Boundaries, Near-field Optics

Transcription

1 Boundares, Near-feld Optcs Fve boundary condtons at an nterface Fresnel Equatons : Transmsson and Reflecton Coeffcents Transmttance and Reflectance Brewster s condton a consequence of Impedance matchng Evanescent Waves

2 BOUNDARY CONDITIONS 1. The frequency of lght does not change across the border, unless one of the meda happens to be a nonlnear medum. In nonlnear meda hgher harmoncs are generated. 2. The wavelength ether expands or contracts, accordng to the rato of the ndces of refracton:

3 BOUNDARY CONDITIONS BOUNDARY CONDITIONS 3. Speakng n terms of electromagnetc theory, the rato of lght energy, contaned n the form of electrc energy, to that n the form of magnetc energy s changed at the boundary dependng on the ntrnsc mpedance of the medum. Energy densty stored n an electrc feld Energy densty stored n a magnetc feld ( ) r o E ε ε ε E u 2 1 2, ε ( ) r o H H u 2 1 2, o n H E η ε η 1 Intrnsc mpedance of the medum : H E u u u ε,, (vacuum) Ω Ω 377 ) 120π ( ε η o o o Index of refracton : r o n ε ε ε / v ε 1 H B v E B o r o r o n v B E H E H E η ε ε ε ε η The mpedance η has to be changed as soon as the lght crosses the boundary. o

4 BOUNDARY CONDITIONS (Maxwell s contnuty condtons) 4. The tangental components of E and H are contnuous across the boundary. 5. The normal components of D and B are contnuous, where D and B are the electrc and magnetc flux denstes. Remnd that the boundary condton at the metal-delectrc nterface! TM wave At the boundary (contnuty of the tangental E x, H y, and the normal D z ): E xm E xd H ym H yd ε me zm ε de zd

5 BOUNDARY CONDITIONS Suppose that at a partcular nstance and at a partcular locaton of the boundary, the oscllaton of the ncdent wave s at ts maxmum; then both reflected and transmtted waves have to be at ther maxma. In other words, the wavelengths along the nterface surface must have the same temporal and spatal varaton. λ λ λ z1 z2 z3 2 π Propagaton constant : β constant λ z β, k, phase, or momentum matchng But all mean the same thng: wavelength matchng at the boundary! Snell s law :

6 TRANSMISSION AND REFLECTION COEFFICIENTS (Normal Incdence) At normal ncdence, both the E and H felds are parallel to the nterface. From the boundary Condton 4, Electrc (magnetc) feld transmsson coeffcent : t E (t H ) Electrc (magnetc) feld reflecton coeffcent : r E (r H )

7 Example 2.1 Calculate the coeffcents of reflecton and transmsson assocated wth the ar-glass nterface for normal ncdence. ( n 1 1, n ) What does a negatve value for the reflecton coeffcent mean? Can the transmsson coeffcent be larger than 1? We set the postve drectons of E 1 and E 3 are the same but those of H 1 and H 3 are opposte Boundary condton for E s satsfed : E (4/5)E 1 Boundary condton for H s satsfed : H (6/5)H 1

8 Example 2.1 E 3 (4/5)E 1 : H 3 (6/5)H 1 When lght s ncdent from a medum of a hgher ntrnsc mpedance (lower ndex of refracton) onto that of a lower ntrnsc mpedance (hgher ndex of refracton), (n ncdent < n transmtted ) the transmtted E feld decreases, whereas the transmtted H feld ncreases (larger than unty!). The lght power s for unt area (A 1, for smplcty) Transmttance T : power rato of transmtted lght to ncdent lght. Reflectance R : power rato of reflected lght to ncdent lght. In terms of the E feld, 96% 4% In terms of the H feld, The same R and T.

9 TRANSMISSION AND REFLECTION COEFFICIENTS (Arbtrary ncdent angle) In-plane (of ncdent) polarzaton Parallel polarzaton TM polarzaton p polarzaton Out-of-plane (of ncdent) polarzaton Perpendcular polarzaton TE polarzaton s polarzaton

10 TRANSMISSION AND REFLECTION COEFFICIENTS (Arbtrary ncdent angle for TM polarzaton) The tangental components of the E and H felds on both sdes of the boundary are equal; From the contnuty of the normal component of D, and from the equalty θ 1 θ 3, TM polarzaton Fresnel s equatons for TM wave

11 TRANSMISSION AND REFLECTION COEFFICIENTS (Arbtrary ncdent angle for TE polarzaton) The tangental components of the E and H felds on both sdes of the boundary are equal; Fresnel s equatons for TE wave TE polarzaton

12 TRANSMISSION AND REFLECTION COEFFICIENTS (Fresnel Equatons) Note that Ths s a drect consequence of Maxwell s contnuty condton, as can be seen by multplyng both sdes by the ncdent feld E 1. Contnuty of E felds.

13 (Ex) (Ex) Reflecton and and transmsson coeffcents as as a functon of of ncdent angle for for ar ar to to glass nterface (n (n , 1.0, n ). 1.5). Transmsson coeff. For a glass lens used to collect a dvergng lght source such as the emsson from a lght-emttng dode (LED). Brewster angle only for TM pol. Best case for mnmal ncrease n reflecton toward the edge of a menscus lens. Reflecton coeff.

14 TRANSMISSION AND REFLECTION COEFFICIENTS (Impedance approach for arbtrary ncdent angle )) Remember that, at normal ncdence, Defne an mpedance out of certan components of E and H chosen such that ther Poyntng vectors pont normal to the nterface. called characterstc wave mpedances referred to the x drecton S x Replacng η 1 and η 2 by η 1x and η 2x above, and then convertng η to n, gves the same results:

15 TRANSMITTANCE AND REFLECTANCE (AT AN ARBITRARY INCIDENT ANGLE) Power per unt area s Poyntng vector, s E H In applyng conservaton of energy, Note that the medum wth the greater mpedance has the larger E feld, snce Let A be assumed to be a unt area,

16 BREWSTER S ANGLE 0 ; when θ 1 + θ 2 90 o Brewster angle only for TM pol. Brewster s condton s a consequence of mpedance matchng wth reference to the vertcal drecton Fgure 2.5 Reflecton and transmsson coeffcents as a functon of ncdent angle for ar to glass nterface. x From Snell s law : n1 snθ 1 n2 snθ2 η2 snθ1 η1 snθ2 η 1x η2x : Brewster s condton The drecton of reflecton f t were to exst (dashed lne) concdes wth the drecton of polarzaton of the transmtted wave. From a mcroscopc vewpont, the reflected wave s generated by the oscllaton of electrc dpoles n the transmsson medum. The oscllatng dpole does not radate n the oscllaton drecton Brewster s condton

17 WAVE EXPRESSIONS OF LIGHT In meda 1 and 2, the plane waves are expressed as Phase dagram Insertng E 1 and E 2 nto the wave equaton, The wavelength matchng condton (phase matchng condton) on the boundary for the z drecton s,

18 EVANESCENT WAVES When the total nternal reflecton takes place, does the feld abruptly become zero on the boundary? Because of the boundary condton of contnuty, the feld n the less dense medum cannot abruptly become zero. n 1 > n 2 So called evanescent wave (surface wave) exsts! when n n 2 /n 1 < snθ 1 Form a standng wave n x drecton The effectve depth h of the penetraton of the evanescent wave, whch s defned as the depth where the ampltude decays to 1/e of that on the boundary, s 1/γ.

19 Transmsson and Reflecton Coeffcents for Total Internal Reflecton (TIR) TIR when n n 2 /n 1 < snθ 1 Snce the magntudes of the denomnator and numerator are dentcal, the absolute value of the reflecton coeffcent s unty. The phase shft dfference can be used to make waveplates. For example, a phase dfference of π/2 results n a quarter-waveplate, whch s qute useful because t s ndependent of wavelength.

20 For Total Internal Reflecton (TIR) Phase dagram n1k n k 1 where tanδ tanδ and represent the rato of momenta n medum 1 and medum 2.

21 For Total Internal Reflecton (TIR) The evanescent wave s a result of lght energy that goes nto the less dense medum only for a short dstance and then comes back agan nto the dense medum. The transmsson coeffcent descrbes the magntude on the boundary but just a short dstance nsde the less dense medum. t I + r

22 r r 2 jδ For Total Internal Reflecton (TIR) 1 2 jδ 2 j δ δ e e t (1 + e ) cos 2 jδ 2 jδ e t 1+ e n n 2cosδ e jδ Wth an ncrease n cosθ 1, the magntude of the transmsson coeffcent nto the second medum ncreases, also the momentum perpendcular to the boundary (kn 1 cosθ 1 ) ncrease. Ths s as f the photon n the optcally dense medum s beng pushed out nto the less dense medum by ths momentum. The transmsson coeffcents reach ther maxma at the crtcal angle (n snθ 1 ). The amount of phase shft, however, reaches zero at the crtcal angle. Ths fact rejects the smple explanaton that the phase delay s the tme needed for the evanescent wave to go out nto the less dense medum and come back nto the dense medum. It cannot safely be sad that the phase delay s due to the round-trp tme. Anyway, the phase shft s needed to match the boundary condton. Goos and Hanchen clarfy these matters.

23 In 1947, Goos and Hanchen looked at the dfference between reflecton from a slver surface and total nternal reflecton at a glass ar nterface. The reflected lght from the slver surface takes the drect path SQ 1 P 1 The lght that has undergone total nternal reflecton takes the route SQ 1 Q 2 P 2, Near the crtcal angle, where the δ s are at ther mnmum, the D s are at ther maxmum. D

24 Evanescent Feld and Its Adjacent Felds In the dense medum, a standng wave E s1 s formed by the ncdent and reflected waves. The poston of maxmum ntensty s shfted downward by δ radans. Because of the contnuty condton, the ampltude E 2 of the evanescent wave has to be dentcal to that of the standng wave E s1 evaluated on the boundary; n 1 > n 2 E-feld expressons near the boundary. Form a standng wave n x drecton

25 Evanescent Feld and Its Adjacent Felds Wth a decrease n the ncdent angle θ 1 from 90 approachng the crtcal angle θ c, both δ and γ decrease and h( γ 1 ) ncreases Ths means that wth a decrease n θ 1, the maxma shft toward the nterface and the evanescent wave shfts nto the optcally less dense medum. In practce, wth the wavelengths 0.85~1.55 m and n 1 ~ 1.55, n 2 ~ 1.54, h 1~10 m.

26 k-dagrams (phase dagrams) for the Graphcal Soluton of of the Evanescent Wave by elmnatng When total nternal reflecton exsts, k 2x s magnary : γ β where K

27 K β dagram : Radus kn1 large θ 1 large β small K large γ fast decay K γ dagram : Radus k n n θ 1 θ c γ 0 very slow decay θ 1 < θ c no evanescent

28 WHAT GENERATES THE EVANESCENT WAVES? (Corrugated metal surface) (Array of metal pns) To satsfy the boundary condton n z-drecton, ( Assume k y 0.) When d < λ/2 (one-half of the free-space wavelength), all component waves become evanescent waves.

29 Example 2.5 A lossy glass s bordered by ar n the x 0 plane. The propagaton constant n the z drecton on the border s β + jα. Fnd the expresson for the evanescent wave n the ar near the boundary. Propagaton constants n ar, Phase matchng along the z drecton Fnally, the expressons for the evanescent feld become not acceptable snce t goes nfntely as x ncreases.