Supplementary Information Free-space creation of ultralong anti-diffracting beam with multiple energy oscillations adjusted using optical pen

Transcription

1 Supplementary Information Free-pace creation of ultralong anti-diffracting beam with multiple energy ocillation aduted uing optical pen Weng et al.

2 Supplementary Figure Normalized light intenity of ideal anti-diffracting light beam along z axi. (a) Beel beam; (b) Airy beam. Supplementary Figure Schematic of the focuing ytem. Ω i the focal phere, with it center at O and a radiu f, namely, the focal length of the obective len (OL). P i the pupil filter in the wavefront of the len. θ i the convergent angle. l ( θ ) 0 denote the electric amplitude of incident Gauian beam. Supplementary Figure 3 Normalized light intenity of quai-airy beam. Quai-Airy beam (a) i generated by the cubic phae (b), the parameter of which are η = 5, = 3 and ϕ0 = 0.75π.

3 Supplementary Figure 4 Light intenitie of quai-airy beam in two ymmetrical plane. (a, b) are the light intenitie in the z= 000λ, while their counterpart in z=000λ plane are hown in (c, d). The correponding tranvere energy fluxe are indicated by the green arrow. Point A denote the firt idelobe of the light beam. Supplementary Figure 5 Quai-Airy beam with different η under the condition of NA=0.095 and = 3. (a) Quai-Airy beam with η = 3 ; (b) Quai-Airy beam with η = 5. All are generated by the phae (e, h), repectively. (f, i) how the light intenitie of both light beam in the z=0 plane. (c, d) and (g, ) preent the correponding experimental reult. The FWHM of (a) and (b) are theoretically approximately 55.6λ and 560λ, repectively, while their experimental reult are (c) 5.λ and (d) 544.9λ, repectively. The light intenitie of quai-airy beam are normalized to a unit value, which are indicated by the color bar. The phae cale of (e, h) are 0~π.

4 Supplementary Figure 6 Quai-Airy beam with different σ under the condition of NA=0.095 and η=5. (a) Quai-Airy beam with σ=.6; (b) Quai-Airy beam with σ=3.6. All are generated by the phae (e, h), repectively. (f, i) how the light intenitie of both light beam in the z=0 plane. (c, d) and (g, ) preent the correponding experimental reult. The FWHM of (a) and (b) are theoretically approximately 3084λ and 4λ, repectively, while their experimental reult are (c) λ and (d) 49.3λ, repectively. The light intenitie of quai-airy beam are normalized to a unit value, which are indicated by the color bar. The phae cale of (e, h) are 0~π. Supplementary Figure 7 Beel beam generated uing high-pa pupil filter. (a) Light intenity of Beel beam in y-z plane; (b) High-pa pupil filter with a ratio of the inner to outer ring radiu r/r=0.8; (c, d) Light intenitie and energy fluxe (green arrow) at point P, P in the z= 300λ, 300λ plane, repectively. The light intenitie of Beel beam are normalized to a unit value, which are indicated by the color bar (e). The color bar (f) indicate the tranmittance of high pa pupil filter (b).

5 Supplementary Figure 8 Schematic of deriving optical pen in focuing ytem. Ω i the focal phere, with it center at O and a radiu f. A, B are the off- and on-axi point in Ω. O i an arbitrary point in the focal region. P i the pupil filter in the wavefront of the obective len (OL). l ( θ ) 0 denote the electric amplitude of incident Gauian beam and θ i the convergent angle of OL. Supplementary Figure 9 Shifting foci in the y-z plane. The location of foci in (a, c, e) are aduted by the phae in (b, d, f), the parameter of which are N=, =, δ =0, x=0, and (a) y= 5, z=0; (c) y=0, z=0; and (e) y=5, z= 0.

6 6 Supplementary Figure 0 Focal array in the focal plane. Identical 4 4 focal array in (a-c) can be created by the different phae in (d-f), repectively. Here, thee focal array are realized by multiplying two phae, which yield 4 focal array along the x- and y-axi. Thu, N=4, z =0, and δ =0. For 4 foci along the x-axi, y =0, with the foci located at x =, x = 3, x 3 = 9 and x 4 = 9. For 4 foci along the y- 3 3 axi, x =0, with the foci located at y =, y = 3, y 3 = 9 and y 4 = 9. Three pupil filter for the 4 focal array along the x- and y-axi are obtained, with =.05, =0.7, 3 =0.9, 4 = 0.8; = = 0.885, 3 = 4 =; and =.05, =0.7, 3 =0.95, 4 =0.33, which are denoted a l ~ l for the x-axi and ly ~ ly3 x x3 x y x y x3 y for the y-axi. Finally, the phae for (d-f) are (d) l l ; (e) l l ; and (f) l l. Supplementary Figure Arbitrary focal pattern in the focal plane. NANO (a) and OPT (b) are obtained by the phae (c, d), repectively. The light intenitie in (a, b) are normalized to a unit value, which are indicated by the color bar (e). The color bar (f) how the phae cale of (c, d). The parameter can be found in the Supplementary Note 3.

7 7 Supplementary Figure Arbitrary focal pattern in the different z plane. (a) O, P, and T are generated in the z= 0λ, z=0 and z=0λ plane imultaneouly uing the phae (b). The parameter can be found in the Supplementary Note 3. Supplementary Figure 3 UAD light beam with different number of energy ocillation under the condition of NA=0.8. (a) zero ocillation (.34λ); (b) one ocillation (3.96λ); (c) two ocillation (70.56λ); (d) three ocillation (99.50λ). All are generated by the phae (e-h), repectively. The parameter can be found in Supplementary Table. (i-k) Light intenitie and energy fluxe (green arrow) of point A, B, and C in the z= 0λ, 0, and 0λ plane. Here, all light intenitie are normalized to a unit value, which are indicated by the color bar (l). The color bar (m) how the phae cale of (e-h). Supplementary Table : UAD Light Beam under the condition of NA=0.8 N y y y 3 z z z η η η 3 δ δ δ 3 Parameter 0 NA=0.8 5 φ 0 =0.75π π 0.95π x = π 0 0.6π =

8 8 Supplementary Note : Definition of Finite Power for an Anti-Diffracting Light Beam In thi paper, the concept of finite power i adopted a an intrinic attribute of the anti-diffracting light beam. Note that the finite power i prevalent in the regime of anti-diffracting light beam: ee reference [-4]. To be conitent with thee reference, the power for an anti-diffracting light beam i defined a the integration of the amplitude quared along the propagation traectory. Therefore, for an ideal anti-diffracting light beam with infinite propagation ditance, infinite power i needed to maintain it hape in free pace; For a quai- anti-diffracting light beam, it only poee finite power in free pace, reulting in a finite anti-diffracting ditance. Taking the ideal Beel beam and Airy beam a example. A hown in Supplementary Figure (a), an ideal Beel beam with an amplitude proportional to J ( k r)exp( ik z) n r z can preerve it hape infinitely without divergence during propagation in free pace. Thu, the power of the ideal Beel beam can be obtained by [] + I = J ( k r)exp( ik z) drdz B n r z. () Similarly, the power of the ideal Airy beam in Supplementary Figure (b) can be obtained by [] A + 3 ( / 4)exp[ ( / /)] I = Ai ξ i ξ ξ dxdz () where = x/ x0, ξ = z / kx0, and x0 i an arbitrary contant. Since the Beel and Airy function in Supplementary Equation, are not quare integrable, both light beam poe infinite power in free pace. However, uch infinite power can only be achieved by the infinite aperture of the len. In practice, finite aperture can only tranfer finite power to an antidiffracting light beam, thereby leading to a finite anti-diffracting ditance in free pace. For thi reaon, the creation of ultralong anti-diffracting (UAD) light beam i alway conidered to be impoible in free pace.

9 9 Supplementary Note : Energy Ocillation of Anti-Diffracting Light Beam Anti-diffracting light beam are pecial olution of the Helmholtz equation (HE). According to their mathematical form, thee light beam can infinitely preerve their hape without divergence during propagation in free pace [-4]. However, due to finite power in free pace, only light beam with finite anti-diffracting ditance can be obtained. It i well known that anti-diffracting light beam are compoed of a mainlobe and idelobe. A the light intenity of idelobe increae, the anti-diffracting ditance increae accordingly. Up to now, little i known about the mechanim underlying thi phyical phenomenon. To reveal thi mechanim, the role of the idelobe and mainlobe in counteracting the diffraction effect in free pace mut be determined. In theory, anti-diffracting light beam are generally obtained by the Fourier tranformation of their Fourier pectra [-4]. The proce of Fourier tranformation can be accomplihed by an obective len in Supplementary Figure [5, 6]. For example, a quai-airy beam can be obtained by focuing a Gauian beam modulated uing a cubic phae plate [], wherea a Beel beam can be created by the Fourier tranformation of a Gauian beam modulated uing a high pa filter []. Baed on the Debye vectorial diffractive theory, the electric and magnetic field of an anti-diffracting light beam near the focu can be expreed a [7, 8] ia p α / E or H = inθco θtcl0 ( θ) exp( ik dθdϕ p 0 V r ) (3) 0 where θ and ϕ are the convergent angle and azimuthal angle, repectively; and A i a normalized contant. In addition, a = arcin(na / n), where NA i the numerical aperture of the obective len, and n i the refractive index in the focuing pace. The wavenumber i k = nπ / λ, where λ i the wavelength of the incident beam, and r = ( rco φ, rin φ, z) denote the poition vector of an arbitrary field point in the focal region. In the pherical polar coordinate, the ign of the unit vector along a ray in the focuing pace i incorrect, a hown in Fig. of Ref.[7]; i.e., = ( inθco ϕ, inθin ϕ,co θ) i ued intead of (inθco ϕ,inθin ϕ,co θ ). Uing with the oppoite ign will lead to ymmetry between the calculated and experimental reult about the focal plane. In addition, T c i the Fourier pectrum of the deired anti-diffracting light beam, and l 0 ( θ ) i the electric amplitude of the incident Gauian beam, which can be expreed a

10 0 l inθ = (4) inα 0( θ) exp[ ( β0 ) ] where β 0 i the ratio of the pupil radiu to the incident beam wait. In Supplementary Equation 3, V repreent the propagation unit vector of the incident beam right after having paed through the len. Here, we take a linearly polarized beam a an example to generate an anti-diffracting light beam. Thu, the electric vector V e and magnetic vector V m can be written a [7] co θ + ( co θ)in ϕ V e = ( co θ)inϕcoϕ ; inθcoϕ ( co θ)inϕcoϕ ( co θ)in ϕ inθinϕ V m =. (5) Eventually, the light intenity ditribution of an anti-diffracting light beam can be obtained uing I = E. Moreover, the time-averaged Poynting vector, namely, the energy flux, can be obtained uing [9] c S = Re( E H ) (6) 4π where c i the velocity of light in a vacuum, and the aterik denote the operation of complex conugation. According to Supplementary Equation 6, one can flexibly explore the energy proce of anti-diffracting light beam during propagation in free pace. Suppoe that P( xyz,, ) and P( xy,, z) are two ymmetrical point on the anti-diffracting light beam. The electric and magnetic field of thee point can be expreed a [7] Ex(x, y, z) = Ex(x, y,z) Hx(x, y, z) = Hx(x, y,z) E (x, y, z) = E (x, y,z) ; H (x, y, z) = H (x, y,z). (7) y E (x, y, z) = E (x, y,z) z z y y H (x, y, z) = H (x, y,z) By ubtituting Supplementary Equation 7 into 6, the relationhip of tranvere energy fluxe between P( xyz,, ) and P( xy,, z) can be expreed a z tp tp z y S = S, (8) which implie that the light beam experience two invere energy procee that tranfer the energy from S to tp S during propagation in free pace. Without a lo of generality, tp S and tp S are the tp energy charge and energy dicharge, repectively. Energy charge and dicharge compoe an entire energy ocillation, which i a directional energy flux that confine the energy of anti-diffracting light beam into an interaction between the mainlobe and

11 idelobe when propagating in free pace. Thu, the light beam would not diverge freely a oberved with a Gauian beam in free pace. If only energy ocillation occur, an anti-diffracting light beam can preerve it hape without divergence in free pace. Even when encountering an obtacle, the mainlobe can carry out elf-healing with the power from the idelobe [0]. That i why an anti-diffracting light beam i naturally compoed of a mainlobe and idelobe. Moreover, ince all anti-diffracting light beam can be created by focuing their correponding Fourier pectra in Supplementary Figure, energy ocillation i therefore a general property hared by all anti-diffracting light beam. Energy ocillation mechanim of an Airy beam Although energy ocillation i a general property hared by all anti-diffracting light beam, different light beam exhibit different form of energy ocillation, which are mainly determined by the Fourier pectrum T c. For example, the cubic phae i the Fourier pectrum of Airy beam, the tranmittance of which can be written a [, 3] T c k in θ 3 3 = exp iη in ( ϕ + ϕ0) + co ( ϕ + ϕ0) in α where η and σ are the parameter that control the period and phae ditribution of the cubic phae plate, repectively. Typically, if σ = 3, then T c denote a tandard cubic phae, and the entire phae can be rotated by an angle ϕ 0. Accordingly, a quai-airy beam generated uing the cubic phae of η can imply be obtained by rotating the cubic phae of η with ϕ = 0 π. That i, light beam with ± η are ymmetrical about the optical axi, thereby leading to η tp η tp S = S (0) (9) where S and η tp S are the tranvere energy fluxe of point P( xyz,, ) for η and η tp η, repectively. From Supplementary Equation 8 and 0, the tranvere energy flux relationhip between quai- Airy beam with ± η can be implified to S = S = S = S, () η η η η tp tp tp tp

12 where the energy charge Sη tp i equal to the energy dicharge S η tp, and the energy dicharge Sη tp i equal to the energy charge S η tp. Thu, quai-airy beam with ± η can be conidered a pair of mutually complementary mode in free pace. Energy interaction between the mainlobe and idelobe In the following imulation and experiment, NA=0.095, n=, and β 0 =. The unit of length in all figure i the wavelength λ, and the light intenity i normalized to the unit value. The anti-diffracting ditance of the UAD light beam i evaluated with the FWHM (full width at half maximum). The quai-airy beam in Supplementary Figure 3(a) generated by the cubic phae with the parameter η = 5, σ = 3 and ϕ = π in Supplementary Figure 3(b) repreent one entire energy ocillation in free pace, which i compoed of energy charge when z<0 and dicharge when z>0. Energy charge and dicharge imply two different energy procee. In the proce of energy charge when z<0, the energy i tranported from the mainlobe to the idelobe a hown in Supplementary Figure 4(a, b). Thu, the mainlobe i the energy ource of the idelobe, and the energy of the mainlobe tend to be tored in the idelobe intead of diverging a oberved with a Gauian beam. Compared with energy charge, the light beam when z>0 experience a totally different energy proce, namely, energy dicharge. Due to the invere energy flux in Supplementary Figure 4(c, d), the energy i tranported from the idelobe to the mainlobe. In thi cae, the idelobe are the energy ource of the mainlobe while the mainlobe become an energy conumer. During propagation when z>0, the mainlobe endure an energy lo caued by the diffraction effect. However, thi energy can be replenihed by the power of the idelobe; thu, the light beam can remain anti-diffracting. Even when encountering an obtacle, the light beam can carry out elf-healing through the power of the idelobe [0]. Due to the energy ocillation mechanim, the power of the quai-airy beam i confined to the interplay between the mainlobe and idelobe. Therefore, the light beam can propagate without ignificant divergence in free pace. However, thi confinement cannot be limitle. Finite power in free pace can only upport a finite energy charge when z<0, thu leading to only a finite energy dicharge when z>0. When the power tored in the idelobe i exhauted, the light beam can no longer maintain it hape at z>0, and the diffraction effect eventually dominate. Conequently, the light beam can propagate only a finite anti-diffracting ditance in free pace. In addition, the power of energy charge i equal to that of energy dicharge becaue of energy converation, thereby generating an equivalent nondiffractive ditance when z<0 and z>0. For thi reaon, a quai-airy beam poee a ymmetrical

13 3 traectory a hown in Supplementary Figure 3(a). Throughout the entire energy ocillation proce, the z=0 plane i the inflection plane in which the role of the mainlobe change from the energy ource to the energy conumer. Specifically, when z<0, the idelobe can receive power from the mainlobe continually during the energy charge proce. However, once the idelobe attain the maximum energy capacity, energy charge can no longer be conducted, and the idelobe begin providing power to the mainlobe when z>0. Thu, the idelobe have maximum light intenity in the z=0 plane. Suppoe that I i the light intenity of idelobe A in the z=0 plane, a hown in Supplementary Figure 4(a). In principle, a larger I indicate tronger energy ocillation, thereby leading to a longer anti-diffracting ditance of the light beam in free pace. That i, if I can be aduted, then the trength of the energy ocillation along with the non-diffractive ditance can be controlled accordingly. Aduting the Strength of Energy Ocillation Two method are propoed to control the trength of energy ocillation via I : adut the period of cubic phae η while σ = 3, and manipulate the cubic phae ditribution via σ while keeping the period η unchanged. For the firt method, quai-airy beam with different value of η are generated a hown in Supplementary Figure 5, where η = 3 in (a, f) and η = 5 in (b, i). Supplementary Figure 5(e, h) depict their correponding cubic phae plate. The theoretical reult how that I increae with the period of the cubic phae η, where I =0.3 for η = 3 [Supplementary Figure 5(f)] and I =0.39 for η = 5 [Supplementary Figure 5(i)]. Therefore, a longer non-diffractive ditance can be achieved when η i larger. The FWHM of Supplementary Figure 5(a) and (b) are 55.6λ and 560λ, repectively. By adopting the ame experimental etup a in Fig. of the main text, quai-airy beam with η = 3 [Supplementary Figure 5(c)] and η = 5 [Supplementary Figure 5(d)] are created under the condition of NA= The FWHM are 5.λ and 544.9λ, which are coincident with thoe of Supplementary Figure 5(a, b) in theory. A for the econd one, Supplementary Figure 6 how the light intenity of quai-airy beam with σ =.6 (a, f) and σ = 3.6 (b, i) when η = 5. A hown in Supplementary Figure 6(f, i), I increae a σ decreae, with I =0.355 [Supplementary Figure 6(f)] and I =0.39 [Supplementary Figure 6(i)]. Similarly, the non-diffractive ditance of σ =.6 i much longer than that of σ = 3.6. In the experiment, the quai-airy beam hown in Supplementary Figure 6(a, f) and (b, i) are created by coding the phae of Supplementary Figure 6(e, h) in the SLM hown in Fig. of the main text. The FWHM value for

14 4 both beam are λ [Supplementary Figure 6(c)] and 49.3λ [Supplementary Figure 6(d)], repectively, which fit well with thoe of σ =.6 (3084λ) and 3.6 (4λ) in theory. Although both method mentioned above are capable of aduting the energy trength via the light intenity of idelobe I, neither method can trengthen the energy ocillation without limitation, which make the ideal Airy beam impoible in practice. However, thi new mechanim till offer the poibility of breaking through the light beam limitation of finite non-diffractive ditance in free pace. Energy ocillation mechanim of a Beel beam Supplementary Figure 7 preent the light intenity and energy flux of a Beel beam, which i generated by focuing a linearly polarized Gauian beam with the modulation of a high-pa pupil filter in Supplementary Figure 7(b) []. Here, the tranmiion of thi pupil filter can be expreed a T h r/ R 0.8 =, () 0 0 r/ R< 0.8 where r and R are the radii of the inner and outer ring of the high-pa pupil filter, repectively. By ubtituting T c for T h in Supplementary Equation 3, the light intenity and energy flux of the Beel beam can be calculated uing Debye vectorial diffractive theory. In the following imulation, NA=0.095, n=, and β 0 =. The unit of length in all figure i the wavelength λ, and the light intenity i normalized to the unit value. A hown in Supplementary Figure 7(c, d), a Beel beam can be divided into two part according to the energy flux: energy charge when z<0 and energy dicharge when z>0. For the light beam when z<0, the energy i tranported from the idelobe to the central mainlobe radially. Thu, the idelobe are the energy ource of the mainlobe, a hown in Supplementary Figure 7(c). Even when encountering an obtacle, the light beam can carry out elf-healing []. In contrat to the light beam when z<0, the light beam when z>0 experience an invere energy proce. A hown in Supplementary Figure 7(d), the energy i tranported from the mainlobe to the idelobe radially. In thi cae, the mainlobe turn into an energy ource, while the idelobe erve a the energy conumer of the mainlobe. Via thi energy interaction between the mainlobe and idelobe, the Beel beam can preerve it hape without a ignificant divergence in free pace. Similar to the quai-airy beam in Supplementary Figure 4, finite power in free pace can upport only a finite energy ocillation, thu leading to a finite anti-diffracting ditance of the Beel beam.

15 5 Supplementary Note 3: Derivation of the Optical Pen Supplementary Figure 8 preent a chematic of deriving optical pen in focuing ytem. A collimate incident vector beam propagating along the +z-axi goe through the pupil filter P before being focued by a len obeying the ine condition. Ω i the focal phere, and it center i at O and it radiu i f, which i the focal length of the len. O i an arbitrary point in the focal region of the len. In principle, only one focu i located at the focal point O, where contructive interference can occur only becaue of the equivalent optical path of the light beam between the point in Ω and point O. If contructive interference doe not occur at point O but rather at arbitrary point O, the focu i generated at point O intead of at O. To thi end, the optical path difference (OPD) for the light beam between the point in Ω and O mut be compenated by the phae of pupil filter P. A hown in Supplementary Figure 8, the OPD for the light beam between the point in Ω and O can be implified to L -L, where L and L are the light path AO and BO, repectively. Here, in the cylindrical coordinate ytem, the point A, B, O and O can be expreed a ( f inθco ϕ, f inθin ϕ, f co θ), (0,0, f ), (0,0,0) and ( ρco ϕ, ρin ϕ, ), repectively. The light path L and L can be decribed a z ( ) ( ) L = f inθcoϕ ρcoϕ + f inθinϕ ρin ϕ + ( f co θ + z) = + ρ + ρinθco( ϕ ϕ ) + coθ f z f fz (3) ( ρ ϕ ) ( ρ ϕ ) L = co + in + ( f z) = ρ f z fz. (4) The OPD for the light beam between the point in Ω and O can be calculated a = L L = ρinθco( ϕ ϕ ) + z( co θ) η + + η + η + + η + η η inθco( ϕ ϕ ) + η coθ r z z r z r z (5) where θ and φ are the convergent angle and azimuthal angle, repectively; η z = z/ f ; and ηρ = ρ / f. Since O i an arbitrary point in the vicinity of the focu, ρ, z<< f. That i, η, η 0. Finally, Supplementary Equation 5 can be implified to = ρinθco( ϕ ϕ ) + z( co θ). (6) To generate a focu at O, mut be compenated by the pupil filter P. Thu, ρ p z =, where

16 6 p i the OPD induced by the pupil filter P. According to the relationhip between the phae and OPD, the phae of pupil filter P i ψ = k p, where k = πn/ λ i the wavenumber, and n i the refractive index in the focuing pace. Conequently, the tranmittance of pupil filter P for one ingle focu can be expreed a T = exp( iψ ). If multiple focue occur in the focal region, the tranmittance of pupil filter P can be further tranformed into N T = exp( iψ ) (7) where N i the number of foci and denote the -th focu. A hown in Supplementary Equation 7, the pupil filter P require both amplitude and phae modulation for the incident light beam. In practice, amplitude modulation alway lead to low light tranformation efficiency and i difficult to implement. However, thi problem can eaily be olved by extracting only the phae of pupil filter P in Supplementary Equation 7. Finally, the phae-only pupil filter P can be obtained uing = N { ( = ) } T = exp i Phae exp[ i( ψ + δ )] (8) Here, we refer to thi pupil filter a the optical pen. For the ake of implicity, thi optical pen can alo be expreed a N { ( = ) } T = exp i Phae PF(, x, y, z, δ ) (9) where x, y, z are the poition of the -th focu in the focal region correponding to ρϕ,, z in Supplementary Equation 6; and and δ are the two weight factor that are reponible for aduting the amplitude and phae of the -th focu, repectively. Manipulation of light field via the optical pen Note that the pupil filter P developed by the OPD compenation i valid for an arbitrary incident vector beam. Here, we take only a linearly polarized beam a an example to verify the function of the optical pen. Baed on Debye vectorial diffractive theory, the light intenity in the vicinity of the focu can be obtained uing Supplementary Equation 3. In the following imulation, NA=0.8, n=, and β 0 =. The unit of length in all figure i the wavelength λ, and the light intenity i normalized to the unit value. In the cae of N =, only one focu i obtained in the focal region, and it poition can be aduted in three dimenion uing the optical pen. A hown in Supplementary Figure 9, foci with different poition in the y-z plane are realized with the different phae of pupil filter P [namely, Supplementary

17 7 Figure 9(b, d, f)], the parameter of which are N =, =, x =, δ = 0 ; (a, b) y = 5, z = 0, (c, d) y = 0, z = 0, and (e, f) y = 5, z = 0. Compared with the original focu in Supplementary Figure 9(c), the hape of the focu remain invariant while the poition can be aduted freely in the focal region. For an optical pen, the focal region can be conidered a drawing board on which an arbitrary pattern can be realized by preciely controlling the number and poition of foci and their correponding weight factor in Supplementary Equation 9. The ize of the optical pen i determined by the focu of the incident linearly polarized beam, which i relevant to the NA. For one particular light pattern, the number and poition of foci determine the hape of the light pattern, which can be manipulated uing the parameter N, x, y, z. Once the hape i confirmed, the amplitude and phae of each focu mut be aduted by and δ o that the deired light pattern can be realized in the focal region. For the ame light pattern, which i determined only by the hape, the number and poition of foci are identical. However, the weight factor and δ have countle poibilitie. A hown in Supplementary Figure 0, a 4 4 focal array in the x-y plane can be realized with the different phae hown in Supplementary Figure 0(d-f), repectively. Supplementary Figure 0(d-f) can be obtained by multiplying two phae, which yield a 4 focal array along the x- and y-axi. Thu, N = 4, z = 0 and δ = 0. For the 4 foci along the x-axi, y = 0 and the foci are located at x = 3, x = 3, x 3 = 9 and x 4 = 9. For the 4 foci along the y-axi, x = 0 and the foci are located at y = 3, y = 3, y 3 = 9 and y 4 = 9. The ditance between each focu along the x- and y-axi i 6λ. The only difference between the phae for the ame 4 foci i the weight factor. For example, three pupil filter for the 4 focal array along the x- and y-axi are obtained, with =.05, = 0.7, 3 = 0.9, 4 = 0.8 ; = = 0.885, 3 = 4 = ; and =.05, = 0.7, = 0.95, = 0.33, which are denoted a lx ~ l x3 for the x-axi and ly ~ l y3 for the y-axi. Finally, the 3 4 phae can be obtained a follow: (d) lx ly; (e) lx ly; and (f) lx3 ly. Clearly, all phae in Supplementary Figure 0(d-f) are different from each other, although they all generate an identical 4 4 focal array in the focal plane. The mot typical array i the phae in Supplementary Figure 0(e), which i a common Dammann grating, and it wa alo obtained in Ref. []. In other word, the optical pen in Supplementary Equation 9 repreent all poible phae, with only the weight factor being different,

18 8 and the Dammann grating i only one of the pecial olution. Thu, additional phae can be obtained by aduting the weight factor. When generating a light pattern in the focal region, the hape i determined by the number and poition of foci, wherea the amplitude and phae of each focu enure the quality of the entire pattern. By controlling the number and poition along with the weight factor and δ, more complex focal pattern can be realized. A hown in Supplementary Figure (a, b), NANO and OPT are created in the focal plane with the phae in Supplementary Figure (c, d). In addition, a three-dimenional focal pattern can alo be realized uing the optical pen. A hown in Supplementary Figure (a), OPT i obtained in the z = 0 λ,0, 0λ plane imultaneouly, with the phae hown in Supplementary Figure (b). All parameter for the above phae can be found in the Parameter of Supplementary Figure and. Once again, the phae for the generation of focal pattern in Supplementary Figure and are not unique, but all can be obtained uing and δ in Supplementary Equation 9. Significance of the Optical Pen The generation of an ultralong anti-diffracting light beam in free pace lead to the precie manipulation of the number, poition, amplitude and phae of foci in the focal region. However, uch preciion cannot be achieved uing previou technique [-7]. The optical pen i therefore developed to olve thi problem. A a veratile optical tool, the optical pen poee an explicit form (Supplementary Equation 9) that can be ued to unify the relationhip between the focal pattern and the phae in the entrance plane. By aduting the parameter of the optical pen, the number, poition, amplitude and phae of foci can be aduted at will in the focal region o that an arbitrary focal pattern can be realized in free pace. Thi advantage make the optical pen a perfect optical tool for the creation of ultralong anti-diffracting light beam in free pace. Parameter of Supplementary Figure and The parameter of the phae in Supplementary Figure (c) Parameter for O N = 6 l = PF(, x, y,0), O = where x = dcoφ, y = dinφ and d = 9 ; φ = π( ) / N, =,,3... N ; 4 = 6,7 = 9 = 6 = ; 5 = 0.97 ; 8,3 =.05 ; 4 =.5 ; 5 = Parameter for N

19 9 d = 7.5 ; d 3 = 0 ; l = PF(, d, d,0) + PF(0.97, d, d,0) + PF(, d, d /,0) + PF(0.95, d /, d /,0) + N PF( 0.97, d, d /,0) + PF( 0.8, d,0,0) + PF(.05,0,0,0) + PF( 0.8, d,0,0) + ; PF( 0.85,, /,0) PF(0.95, /, /,0) PF( 0.88,, /,0) + PF(0.88,,,0) + 3 d d3 + d d3 + d d3 d d3 PF(, d, d,0) + PF( 0.,0,5,0) + PF( 0., d /, d,0) 3 3 l = PF(0.98, d, d,0) + PF(0.93, d, d,0) + PF( 0.88, d, d /,0) + PF(0.95, d /, d /,0) + N PF( 0.85, d, d3 /,0) + PF( 0.8, d,0,0) + PF(.05,0,0,0) + PF( 0.8, d,0,0) + PF( 0.9, d, d3 /,0) + PF(0.943, d /, d3 /, 0) + PF( 0.93, d, d3 /,0) + PF(0.9, d, d3,0) + PF(, d, d,0) + PF( 0., d /, d,0) Parameter for A d 4 = 0 ; d 5 = 7.5 ; 3 3 l = PF(0.83,0, d,0) + PF( 0.9, d / 4, d /,0) + PF( 0.9, d / 4, d /,0) + PF(0.9, d /,0,0) + A PF(0.9, d /,0,0) + PF(0.8, d, d /,0) + PF(0.89, d, d /,0) + PF( 0.8, d / 4, d /,0) PF( 0.83, d / 4, d /,0) + PF(0.8, d, d,0) + PF(0.83, d4, d4,0) Final phae d 6 = 5 ψ = Phae[PF(.3, d, d,0) l + PF(.3, d, d,0) l + PF(, d, d,0) l + PF(0.9, d, d,0) l ]; l pf 6 6 N 6 6 N 6 6 A 6 6 O pf = exp( iψ ). pf The parameter of the phae in Supplementary Figure (d) Parameter for O N = 6 l = PF(, x, y,0), O = where x = dcoφ, y = dinφ ; d = 9 ; φ = π( ) / N, =,,3... N ;,3 = 6,7 = ;, =.03 ; 4 =.0 ; 5 =.5 ; 8 =.; 9 =.08 ; 0 =.05 ; 3 =.4 ;,4 6 =.0. Parameter for P 7 l P = PF(,0, d,0) PF( + 8,3,0,0) + PF( 9,6,0,0) + PF( 0,3,9,0) + PF(,6,9,0) + =, PF(,8,6.75,0) + PF(,8,.5,0) + PF(,3, 9,0) 3 4 where d = ( 4) 3 ; =.5 ; =. ; 3,5 =.05 ; 4 =.505 ; 6 = ; 7,3 =. ; 8 =.6 ; 9 =.4 ; 0 =.45 ;, =. ; 4 = 0.3. Parameter for T ; ;

20 0 7 9 T = PF(,0,,0) PF(,,9,0) PF(, + 8 0,9,0) = + =, = l d D D + where d = ( 4) 3 ; D = ( 0) 3 and =.4 ; =.6 ; 3 =.3 ; 4 =. ; 5 =.5 ; 6 =.35 ; 7 =.55 ; 8 = 0.95; 9,0 =.46 ; =.5. Final phae ψ = Phae[PF(.45, 0,0,0) l + PF(., 3,0,0) l + PF(., 0,0,0) l ] ; l pf pf O P T = exp( iψ ). pf The parameter of the phae in Supplementary Figure Parameter for O N = 6 l = PF (, x, y,0), O = where x = dcoφ, y = dinφ and d = 9 ; φ = π( ) / N, =,,3... N ; = 0.95 ; 3 =. ;,4,6 8,4 6 = ; 3,5,9 =.. Parameter for P 7 l P = PF(,0, d,0) PF( + 8,3,0,0) + PF( 9,6,0,0) + PF( 0,3,9,0) + PF(,6,9,0) + =, PF(,8,6.75,0) + PF(,8,.5,0) 3 where d = ( 4) 3 ;,7 = 0.8 ;,0, =.3 ; 3 =.3 ; 4 =.6 ; 5 =.4 ; 6,9 =. ; 8 =.4 ; =.5 ; 3 =. Parameter for T 7 9 T = PF(,0,,0) PF(,,9,0) PF(, + 8 0,9,0) = + =, = l d D D + where d = ( 4) 3, D = ( 0) 3 ; =.45 ; =.35 ; 3 =.35 ; 4 = 0.8 ; 5 =.3 ; 6 =.5 ; 7 =.4 ; 8 = ; 9,0 =.3; = Final phae ψ = Phae[PF(.5,,, 0) l + PF(0.98,0,0,0) l + PF(,,, 0) l ] ; l pf pf O P T = exp( iψ ). pf

21 Supplementary Note 4: UAD light beam under the condition of NA=0.8 Supplementary Figure 3 how UAD light beam with 0,,, and 3 energy ocillation in the y-z plane, and they are generated by focuing a linearly polarized Gauian beam with NA=0.8. The correponding phae are hown in Supplementary Figure 3(e-h), and the parameter can be found in Supplementary Table. Notably, the anti-diffracting ditance of the UAD light beam for zero and one energy ocillation are only.34λ and 3.96λ, repectively. However, uing the multiple energy ocillation mechanim, UAD light beam can be eaily achieved by imply increaing the number of energy ocillation via modulation of the optical pen. A hown in Supplementary Figure 3(c, d), the antidiffracting ditance of two and three energy ocillation are 70.56λ and 99.50λ, repectively. The multiple energy ocillation mechanim can lead to a peculiar energy flux at the witch point between adacent energy ocillation a indicated by the cro-haped idelobe in Supplementary Figure 3(). The bottom idelobe correpond to an energy dicharge proce imilar to that at point A in the initial energy ocillation a hown in Supplementary Figure 3(i). Although the energy dicharge at point B will oon be completed, the upper idelobe can provide an additional energy recharge o that the light beam can propagate further. That i, at point B, the light beam experience not only an energy dicharge in the initial energy ocillation but alo an energy charge imilar to that at point C in Supplementary Figure 3(k) in the econd energy ocillation. Thu, the UAD light beam can maintain it hape without ignificant divergence over a uper-long range.

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