Supporting Information

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Damian Watkins
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1 Supporting Information Multiple-beam interferene enable broaban metamaterial wave plates Junhao Li, Huijie Guo, Tao Xu, 3 Lin Chen,, * Zhihong Hang, 3 Lei Zhou, an Shuqi Chen 4 Wuhan National Laborator for Optoeletronis, Huahong Universit of Siene an Tehnolog, Wuhan 4374, China. State Ke Laborator of Surfae Phsis an Ke Laborator of Miro an Nano Photoni Strutures (Ministr of Euation), Fuan Universit, Shanghai 433, China. 3 College of Phsis, Optoeletronis an Energ an Collaborative Innovation Center of Suhou Nano Siene an Tehnolog, Soohow Universit, Suhou 56, China. 4 Laborator of Weak Light Nonlinear Photonis Ministr of Euation Shool of Phsis, Tea Applie Phsis Institute, Nankai Universit, Tianjin 37, China. * Coesponing author: hen.lin@mail.hust.eu.n

2 Note : Derivation proess of Eqs. (5) an (6) The erivation proesses of Eqs. (5) an (6) are as follows Δ = arg i 3 e =arg( 3 os i 3 sin ) os 3 = arot sin Then the erivative of Δ with respet to f, is given as os 3 sin Δ = os f 3 + ( ) sin 3 sin ( + sin ) ( os )os f r 3 r3 = (sin ) + ( os ) 3 3 sin + 3 (os 3 ) f = ( sin ) + ( os ) 3 3 The funtional form of Δ is the same as that of Δ. Thus, the erivative funtion of Δ an be written b replaing the oesponing smbols in Eq. (S) as r3 r34 sin + r3r34 (os r3r34 ) f Δ = (S3) ( r r sin ) + ( r r os ) (S) (S)

3 Note : Derivation proess of an an be written as = π nh +, sine 3 3 r r3 ik h ik ( h r r t r t e r r3e ) 3 3 ( ikh ikh = r3) into r3 = r3 + t3rt3e ( rr3 e ), 3 3 ( ik h ik h 3 ) 3 = kh +. Aoring to Eq. () of the main tet, = +. B inorporating r 3 = r 3 an t t r an be reue to r = r + r e r r e an its argument an thus be represente b ikh r3 + re r = arg 3 ikh e 3 r3 + r os + ir sin = arg (S4) 3 os 3 sin i r sin rr3 sin = artan artan r3 + r os rr3 os r sin rr3 sin r3 + r os rr3 os f f = r3 r sin rr3 sin (S5) + + r3 + r os rr3 os 3 os + ( r ) 3 os ( 3) = ( r3 + r os ) + ( r sin ) ( rr3 os ) + ( rr3 sin ) Sine avit one interferes onstrutivel along the -polariation iretion, we an get os = an sin =. B substituting os an sin into Eq. (S5), we an get = r [ r r + ( r ) ] [( r r )( r r )]. Then, an be written as r π nh r[ + ( r ) ] = + ( r r3 )( rr3 ) an be written b replaing the oesponing smbols of Eq. (S5): 3 os + ( r) 3 os ( 3) = (S7) r 3 ( r 3 r os ) ( sin ) ( 3 os ) ( 3 sin ) + + r r r + r r Sine avit one interferes estrutivel along the -polariation iretion, we an get os = an sin =. B substituting os an sin into Eq. (S7), we an get that = r [ r r + ( r ) ] [( r r ) ( r r )]. Then, an be written as r π nh r[+ + ( r ) ] = + ( r + r3 )( + rr3) (S6) (S8) 3

4 Note 3: Theoretial alulation of the ispersion relation of the SSP moe supporte b a one-imensional anisotropi MTM ompose of metal/ieletri multilaer =-W/ =W/ =-P/ =P/ W o =-P / =-t / o =t / PEC =P / Air n Air P FIG. S. Shemati of the one-imensional anisotropi MTM. The anisotropi MTM is ompose of alternating metal/ieletri multilaer with finite with, W, suoune with air. The thiknesses of metal an ieletri laers are represente b t m, an t, respetivel, an the lattie onstant along iretion is enote as P ( = t + t ). The lattie onstant along m iretion is represente b P. In the mirowave an terahert omain, the metal an be treate as perfet eletri onutor. We firstl write out the EM fiels in all the regions eept in the metal region: ik ik ik ik [ ρ+ e e + ρe e ] P ( P/ < <W /) E = (S9) ik ik ik ik [ ρ+ ' e e + ρ' e e ] P ( W / < < P/) ik ik ik ik k[ ρ+ e e + ρe e ] ( k P) ( P/ < <W /) H = (S) ik ik ik ik k[ ρ+ ' e e + ρ' e e ] ( k P ) ( W / < < P/) ink ink [ Ce + + Ce ] t ( t / < < t /, < W /) E = (S) ( t / < < P /, < W /) ink ink H = [ nc+ e + nce ] t ( t / < < t /, < W / ) (S) where ρ ±, ρ ± ' an C ± are unsolve oeffiients relate to moe amplitues in ifferent regions. One an onnet Eqs. (S9)-(S) with the Mawell s bounar onitions an the perioi bounar onitions, ieling the following relations ρ + + ρ + = ( C+ ' + C ') S ( E +, =W /) ρ+ ' + + ρ' = ( C+ + ' + C ') S ( E, = W /) ρ + Sk / k ρ + Sk / k = ( C+ ' C ') n ( H +, =W /) (S3) ρ+ ' + Sk / k ρ' Sk / k = ( C+ + ' C') n ( H, = W /) P P P P ρ ρ + = ρ+ ' + + ρ' ( E, = P/) P P P P ρ + - ρ + = ρ+ ' + ρ' ( H, /) = P ikw / where = ± ± e, in k W / ' = ± ± e, P ikp/ = ± ± e an S = t Psin( kt/ ) ( kt ). B eliminating ρ ±, ρ ± ' an C ± in Eq. (S3), we obtain nkw kg nkw kg ± A [tan( ) tan( )] = A+ [tan( ) + tan( )] (S4) where g = P W an A = ks k ± ± n : 4

5 () If A [tan( nkw ) tan( kg )] = A [tan( nkw ) + tan( kg )], then + n tan( nkw ) = ( k k) S tan( kg ). B using k = i k k, the ispersion relation an be epresse as k k g e nkw k k =ks ot( ) / n (S5) k k g + e k k g k k g for asmmetrial SSP moe. As g +, ( e ) ( + e ), Eq. (S5) will be reue to the ispersion relation for the asmmetrial SSP moe. () If A [tan( nkw ) tan( kg )] = A+ [tan( nkw ) + tan( kg )], then n tan( k g ) = ( k k ) S tan( n kw ). B using k = i k k, we an get k k g + e nkw = tan( ) / k k g k k k S n (S6) e k kg k kg for smmetrial SSP moe. As g +, ( e ) ( + e ), Eq. (S6) will be reue to the ispersion relation for the smmetrial SSP moe. The alulate ispersion urves [Fig. S(a)] an eletri fiel iagrams [Fig. S(b)] notabl reflet the oupling effet of the SSP moes between the ajaent units. A higher effetive refrative ine of the SSP moe oul be ahieve b tuning the geometrial parameters of the anisotropi MTM or using a smaller gap separation to enhane the oupling between the ajaent SSP moes. It shoul be note here, espite the great apabilit of the asmmetrial SSP moe in tuning the ispersion relation, it is not suitable for the esign of funtional evies ue to ero oupling with the inient waves. (a) f (/ W ) f f asmmetri moe smmetri moe Theor Simulation Κ ( π/p) (b) -8 (P ) (P) (P) smmetri asmmetri FIG. S. Dispersion urves an fiel istributions of smmetri an asmmetri SSP moes of one imensional anisotropi MTM. (a) Dispersion urves alulate for t =.4 mm, t =. mm, w = 6 mm, P = 8 mm, an n =.7. (b) Distributions of the real part of E of smmetri an asmmetri SSP moes for the D anisotropi MTM at f =.3( W) an f =.9( W), respetivel. - m 5

6 Note 4: The broaban QWP in the terahert regime (a) 3 (b) 8.8 Tran. Ampl polariation -polariation total insertion loss.5.5 f (TH) Insertion Loss (B) PD ( egrees ) f (TH) FIG. S3. (a) Transmission amplitues for the - an - polariations, total transmission amplitue, an insertion loss versus f. (b) PD (re) an DoLP (blue) as a funtion of f. For the broaban QWP in the terahert regime, the anisotropi MTM is ompose of Al/GaAs multilaer ( n GaAs = 3.6 ). Eah Al an GaAs laer is with a thikness of t =.5 μm an t m =.5 μm, respetivel. There are 4 pairs of Al/GaAs laers, resulting in a total thikness of 4 μm for the anisotropi MTM ( h = 4 μm ). The ross-setional imension of the anisotropi MTM is set as W = 9 μm an W =.8 μm, an the lattie onstants along the - an - aes are P = P = μm. The anisotropi MTM is eposite on a semi-infinite PDFE substrate with the refrative ine of n 3 =.38. We esigne the strutural parameters to make onstrutive interferene for polariation an estrutive interferene for -polariation enounter at aroun.9 TH. It an be infee from the transmission amplitues in Fig. S3(a) that t 3 an t 3 reahes the maimum an minimum values at aroun.9 TH, respetivel. The esigne QWP an keep the rosspolariation phase ifferene, PD, within the range of 9± over a wie wavelength ban of.5.9 TH (ispersionless ban), 35.8% with respet to.9 TH [Fig. S3(b)]. The total transmission amplitue ( 3 t3 ( t + ) ) reahes up to approimatel 65% [Fig. S3(a)], assoiate with the insertion loss of less than.4 B [Fig. S3(a)], an the DoLP is above 98% [Fig. S3(b)] in the ispersionless ban. DoLP 6

Chapter 2: One-dimensional Steady State Conduction

Chapter 2: One-dimensional Steady State Conduction 1 Chapter : One-imensional Steay State Conution.1 Eamples of One-imensional Conution Eample.1: Plate with Energy Generation an Variable Conutivity Sine k is variable it must remain insie the ifferentiation