steve Granick, François Amblard Steve Granick, François Amblard

Email: [email protected] and [email protected] This PDF file includes: Supplementary text Figs. S1 to S14 References for SI reference citations

1 www.pnas.org/cgi/doi/10.1073/pnas.1803652115

Supplementary Information Text a) Definition of focal shift. Focusing a wave primarily means that the cross-section of the flow of its energy somehow shrinks. As a consequence, at the center of the focal region, one expects the energy density to reach a global maximum in all directions, and the focus can possibly be defined as the position of maximum intensity along the optical axis. Energy concentration in transversal planes is also a defining feature of the focus, and the focal plane could alternatively be defined as the plane of maximal concentration. However, a few reports show that the notion of encircled energy is rather complex. Indeed, the radius of the energy envelope (see main text) reaches a minimum at a plane that depends on the energy fraction considered to define that envelope. In other words, the focal shift is uniquely define from the axial intensity profile, but not from the radius of energy envelopes. However, for energy fractions corresponding to the typical definition of the beam waist, e.g. FWHM or the 𝑒 ratio, the experimental difference between the focal shifts seen with these two definitions is negligible. In addition, from the theory point of view, if the focal shift is defined from the transverse energy distribution, it can only be assessed by a secondary spatial integration of the diffraction integral. This can be a rather complex task for which we know no solution. While the only experimental results on the shape of the energy envelope have not yet been modeled, the heuristic arguments in this paper apply.

b) Circular spatial focusing Although spatial and temporal focusing has been studied to some extent (1-4), it is still not clear yet what determines its axial resolution. In this first note, we study theoretically the question of focal shift and axial resolution, in the context of the optical set-up described in this paper (Fig. S4 and S9). A first consideration has to do with an index mismatch interface. Indeed, the laser beam is focused through a plane interface between two isotropic media with distinct refractive indices (n1 and n2) (Fig. S13). Let’s first consider the case when the back aperture is illuminated by an optical field that is generically described by its complex amplitude 𝑈 𝜉, 𝜂, 𝜔 defined as a function of Cartesian coordinates over the aperture and of the light frequency 𝜔. We treat this problem in the context of the scalar diffraction theory within the paraxial approximation. In the focal region, the diffraction integral reads (5): 𝑈 𝑥, 𝑦, 𝑧, 𝑡

∬𝒜 𝑈 𝜉, 𝜂, 𝜔 𝑒

,

𝑒

𝑑𝜉𝑑𝜂𝑑𝜔

(S1)

where 𝒜 is the aperture surface, k is the wave number corresponding to the central wavelength of the spectrum, 𝛺 is the spectral bandwidth, (x,y,z) the Cartesian coordinates in the focal region, 𝑧 𝑧 𝑧 ⁄𝑛 , 𝑧 𝑧 𝑛 𝑓 𝑧 and 𝑛 𝑛 ⁄𝑛 . Using the Fresnel approximation, 𝑧 equation (S1) yields: 𝜓 𝜉, 𝜂

𝜓 𝜉, 𝜂

𝑧

𝑓

𝑛 𝑧

𝑧

(S2)

where 𝒜 is the aperture surface, k is the wave number corresponding to the central wavelength of 𝑛 ⁄𝑛 1, then equation S2 becomes: the spectrum. If we consider 𝑛 𝜓 𝜉, 𝜂

𝑧

𝑓

(S3)

At that stage, the classical treatment of the diffraction integral approximation would approximate this phase function by its linearized version, in other words, to consider that:

2

(S4) Not making this “linear pupil approximation” of the pupil phase function is the key point for finding the accurate focal shift. Interestingly, linearization is needed to transform the diffraction integral into a Fourier transform integral. In other words, in the framework in which one can study the focal shift, the Fourier formalism cannot apply, except for z=f. If the input laser beam is monochromatic or if its spectral components are all carried by the same wavefront, and if the beam has a circular cross-section with radius a, the input field can be simplified as 𝑈 𝜌, 𝜗 1, where 𝜌 𝜉 𝜂 , 𝜗 tan 𝜂 ⁄𝜉 . The (x,y,z) coordinates in the focal region can be replaced by (𝑟, 𝜑, 𝑧 . The complex amplitude then becomes: 𝑈 𝑟, 𝜑, 𝑧

𝑈 𝜌, 𝜗 𝑒

,

𝜌𝑑𝜌𝑑𝜗

S5

𝜌𝑑𝜌

(S6)

leading to the field intensity given by : 𝐼 𝑟, 𝜑, 𝑧

𝐽

For observations on the 𝑧

𝑒

𝑧

plane, equation (S6) yields:

𝐼 𝑟, 𝜑, 𝑧

∙

∙

(S7)

𝑅⁄𝑓 is the convergence angle under the paraxial approximation. For observation where 𝜃 points on the 𝑟 0 axis, the axial intensity distribution is: sinc

𝐼 𝑧 where 𝒩

(S8)

𝑎 ⁄𝜆𝑓 is the Fresnel number.

On the condition of focusing through the planar media interface, the focal shift is ∆𝑓 where 𝜇

2⁄ 𝜋𝑁 , 𝜎

𝜋𝑁 𝑓

𝑧 ⁄ 2𝑧

𝑛 𝑓

(S9)

is the solution of the equation tan 𝜎

𝜎 1

𝜇𝜎 .

c) Non-circular spatial focusing For non-circularly filled apertures, when light is focused on the back aperture by a cylindrical lens followed by a spherical lens, equation (S1) yields: 𝑑𝜉𝑑𝜂

∬𝒜 𝑈 𝜉, 𝜂 𝑒

𝐼 𝑥, 𝑦, 𝑧

(S10)

and 𝑈 𝜉, 𝜂

𝑒

(S11)

where √2 ln 2 𝑎 and √2 ln 2 𝑎 are the values of the beam radii at FWHM in both directions. From (S10), the light intensity on the (0, 0, z) axis becomes: 𝐼 𝑧

(S12) ⁄

⁄

3

1⁄𝑧 1⁄𝑓, 𝑁 where 1⁄𝑧 𝑎 ⁄𝜆𝑓 and 𝑁 𝑎 ⁄𝜆𝑓 are the Fresnel numbers in the zx- and zy-planes, respectively. It should be noted here that the case of elliptical apertures has only been treated for very small Fresnel numbers (0.3 𝑁 10) (6), and never through an interface for noncircular apertures. d) Temporal focusing In temporal focusing, the spectral components of the ultrashort laser are first spatially dispersed and then collected by a collimating lens. The complex amplitude at the aperture can therefore be written as (3): 𝑈 𝜉, 𝜂, 𝜔

𝑒

(S13)

From equation (S1) the field amplitude in the focal region can be written as 𝑒

𝑒

𝑈 𝑥, 𝑦, 𝑧, 𝑡

𝑒

𝑑𝜔

(S14)

and its value on the z-axis is 𝑒

𝑈 𝑧, 𝑡

𝑒

𝑒

𝑑𝜔

(S15)

where Γ

Ω

(S16)

and 𝛾

(S17)

Integrating the squared intensity 𝐼 𝑈𝑈 ∗ over time t and the transverse plane yields the linear density of the two-photon excitation (TPE) signal projected on the z-axis: 𝑇𝑃 𝑧

𝐼 𝑧, 𝑡 𝑑𝑡

(S18)

where 𝒩 𝑎 ⁄𝜆𝑓 and 𝒩 𝑎 𝛼 Ω ⁄𝜆𝑓 are the Fresnel numbers for the spatial focusing and the temporal focusing, respectively. e) Approximation of the focal shift and axial FWHMs in line-temporal focusing Let’s now consider the initial beam to be a non-circular light beam with beam sizes of ax and ay along x-axis and y-axis, respectively. Let us introduce the unidirectional Fresnel numbers 𝑁 𝑎 ⁄𝜆𝑓 . The two corresponding unidirectional waists are 𝑤 , 𝑎 ⁄𝜋𝑁 and 𝑎 ⁄𝜆𝑓 and 𝑁 ⁄ ⁄ ⁄ 𝑎 𝜋𝑁 , and the Rayleigh lengths 𝑍 , 𝑓 𝜋𝑁 and 𝑍 , 𝑓 𝜋𝑁 . The intensity is then 𝑤, given by 𝐼 𝑥, 𝑦, 𝑧

𝐼 𝑧 𝑒

(S19)

4

where the beam extension in the x and y direction is given by 𝑤 𝑧

𝑤

,

1

𝑤 𝑧

𝑤

,

1

(S20)

,

(S21)

,

Because we assumed no loss of the optical power Ptot along the axis, the axial intensity 𝐼 0,0, 𝑧 that we want to know can be inferred given from the following conservation equation 𝐼 𝑧

𝑃

∬

𝑑𝑥𝑑𝑦

𝑒

(S22)

Because of equations S20 and S21, the axial intensity distribution is proportional to the product of two functions ℒ 𝑧 and ℒ 𝑧 that reflect the contribution of the two focusing planes zx and zy, and its overall shape can be written as: ℒ 𝑧

ℒ 𝑧 ℒ 𝑧

(S23)

where ℒ 𝑧 ℒ 𝑧

1 1

⁄

𝜋 𝑁 𝑧 ⁄𝑓 𝜋 𝑁 𝑧 ⁄𝑓

(S24)

⁄

(S25)

How much the axial intensity profile spreads when the aperture is reduced in the x direction is classically described by ℒ 𝑧 . Seen from Fig. 3B and S7, it is obvious that the temporal focusing along x-axis is independent of spatial focusing along y-axis. Our purpose here is to evaluate the impact of focal shift to the axial spread in line-temporal focusing. An exact analysis is not known. But from Fig. S4, it can be seen that the focal shifts and axial FWHMs in non-circular spatial focusing and temporal focusing are closely similar. To make an easy and simple evaluation, we use equations S24 and S25 to approximately express the axial intensity expression for spatial and temporal focusing, respectively. In this condition, we propose here to introduce the focal shift effect by a simple translation of the argument of the axial intensity distributions by a shift Δf into the intensity profile. This becomes: ℒ

𝑧

1

𝜋 𝑁 𝑧 ⁄𝑓

⁄

∙ 1

𝜋 𝑁 𝑧

∆𝑓 ⁄𝑓

⁄

(S26)

From equations S26 we can numerically compute values of the axial FWHMs and the focal shifts for different sizes of line-temporal focusing. f) Group velocity dispersion in temporal focusing In the condition of SI Appendix c, if there is group velocity dispersion in the laser pulses, the second-order dispersion Π of the chirped pulses results in a curved wavefront, which acts as an extra lens that induces the temporal focal plane to shift. The shift of the temporal focal plane is Δ𝑧 Π 𝑧 , where 𝑧 𝜆𝑓 ⁄ 𝜋 𝑎 𝛼 Ω . In our experiment, the chirp created by unwanted sources of dispersion can be compensated by a prism pair. In spatial and temporal focusing modes, the geometry of the quadratic intensity distribution in the focal region was measured by imaging the two-photon excited response of a fluorescent solution, and curves were fitted for the peak intensity position and the FWHM (full axial width at half maximum).

5

g) Line-temporal focusing In line-temporal focusing, when the focal planes SF and TF (see Fig. 3A) corresponding to spatial (zy-plane) and temporal focusing (zx-plane) are made to coincide by fulfilling the back aperture of the objective lens, the quadratic intensity profile is proportional to (7): 𝐹

1

𝑧⁄𝑧

⁄

1

𝑧 ⁄𝑧

⁄

(S27)

where the Rayleigh lengths zS and zT correspond to the temporal and non-circular spatial focusing, respectively. The temporal Rayleigh length is given by 𝑧 𝐴 𝜏⁄ 𝐵 ∙ 𝜏 𝐶 ∙ 𝑀 ∙ 𝑁𝐴 , where A, B and C are constants that depend on the system parameters. In our experiment, the fitting parameters A=0.165, B=0.89, C=8.2. In the circular aperture case with spatial focusing, the light intensity exhibits axial symmetry about the optical axis and is also symmetric about the focal plane, provided the aperture is large enough (Fig. S1A). However the symmetry about the focal plane is broken because of the shift of the intensity peak. In addition, the intensity distribution becomes skewed, thus breaking the symmetry about the shifted peak plane (Fig. S1B). In the case where the back aperture is filled with non-circular spatial focusing (Fig. S2), a line focus is obtained along the x-axis with a length that is longer than the Rayleigh length of the zy focus (Fig. S2A). When the back aperture is maximally filled in the y direction, the intensity distribution is symmetric about the focal plane, but that symmetry is broken for smaller y-apertures. (Fig. S2B). By comparing the situations of circular aperture vs. elliptic aperture, the focal shift and the axial-FWHM increase obviously faster in the latter case (Fig. S4).

6

Fig. S1. Simulated 3D intensity for circular spatial focusing. Numerical simulations of the squared intensity distribution produced by focusing a circular collimated laser beam with full diameter at FWHM of 7 mm. (A) through a spherical lens (f = 18 mm), or (B) through the afocal combination of a first (f = 300 mm) and a second spherical lens (f = 18 mm).

7

Fig. S2. Simulated 3D intensity in non-circular spatial focusing. (A) Numerical

simulations of the squared intensity distribution produced by focusing a circular collimated laser beam with full diameter at half maximum of 7 mm through (A) an x-axis cylindrical lens (f=300 mm) and then focused by a second spherical lens (f=18 mm) with a 318 mm distance between them. (B) A slit is inserted before the spherical lens in (A) that reduces the y-aperture size to 0.5 mm.

8

Fig. S3. Setup of focal shift measurement with non-circular apertures. (A)-(C) A femtosecond incident laser beam was reflected by a blazed grating (600/mm) or a reflecting mirror, and collected by a cylindrical (Cyl) lens or spherical (Sph) lens with the same 300 mm focal length for both options. The beam size was controlled by a 2D slit at the back aperture of the focusing objective. The laser pulses were focused by an objective lens into a cuvette sealed with a cover glass and filled with rhodamine B dye. The fluorescence emission was collected by a perpendicular objective lens and imaged through a tube lens and an emission filter into an EMCCD. BA, back aperture of the objective lens. Right shows the light illumination at BA and fluorescence distribution in the xy plane.

9

Fig. S4. Focal shifts and Axial-FWHMs for circular vs. non-circular spatial focusing.

From the quadratic intensity distributions simulated and shown on Fig. S1 and S2, the axial FWHM (dash line) and the focal shift (solid line) are plotted for circular apertures (circles, from Fig. S1) and for elliptic apertures (triangles, from Fig. S2), as a function of the FWHM of the diameter or the aperture size along the y-direction which is the long side of the ellipse (see Fig. 2).

10

Fig. S5. Focal shifts and Axial-FWHMs for temporal focusing vs. non-circular spatial focusing. The focal shifts and axial-FWHMs are shown (solid lines and dashed lines, respectively) for the quadratic intensity of the temporal (triangles) and non-circular spatial (circles) focusing as a function of the y-aperture size for spatial focusing and xaperture (spectral dispersion) for temporal focusing (see Fig. 2).

11

Fig. S6. Focal shifts in temporal focusing with different group velocity dispersions

(GVDs). The axial fluorescence peak position and FWHM are shown. The upper panel displays the fluorescence zx-images with different GVD values; for each of these GVDs, the bottom panel shows how the focal shifts changes with aperture.

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Fig. S7. Focal plane gap and axial-FWHMs in line-temporal focusing with different

GVDs. The back aperture of the objective is filled with the rainbow beam shown in Fig. 3B. (A) SFP/TFP split focus on zx-images for different GVDs. (B) The axial-FWHMs for spatial focusing (blue dashed lines) and temporal focusing (red solid line) with different gaps between SFP and TFP. (C) Data analysis with the method discussed in SI Appendix e.

13

Fig. S8. Focal shift and axial-FWHM in line-temporal focusing with different y(spatial) fill factors. The back aperture of the objective is filled with the rainbow beam shown in Fig. 3B. Focal shift and axial-FWHMs on zx-images for different y- (spatial) fill factors.

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Fig. S9. Diagram of two-photon imaging system with line-temporal focusing

microscopy and point-scanning microscopy and line-scanning microscopy. Laser power is controlled by a half waveplate (HWP) and a Faraday isolator (FI), laser spectrum is measured by a fiber spectrometer (FS). The laser beam passes through a telescope (T), reflects from the spatial light modulator (SLM) and propagates in two optical exclusive branches, namely a first “Line-temporal focusing microscopy” branch (yellow), or a second “point-scanning microscopy” branch (green). 1/ In the yellow branch , the laser beam is reformed along y-axis onto 1D galvanometric mirror (GM1) by two cylindrical lenses (CL1 and CL2), focused by cylindrical lens (CL3), diffracted by a blazed grating (G), reflected from a mirror (M1), and a collimating lens (LC) collects laser frequencies into the objective lens. A schematic diagram is inserted to show how the laser spectrum is modulated by a computer interface, with an intra-cavity prism pair and a controlled mode-locking aperture. 2/ In the green branch, the laser beam is reflected by a switchable mirror (FM1), focused by a spherical lens (L1), reflected by a mirror (M2) and then collimated by a spherical lens (L2) onto a two-dimensional galvanometric mirror (GM2). The beam is then focused by a spherical lens (L3), reflected by a mirror (M3), collimated by a spherical lens (L4) and then reflected by another switchable mirror (FM2) that reinjects the light in the common part of the set-up. The laser beam from two branches is reflected by a dichroic mirror (DM) into the objective lens (Obj). The fluorescence emission is collected back from the same focusing objective, and is imaged through a dichroic mirror (DM) and a tube lens (TL) onto an EMCCD camera (see Methods).

15

Fig. S10. Line-temporal focusing images taken with different cylindrical lenses (CL3) and collimating lenses (LC) and a constant laser spectrum. See Fig. S9, immobilized 0.5 μm diameter fluorescence beads were imaged by line temporal focusing microscopy using a fixed spectrum (𝛀 𝟔. 𝟔 nm) and different cylindrical lens CL3 (fC) and collimating lens LC (fL), with the objective (10x NA=0.45, see Methods). The transverse resolution is best seen by the xy image, while xz and yz images show the axial resolution.

16

Fig. S11. Optical section images taken with different laser spectra. Spectrum of the

laser output for different values of the intracavity-modulated spectral bandwidth (𝟔. 𝟔 𝛀/𝐧𝐦 𝟐𝟏. 𝟒). (A)-(I). Immobilized 0.5 μm diameter fluorescence beads were imaged by line-temporal focusing microscopy, using fC (CL3) = fL (LC )= 500 mm. xz images are shown for different values of 𝛀/𝐧𝐦 : 6.6, 7.7, 9.0, 11.2, 12.7, 14.3, 16.7, 19.5 and 21.4, for respectively (A) to (I). Scale bars, 5 μm.

17

Fig. S12. Comparison between line-temporal focusing microscopy and point-

scanning microscopy. Immobilized 0.5 μm diameter fluorescence beads were imaged by two photon excitation using line-temporal focusing microscopy and point-scanning microscopy with the same objective (10X NA0.45). The circular back aperture of the objective is filled. The laser spectrum bandwidth in both experiments is Ω = 21.4 nm. Orthogonal images are shown for both home-built microscopies. Scale bars, 10 μm.

18

Fig. S13. Improved axial resolution in line-temporal focusing microscopy. Fluorescence response of 0.5 µm diameter microspheres and xz images of a fluorescent mouse lung sample imaged with line-temporal focusing microscopy (LTFM) and pointscanning microscopy (PSM). (A)-(B) Axial FWHMs of LTFM (red) as a function of the rainbow-beam size, compared with PSM (blue line) and its theoretical near-diffractionlimited resolution (black dotted line). Laser bandwidth Ω was fixed to 6.6 nm and the focal length fC and fL changed from 200 to 1000 mm in (A); fC = fL = 500 mm and Ω changed from 6.6 to 21.4 nm in (B). (C) Axial fluorescence profiles of LTFM and PSM with fC = fL = 500 mm and Ω=21.4 nm. (D) xz-images of a mouse lung sample by LTFM and PSM in the same region, right figures show the 2D Fourier transforms of the correspondent xz-sections.

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Fig. S14. Focusing through a plane interface. Schematic representation of the light

focused through an interface by a lens with focal length f, through a succession of two homogeneous and isotropic media, with refractive indices n1 and n2, separated by a planar interface at position z0. The effective focus is shifted to the position zapp beyond the expected focal plane if n1< n2.

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References

1. 2. 3. 4. 5. 6. 7.

Born M & Wolf E (1999) Principles of optics : electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, Cambridge ; New York) 7th expanded Ed pp xxxiii, 952 p. Self SA (1983) Focusing of spherical Gaussian beams. Appl. Opt. 22(5):658-661. Zhu G, van Howe J, Durst M, Zipfel W, & Xu C (2005) Simultaneous spatial and temporal focusing of femtosecond pulses. Opt. Express 13(6):2153-2159. Dana H & Shoham S (2011) Numerical evaluation of temporal focusing characteristics in transparent and scattering media. Opt. Express 19(6):4937-4948. Stamnes JJ (1986) Waves in focal regions : propagation, diffraction, and focusing of light, sound, and water waves (A. Hilger, Bristol ; Boston) pp xviii, 600 p., 604 p. of plates. Kathuria YP (1987) Fresnel and Far-Field Diffraction of a Truncated Elliptic Gaussian-Beam Due to an Elliptic Aperture. J. Mod. Optic. 34(8):1085-1092. Dana H, Kruger N, Ellman A, & Shoham S (2013) Line temporal focusing characteristics in transparent and scattering media. Opt. Express 21(5):5677-5687.

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