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Leaky-mode emission of luminescent thin films on transparent substrates
Optics Communications 229 (2004) 279–290 www.elsevier.com/locate/optcom
Leaky-mode emission of luminescent thin films on transparent substrates A. Penzkofer
a,*
, W. Holzer a, H. Tillmann b, H.-H. H€ orhold
b
a
b
Institut II – Experimentelle und Angewandte Physik, Faculty of Physics, Universit€at Regensburg, Universit€atsstrasse 31, D-93053 Regensburg, Germany Institut f€ ur Organische Chemie and Makromolekulare Chemie der Universit€at Jena, Humboldtstrasse 10, D-07743 Jena, Germany Received 31 July 2003; received in revised form 25 September 2003; accepted 26 October 2003
Abstract The angle, wavelength, and thickness dependent fluorescence emission of a luminescent film on a transparent substrate into the substrate and the air is studied theoretically by Fresnel type transmittance calculations for s- (TE) and p- (TM) polarised emission. A micro-cavity behaviour due to Fabry–Perot like constructive and destructive interference is observed. The Fabry–Perot like interference strongly modifies the grazing-angle in-substrate spontaneous emission spectrum compared to the normal-incidence spontaneous emission. The wavelength selective constructive leaky-mode emission allows the determination of the refractive indices of luminescent thin films for TE and TM polarised light. Sublaser-threshold grazing-angle in-substrate emission is analysed for molecular 4-methyl-TPD and polymeric TPD(4M)MEH-P-PPV thin films. Ó 2003 Elsevier B.V. All rights reserved. PACS: 78.65; 42.80.L; 42.70.G Keywords: Leaky mode emission; Asymmetric wave-guide; Luminescent polymers; Multiple-beam reflection; Refractive index dependence
1. Introduction In luminescent asymmetric wave-guiding thin films the light propagation is dependent on the emission angle. Beyond the critical angle of total
* Corresponding author. Tel.: +499419432107; fax: +499419432754. E-mail address: [email protected] (A. Penzkofer).
internal reflection all emission is wave-guided in the neat film [1,2]. Slightly below the critical angle of total internal reflection the emission leaks into the substrate, propagates at a small angle to the film–substrate interface and escapes at the substrate edge (substrate edge emission). The leakedout emission interferes with the light which is reflected at the substrate–film interface, then reflected at the film–air interface, and finally transmitted at the substrate–film interface. The Fabry–Perot like multiple light interference gives
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.10.039
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rise to a wavelength, angle, film thickness, and polarisation dependent in-substrate light emission. The occurrence of constructive and destructive interference for light propagating at the film– substrate interface was observed in the case of nearly in-plane photoluminescence studies on asymmetric semiconductor micro-cavities [2]. Leaky-mode propagation of self-trapped exciton emission in anatase thin films on sapphire substrates was observed and analysed in [3]. The wavelength and angle dependent emission behaviour was explained by subtotal internal reflection constructive and destructive interference of the leaking radiation. The wavelength selectivity for the TE and TM modes was clarified by the different phase changes for TE and TM polarised light [4]. The occurring multi-beam interference modified the spontaneous emission spectrum. The constructive and destructive Fabry–Perot like asymmetric thin-film waveguide behaviour at grazing angle is quite similar to the constructive and destructive Fabry–Perot like behaviour of thin-film micro-cavities. In thin-film micro-cavities metallic mirrors or Bragg reflectors are used for high reflectivity. In asymmetric air–film–substrate waveguides at grazing-angle of light escape to the substrate there is total internal reflection at the film–air interface and high reflectivity at the film– substrate interface. Therefore, all phenomena of micro-cavities [5–7] are present. The micro-cavity quality of the waveguides increases with increase of reflectivity as the film-substrate angle of incidence approaches the angle of total internal reflection (grazing-angle leaky-mode emission). Micro-cavity effects in polymeric light-emitting diodes allow the control of the emission characteristics [8–10]. A wave-length tuning of the emission is achieved by cavity thickness variation [11,12]. Optically pumped organic polymeric thinfilm micro-cavities have been realised [13–18]. The laser emission of these mirror - thin-film – mirror devices with mirror spacing in the few 100 nm range shows the typical micro-cavity resonance effects (spectral line narrowing, thickness and angle dependent laser wavelength). In non-waveguiding thin luminescent polymer films on glass substrates (film thickness below the critical thickness, ‘crit , necessary for wave-guiding [1]) spectral changes [19] and spectral line-narrow-
ing [19–21] by leaky-mode emission were observed. Grazing-angle emitted light from polyfluorene derivative thin films of TE and TM polarisation was studied experimentally and theoretically in [22]. The influence of the cut-off film thickness on the amplified spontaneous emission dependence was analysed in [23]. Amplified spontaneous emission wavelength tuning by cut-off wavelength tuning with the film thickness was achieved [24,25] (critical thickness lowers with increasing film refractive index which is higher at lower wavelength). The amplified spontaneous emission of transversally pumped asymmetric wave-guiding luminescent polymeric thin films dominantly occurs a few degrees into the substrate plane [26] (film thickness df > ‘crit [1]). The wave-guided amplified spontaneous emission leaks out evanescently into the substrate along the interface and is edge emitted a few degrees off-axis the in-plane film–substrate interface towards the substrate side, while the waveguided emission in the film is edge-emitted in all angles in the plane perpendicular to the film plane and including the light propagation direction in the film since the film thickness is smaller than the diffracting limit (df < k; k wavelength of light in vacuum, full divergence angle is Dh ¼ 2p). Here the angle and wavelength dependent transmission of fluorescence light from a luminescent thin film into the substrate (bottom cladding) and into the air (top cladding) is determined theoretically by applying the Fresnel equations of reflection and transmission to multiple beam reflection and transmission in an asymmetric waveguide structure [27– 30]. The dependence of the constructive and destructive Fabry–Perot like interference on the refractive index allows the determination of the refractive indices of thin films for s-polarised and p-polarised light. The sub-laser-threshold leakymode emission behaviour is studied for molecular 4-methyl-TPD thin films [31,32] and polymeric TPD(4M)-MEH-P-PPV thin films [31,32].
2. Theory The light emission from an emitting thin film on a substrate is studied. The emission of light to the air side and to the substrate side is considered. The
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air-side thickness and the substrate thickness are unlimited. Any absorption and any amplification by stimulated emission is neglected. A schematic drawing is shown in Fig. 1(a). S-polarised light (TE mode propagation, electric field vector parallel to film plane) and p-polarised light (TM mode propagation, electric field vector in plane of propagation perpendicular to film plane) are considered. The situation of reflection and refraction at an interface, J, separating the media i and j with refractive indices ni > nj is illustrated in Fig. 1(b). The amplitude coefficients of reflection, rij , and transmission, tij , of the electrical field strength at an angle of incidence, ui , less than the angle of total internal reflection, ui;t , and at an angle of refraction, uj , are given by [27–29]
281
rijs ¼
ni cosðui Þ nj cosðuj Þ ; ni cosðui Þ þ nj cosðuj Þ
ð1aÞ
rijp ¼
nj cosðui Þ ni cosðuj Þ ; nj cosðui Þ þ ni cosðuj Þ
ð1bÞ
tijs ¼
2ni cosðui Þ ; ni cosðui Þ þ nj cosðuj Þ
ð2aÞ
tijp ¼
2ni cosðui Þ ; nj cosðui Þ þ ni cosðuj Þ
ð2bÞ
whereby
ui ¼ arcsin
nj sin uj : ni
ð3Þ
The superscripts s and p stand for s- and ppolarised light, respectively. For ni > nj the angle of total internal reflection, ui;t , is nj ui;t ¼ arcsin : ð4Þ ni For parallel polarised light the Brewster angle, uj;B , in medium j is ni uj;B ¼ arctan : ð5Þ nj In the case of ni > nj and ui > ui;t the electric field amplitude transmission coefficients are tijs ¼ tijp ¼ 0;
ð6Þ
and the amplitude reflection coefficients are [27–29] rijl ¼ exp 2i/lij ; l ¼ s; p ð7Þ with the phase shifts 2h i1=2 3 2 2 2 n sin ðu Þ n i i j 6 7 /sij ¼ arctan 4 5; ni cosðui Þ
ð8aÞ
and Fig. 1. (a) Schematic of ray path through thin film on substrate caused by light emission in film. Transmissions at interface A to air and at interface B to the substrate are indicated. (b) Detailed drawing of ray path by direct transmission at interface J and after reflection at J and I.
2 h i1=2 3 2 2 2 n i ni sin ðui Þ nj 6 7 /pij ¼ arctan 4 5: n2j cosðui Þ
ð8bÞ
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For ui < ui;t , the phase changes at the interfaces by light refraction are 0 for rijl P 0; /lij ¼ l ¼ s; p ð9Þ p for rijl < 0; rijl
as is seen by taking the phase part of ¼ jrijl j expði/lij Þ. The phase change between equal wave fronts of a directly refracted ray at interface, J, and a ray refracted at J after reflections at J and I (see Fig. 1(b)) is given by /path ¼ 2s1 ki s3 kj ;
ð10Þ
where s1 ¼
df ; cosðui Þ
ð10aÞ
s2 ¼ df tanðui Þ;
ð10bÞ
s3 ¼ 2s2 sinðuj Þ ¼ 2df tanðui Þ sinðuj Þ;
ð10cÞ
kl ¼ nl 2p~ m ¼ nl 2p=k;
ð10dÞ
l ¼ i; j;
l ¼ Tij;J
Ij nj cosðuj Þ l 2 jt j : ¼ Ii ni cosðui Þ ij;J
ð13Þ
3. Simulations In the simulations a luminescent film on a transparent substrate is considered. The transmission from the film to the air at the film–air interface, A, and the transmission from the film to the substrate at the film–substrate interface, B, are simulated (Fig. 1(a)). The phase changes are determined. The refractive indices are fixed to na ¼ 1 for the air, nf ¼ 1:7 for the film, and ns ¼ 1:5 for the substrate. The angles of transmission, ua (at air side) and us (at substrate side), the film thickness, df , and the wavelength, k, are varied. In Fig. 2 the intensity reflectance at the film–air 2 interface, RA;s ¼ jrihs j of s-polarised light, RA;p ¼ p 2 jrih j of p-polarised light, and the intensity reflectance at the film–substrate interface, RB;s ¼ jrijs j2 of 2 s-polarised light, RB;p ¼ jrijp j of p-polarised light is displayed. The zero reflectance at Brewster angle for p-polarised light, and the rise in reflectance
kl is the wave vector in medium l; m~ is the wave number in vacuum, and k is the wavelength in vacuum. The total phase difference between a beam being reflected once at both interfaces, J and I, before transmitted at J and a beam directly transmitted at interface J is D/lJ ¼ /lJIJ /lJ ¼ /lij þ /lih þ /path :
ð11Þ
Constructive interference occurs if D/lJ ¼ 2pm;
m ¼ 0; 1; 2; 3; . . . ;
ð12aÞ
while destructive interference occurs if D/lJ ¼ 2pðm þ 1=2Þ;
m ¼ 0; 1; 2; 3; . . . ;
ð12bÞ
m is the interference order. l , at inThe total electric field transmission, tij;J terface J is obtained by the geometric row of the direct transmission and the transmissions after multiple reflection according to l tij;J ¼
tijl : 1 expði/path Þrijl rihl
ð12Þ
The intensity transmission from i to j at interface J is given by [27,28]
Fig. 2. Single-interface intensity reflectance versus angle of incidence for propagation from optical denser to optical thinner medium. Situations are presented for s- and p-polarised light at film–air interface, A, and at film–substrate interface, B. Applied refractive indices are na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5.
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towards 1 near the critical angle of total internal reflection are clearly seen. In Figs. 3 and 4 the emission behaviours from film to air and from film to substrate at normal incidence ðua ¼ us ¼ 0) are illustrated. In Fig. 3 the film thickness is varied at constant wavelength, while in Fig. 4 the wavelength is varied at constant film thickness. There is 2p phase difference between s- (TE) and p- (TM) polarised light which gives no difference in the transmission of s- and p-polarised light. The transmission modulates with film thickness (Fig. 3(a)) and wavelength (Fig. 4(a)) due to constructive and destructive multi-beam interference at the interfaces. The transmission at the film– air interface is lower than at the film–substrate interface because of the larger refractive index step at the film-air border than at the film-substrate border. The modulation depths are moderate since the single-path reflections are small (0.06722 at interface A, and 0.003906 at interface B). At constructive interference the intensity transmission to
Fig. 3. Simulation of thickness dependence of transmission at film–air interface, TA , and at film–substrate interface, TB , in the case of normal incidence, uf ¼ 0. Used parameters: Wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions (no difference between s- and p-polarised light). (b) Phase differences, D/s and D/p , between double reflected beam before transmission and directly transmitted beam (Eq. (11), no differences at interfaces A and B). The interference order m is indicated.
283
Fig. 4. Simulation of wavelength dependence of transmission at film–air interface, TA , and at film–substrate interface, TB , in the case of normal incidence, uf ¼ 0. Used parameters: Film thickness df ¼ 400 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions (no difference between s- and ppolarised light). (b) Phase differences, D/s and D/p , between double reflected beam before transmission and directly transmitted beam (Eq. (11), no differences at interfaces A and B). The interference order m is indicated.
the substrate increases beyond 100% because of directional collection (light emitted to A and partially reflected there constructively interferes with light directly emitted to B). In Figs. 5 and 6 the emission behaviours from film to air and from film to substrate at grazing angle of transmission ðua ¼ us ¼ 89°) are illustrated. In Fig. 5 the film thickness is varied at constant wavelength, while in Fig. 6 the wavelength is varied at constant film thickness. There is p phase difference between s- (TE) and p- (TM) polarised light at the film–air interface A since the angle of incidence is above the Brewster angle at interface A and below the Brewster angle at interface B. Therefore, the s- and p-polarised light transmitted at A to air are 180° out of phase (where TA;s is maximal there TA;p is minimal, and vice versa). The transmission at A is small since the grazing transmission corresponds to near total internal reflection where nearly all light is reflected at A, and at the film–substrate interface B the
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Fig. 5. Simulation of thickness dependence of transmission and phase change at film–air interface A and at film–substrate interface B in the case of grazing angles of transmission, uh ¼ ua ¼ 89° and uj ¼ us ¼ 89°. Used parameters: Wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions. (b) Phase differences. Interference order m is indicated.
reflection is small (far away from angle of total internal reflection). At grazing angle of transmission at the film–substrate interface there occurs strong multiple-beam interference since all light is reflected at interface A and nearly all light is reflected at interface B. Over wide thickness ranges (Fig. 5(a)) and over wide wavelength ranges (Fig. 6(a)) the grazing angle emission is suppressed, and over small thickness and wavelength ranges the grazing angle transmission is strongly enhanced (Fabry–Perot effect). The intensity transmission at constructive resonance increases beyond 1 because of angular compression of the emission towards the forward direction (ratio of cross-sectional area in film to cross-sectional area in substrate is cosðuf Þ= cosðus Þ, with cosðus Þ ! 0 for us ! 90°). In Fig. 7 the dependences of the phase difference, D/A , and of the transmission, TA , on the angle of transmission, ua , at the film–air interface, A, are illustrated. The applied parameters are k ¼ 540 nm, df ¼ 400 nm, na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5. Below the Brewster angle ðua < uA;Br Þ
Fig. 6. Simulation of wavelength dependence of transmission and phase change at film–air interface A and at film–substrate interface B in the case of grazing angles of transmission, uh ¼ ua ¼ 89° and uj ¼ us ¼ 89°. Used parameters: Film thickness df ¼ 400 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions. (b) Phase differences. Interference order m is indicated.
Fig. 7. Simulation of angle dependence of transmission at film– air interface A (angle of transmission ua Þ. Used parameters: Film thickness df ¼ 400 nm; wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions, TA , of s- and p-polarised light. (b) Phase differences, D/A , of s- and ppolarised light. uA;Br indicates Brewster angle at air–film interface A. Interference order, m, is indicated.
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the phase change between s- and p-polarised light is 2p. This phase change reduces to p above the Brewster angle. The transmission of the p-polarised light is 100% at the Brewster angle (Fig. 7(a)). The transmission approaches zero at high angle of transmission (total internal reflection at A). The interface B has only weak influence on the transmission at A because of only weak reflection at B for the involved angle range in the film. In Fig. 8 the dependences of the phase difference, D/B , and of the transmission, TB , on the angle of transmission, us , at the film–substrate interface B are illustrated. The applied parameters are k ¼ 540 nm, df ¼ 400 nm, na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5. Below the Brewster angle at interface A ðua < uA;Br Þ the phase change between s- and p-polarised light is 2p (corresponding us position is denoted as Br,A in Fig. 8(b)). This phase change reduces to p between the Brewster angle and the angle of total internal reflection at A ðuA;Br < ua < uA;t Þ. Then the phase change between s- and p-
Fig. 8. Simulation of angle dependence of transmission at film– substrate interface B (angle of transmission us ). Used parameters: Film thickness df ¼ 400 nm; wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions, TB , of s- and p-polarised light. (b) Phase differences, D/B , of s- and p-polarised light. Br,A indicates us angle position for Brewster angle at air–film interface A; TR,A indicates us angle position for angle of total reflection at film–air interface; Br,B indicates us;Br the Brewster angle at the film–substrate interface B.
285
polarised light increases due to the phase change at A above uA;t until the Brewster angle at the film– substrate interface B is reached, where the phase difference is reduced by p. The complex changes of D/B influence the intensity transmission, TB , from film to substrate (Fig. 8(a)). The transmission of the p-polarised light is 100% at the Brewster angle us ¼ uB;Br (Fig. 8(a)). The transmission has maxima whenever the phase differences D/B approach towards integer multiples, m, of 2p, and the transmission has valleys around uneven integers of p. The explicit transmission, TB ðus Þ, depends strongly on the explicit values of the film thickness, the wavelength, and the refractive indices of film and substrate. In Fig. 9 the transmission peaks, Tpeak , and the corresponding wavelengths, kpeak , due to constructive interference at the film–substrate interface B for s- and p-polarised light are presented versus angle of transmission us . The fixed parameters are df ¼ 400 nm, na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5. The interference order is m ¼ 2. The wavelengths of constructive interference shift to
Fig. 9. Simulation of angle dependence of peak transmission, Tpeak , (a) and corresponding peak wavelength, kpeak , (b) for emission at film–substrate interface B (angle of transmission us Þ. Order of interference m ¼ 2. Used parameters: Film thickness df ¼ 400 nm; wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. Curves are shown for s- (TE) and p- (TM) polarised light.
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shorter wavelength with increasing angle of transmission due to the angle dependence of /path (Eq. (10)). The constructive transmission resonance becomes sharper and the transmission peak higher with rising angle of transmission since the reflectance at the interface A is 100% for us P arcsin½sinðuA;t Þnf =ns and the reflectance at the interface B approaches 100% as us approaches 90° (situation of total internal reflection at B). This is the situation of a Fabry–Perot interferometer of high finesse [27–29] (At resonance more and more light becomes transmitted as us approaches 90°, and the intensity transmission rises towards infinity because of cross-sectional contraction towards zero. In reality a singularity does not occur because of increasing diffraction with cross-sectional contraction). The dependence of the wavelength position, kpeak , of peak constructive transmission at grazing emission to the substrate on the film refractive index, nf , is illustrated in Fig. 10. Curves are presented for interference orders m ¼ 1, m ¼ 2, and m ¼ 3, film thickness of df ¼ 400 nm, and a filmsubstrate angle of transmission of us ¼ 89:5°, both
for s- and p-polarised light. The fixed refractive indices are na ¼ 1, and ns ¼ 1:5. There is a strong dependence of kpeak on nf . For grazing emission, us > 85°, the position kpeak depends only weakly on the actual value of us ðkpeak shifts approximately 2 nm to shorter wavelengths for us ¼ 85° compared to us ¼ 90°). The wavelength of constructive interference, kpeak , is proportional to the film thickness. The strong dependence of kpeak on nf at grazing angle of transmission may be used to measure accurately the refractive indices of thin films for s- and p-polarised light.
4. Relation of leaky-mode emission to wave-guiding The occurrence of grazing angle emission of optically pumped luminescent polymeric thin films on substrates with film thicknesses below and above the critical thickness of wave-guiding was already reported in Section 1. For wave-guiding light of wavelength, kL , in asymmetric thin films the film thickness, df , has to be larger than a critical thickness, ‘crit , which is given by [1] 2 1=2 kL ns 1 ‘crit ¼ arctan : ð14Þ n2f n2s 2pðn2f n2s Þ Light of any wavelength, k < kL , is guided for uf > uB;t , each wavelength at a particular angle, uf ðkÞ > uB;t , for constructive interference (see for example Eq. (8) in [33]). In the case of our simulations with k ¼ kASE;max ¼ 540 nm, nf ¼ 1:7, and ns ¼ 1:5 it is ‘crit ¼ 127:53 nm. Having a film of thickness df light is wave-guided for k < kC , the cut-off wavelength, which is obtained by re-writing Eq. (14) to kC ¼
Fig. 10. Simulation of the dependence of the wavelength, kpeak , of constructive grazing-angle emission from film to substrate on the film refractive index, nf . The fixed parameters are na ¼ 1, ns ¼ 1:5, us ¼ 89:5°, and df ¼ 400 nm. The interference orders are m ¼ 1, 2, and 3. The solid curves apply to s-polarised light, and the dashed curves apply to p-polarised light.
2pðn2f n2s Þdf 2 1=2 : ns 1 arctan n2f n2s
ð15Þ
For df > ‘crit amplified spontaneous emission (ASE) occurs above a threshold pump pulse energy density, w0L;th , with wavelength of maximum amplitude, kASE;max < kC , determined by the position of maximum effective stimulated emission cross-section [26].
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For grazing-angle leaky-mode emission the wavelength of constructive peak emission, kpeak ðus Þ, is fixed by the Fabry–Perot constructive interference condition, D/B ¼ 2pm (m integer, see Eq. (12a)). Using df ¼ ‘crit ¼ 127:53 nm for nf ¼ 1:7, ns ¼ 1:5, and na ¼ 1, and k ¼ kASE;max ¼ 540 nm, we calculate kpeak;s (us ¼ 89°) ¼ 292.5 nm (m ¼ 1), ¼ 120 nm (m ¼ 2), and kpeak;p (us ¼ 89°) ¼ 354 nm (m ¼ 1), ¼ 129 nm (m ¼ 2) at the critical film thickness. The calculated examples show that at the critical film thickness, ‘crit , the wavelength, kpeak , of constructive leaky-mode emission is below the wavelength of amplified spontaneous emission, kASE;max . If kpeak is outside the range of the fluorescence emission of the film then no constructive leaky-mode emission occurs. In wave-guiding thin films the effective interaction length for amplified spontaneous emission, ‘ASE , is approximately given by the absorption length, ‘abs ¼ 1=af ðkASE;max Þ, where af is the absorption coefficient of the luminescent film material at the wavelength of peak amplified spontaneous emission [26], i.e., 1 ‘ASE ¼ ‘abs ¼ : ð16Þ af ðkASE;max Þ In the case of grazing-angle in-substrate leakymode emission the effective interaction length of amplification of spontaneous emission is the shorter of the absorption length, ‘abs , and the leaky-mode length, ‘lm ¼ 2df =½cosðuf Þð1 RB;l Þ l ¼ s, p; i.e., ‘ASE ¼ minð‘abs ; ‘lm Þ: ð17Þ Except very near to the angle of total internal reflection, uB;t , (us very near to 90°), in good lasing materials it will be ‘abs > ‘lm and the amplification length will be longer for wave-guided emission than for leaky-mode emission. The wave-guided travelling-wave lasing will have the lower threshold than the leaky-mode lasing, and the waveguided travelling-wave laser action will suppress efficient leaky-mode laser action. In wave-guiding thin films with grating structure the constructive interference condition is given by the grating equation (Bragg condition) [27–29]. This interference forces the broad-band travelling-wave laser operation into the narrow-band wave-guided distributed feed-back laser operation [33–36].
287
5. Experimental sub-laser-threshold leaky-mode emission The modification of the spontaneous emission in grazing-angle direction due to multiple-beam interference (micro-cavity effect [5–7]) has been reported shortly in [31] for the laser-active molecule 4-methyl-TPD [31,32] on a glass substrate. Here more detailed results are given for 4-methylTPD and the laser-active polymer TPD(4M)MEH-P-PPV [31,32]. The structural formulae of the molecule and the polymer are shown in Fig. 11. In Fig. 12 results are presented for two 4methyl-TPD thin films of different thickness on glass substrates. In Fig. 12a the film thickness is df 360 nm. The dotted line shows the fluorescence spectrum, SF;n ðkÞ, emitted normal to the film surface where interference effects are negligible (see Fig. 3). A cavity-free spontaneous emission spectrum is observed. The solid curve shows the edge emitted light spectrum, SF;gr ðkÞ, well below the threshold of wave-guided travelling-wave lasing (w0L ¼ 5 lJ cm2 , w0L;th 20 lJ cm2 [31]). The spectrum was measured without a polarizer in the detection path. The collection angle of edge emitted light was approximately 10°. The edge-emitted spectrum looks quite different from the normal surface emitted spectrum (micro-cavity modification of spontaneous emission). The peak at 435 nm is interpreted as TM mode resonance
CH3
CH3
N
N
4-methyl-TPD CH3
CH3 O
N
N
H C
C H
n
H3CO
TPD(4M)-MEH-P-PPV
Fig. 11. Structural formulae of 4-methyl-TPD and TPD(4M)MEH-P-PPV (for details see [31,32]).
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Fig. 12. Normalized absorption spectrum, ra ðkÞ=ra;max , normalized spontaneous emission spectra S F =S F;max and normalized wave-guided amplified spontaneous emission spectra S ASE =S ASE;max of two 4-methyl-TPD films on glass substrates. Exposed film area by picosecond pump laser is 0.19 mm 5 mm (for details see [31,32]). (a) Film thickness df ¼ 360 nm. Curves are explained in the text. (b) Film thickness df ¼ 195 nm. Curves are explained in the text.
corresponding to a film refractive index of nf;p ¼ 1:803. The peak at 412 nm is interpreted as TE mode resonance corresponding to a film refractive index of nf;s ¼ 1:820 (substrate refractive angle ns ¼ 1:53). The dash-dotted curve (TE polarised edge-emission collected) and the dashedtriple-dotted curve (TM polarised edge emission collected) were obtained well above laser threshold (w0L ¼ 160 lJ cm2 ). They show the wave-guided amplified spontaneous emission spectra. The spectral peak occurring at the first vibronic peak of the normal spontaneous emission spectrum is determined by the position of the highest effective stimulated emission cross-section. The position of the highest normal spontaneous emission does not have the highest gain because of residual groundstate absorption. The shape of the absorption cross-section spectrum, ra ðkÞ, is shown by the dashed curve in Fig. 12(a). In Fig. 12(b) the spontaneous emission behaviour is shown for a 4-methyl-TPD film of df 195 nm film thickness. The dotted line shows the
fluorescence emitted normal to the surface, SF;n ðkÞ, and the solid curve shows the edge-emitted fluorescence, SF;gr ðkÞ, at a pump pulse energy density of w0L ¼ 7:8 lJ cm2 . The wavelength of the edge emission peak is determined by the absorption edge of the polymer and not by the occurrence of constructive grazing-angle interference (resonance is expected in the absorption band). In Fig. 13 results are shown for a 430 nm thick TPD(4M)-MEH-P-PPV film on a glass substrate. The dotted curve shows the normal-incidence spontaneous emission spectrum, SF :n ðkÞ. The solid curve displays the s-polarised leaky-mode edge emission, SF ;s;gr ðkÞ, below laser threshold (w0L ¼ 15 lJ cm2 , w0L;th 17 lJ cm2 [31]). The emission peak around 540 nm corresponds to a refractive index of nf;s (540 nm) 1.871 using a substrate refractive index of ns ¼ 1:518. The dashed curve gives the p-polarised leaky-mode edge emission, SF;p;gr ðkÞ ðw0L ¼ 10 lJ cm2 ). The emission peak around 590 nm fits the refractive index of nf;p (590 nm) 1.867 The dash-dotted curve shows the wave-guided amplified spontaneous emission, SASE ðkÞ, well above laser threshold (w0L 200 lJ cm2 ). The ASE peak fits to the emission peak of
Fig. 13. Normalized absorption spectrum, ra ðkÞ=ra;max , normalized spontaneous emission S F =S F;max and normalized waveguided amplified spontaneous emission spectra S ASE =S ASE;max of a TPD(4M)-MEH-P-PPV film on a glass substrate. Film thickness, df ¼ 430 nm. Curves are explained in the text.
A. Penzkofer et al. / Optics Communications 229 (2004) 279–290
the normal (cavity-free) spontaneous emission. The long-wave-length absorption cross-section spectrum, ra ðkÞ, normalized to ra (360 nm) ¼ 1.63 1016 cm2 [31], is shown by the dashed curve. It indicates that the absorption edge does not influence the wavelength peak positions of the s- and p-polarised grazing-angle spontaneous emission for the applied film thickness.
6. Conclusions The light transmission of luminescent films on transparent substrates of lower refractive index than the films was studied theoretically. The grazing-angle constructive and destructive TE and TM light emission into the substrate was analysed in detail. The micro-cavity behaviour was worked out. The results were set in context with the waveguided amplified spontaneous emission (travellingwave laser action). The leaky-mode emission was discussed for luminescent organic neat films on transparent substrates, and results were presented for an organic molecular thin film (4-methyl-TPD) and an organic polymeric thin film (TPD(4M)-MEH-P-PPV). The presented theory applies to all transparent films on transparent substrates with higher film refractive index than substrate refractive index. The wavelength selective constructive leakymode emission was shown to allow the determination of the refractive indices of luminescent thin films for TE and TM polarised light.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
[15]
[16] [17] [18]
[19] [20] [21] [22] [23] [24] [25]
Acknowledgements
[26]
The authors thank Prof. Ilmo Sildos, University of Tartu, Estonia, for stimulating discussions.
[27]
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K. M€ ullen, A. Gombert, V. Wittwer, Adv. Mater. 10 (1998) 920. [35] G.A. Turnbull, P. Andrew, M.J. Jory, W.L. Barnes, I.D.W. Samuel, Phys. Rev. B 64 (2001) 122. [36] G. Kranzelbinder, E. Toussaere, J. Zyss, A. Pogantsch, E.W.J. List, H. Tillmann, H.-H. H€ orhold, Appl. Phys. Lett. 80 (2002) 716.
Leaky-mode emission of luminescent thin films on transparent substrates A. Penzkofer
a,*
, W. Holzer a, H. Tillmann b, H.-H. H€ orhold
b
a
b
Institut II – Experimentelle und Angewandte Physik, Faculty of Physics, Universit€at Regensburg, Universit€atsstrasse 31, D-93053 Regensburg, Germany Institut f€ ur Organische Chemie and Makromolekulare Chemie der Universit€at Jena, Humboldtstrasse 10, D-07743 Jena, Germany Received 31 July 2003; received in revised form 25 September 2003; accepted 26 October 2003
Abstract The angle, wavelength, and thickness dependent fluorescence emission of a luminescent film on a transparent substrate into the substrate and the air is studied theoretically by Fresnel type transmittance calculations for s- (TE) and p- (TM) polarised emission. A micro-cavity behaviour due to Fabry–Perot like constructive and destructive interference is observed. The Fabry–Perot like interference strongly modifies the grazing-angle in-substrate spontaneous emission spectrum compared to the normal-incidence spontaneous emission. The wavelength selective constructive leaky-mode emission allows the determination of the refractive indices of luminescent thin films for TE and TM polarised light. Sublaser-threshold grazing-angle in-substrate emission is analysed for molecular 4-methyl-TPD and polymeric TPD(4M)MEH-P-PPV thin films. Ó 2003 Elsevier B.V. All rights reserved. PACS: 78.65; 42.80.L; 42.70.G Keywords: Leaky mode emission; Asymmetric wave-guide; Luminescent polymers; Multiple-beam reflection; Refractive index dependence
1. Introduction In luminescent asymmetric wave-guiding thin films the light propagation is dependent on the emission angle. Beyond the critical angle of total
* Corresponding author. Tel.: +499419432107; fax: +499419432754. E-mail address: [email protected] (A. Penzkofer).
internal reflection all emission is wave-guided in the neat film [1,2]. Slightly below the critical angle of total internal reflection the emission leaks into the substrate, propagates at a small angle to the film–substrate interface and escapes at the substrate edge (substrate edge emission). The leakedout emission interferes with the light which is reflected at the substrate–film interface, then reflected at the film–air interface, and finally transmitted at the substrate–film interface. The Fabry–Perot like multiple light interference gives
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.10.039
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rise to a wavelength, angle, film thickness, and polarisation dependent in-substrate light emission. The occurrence of constructive and destructive interference for light propagating at the film– substrate interface was observed in the case of nearly in-plane photoluminescence studies on asymmetric semiconductor micro-cavities [2]. Leaky-mode propagation of self-trapped exciton emission in anatase thin films on sapphire substrates was observed and analysed in [3]. The wavelength and angle dependent emission behaviour was explained by subtotal internal reflection constructive and destructive interference of the leaking radiation. The wavelength selectivity for the TE and TM modes was clarified by the different phase changes for TE and TM polarised light [4]. The occurring multi-beam interference modified the spontaneous emission spectrum. The constructive and destructive Fabry–Perot like asymmetric thin-film waveguide behaviour at grazing angle is quite similar to the constructive and destructive Fabry–Perot like behaviour of thin-film micro-cavities. In thin-film micro-cavities metallic mirrors or Bragg reflectors are used for high reflectivity. In asymmetric air–film–substrate waveguides at grazing-angle of light escape to the substrate there is total internal reflection at the film–air interface and high reflectivity at the film– substrate interface. Therefore, all phenomena of micro-cavities [5–7] are present. The micro-cavity quality of the waveguides increases with increase of reflectivity as the film-substrate angle of incidence approaches the angle of total internal reflection (grazing-angle leaky-mode emission). Micro-cavity effects in polymeric light-emitting diodes allow the control of the emission characteristics [8–10]. A wave-length tuning of the emission is achieved by cavity thickness variation [11,12]. Optically pumped organic polymeric thinfilm micro-cavities have been realised [13–18]. The laser emission of these mirror - thin-film – mirror devices with mirror spacing in the few 100 nm range shows the typical micro-cavity resonance effects (spectral line narrowing, thickness and angle dependent laser wavelength). In non-waveguiding thin luminescent polymer films on glass substrates (film thickness below the critical thickness, ‘crit , necessary for wave-guiding [1]) spectral changes [19] and spectral line-narrow-
ing [19–21] by leaky-mode emission were observed. Grazing-angle emitted light from polyfluorene derivative thin films of TE and TM polarisation was studied experimentally and theoretically in [22]. The influence of the cut-off film thickness on the amplified spontaneous emission dependence was analysed in [23]. Amplified spontaneous emission wavelength tuning by cut-off wavelength tuning with the film thickness was achieved [24,25] (critical thickness lowers with increasing film refractive index which is higher at lower wavelength). The amplified spontaneous emission of transversally pumped asymmetric wave-guiding luminescent polymeric thin films dominantly occurs a few degrees into the substrate plane [26] (film thickness df > ‘crit [1]). The wave-guided amplified spontaneous emission leaks out evanescently into the substrate along the interface and is edge emitted a few degrees off-axis the in-plane film–substrate interface towards the substrate side, while the waveguided emission in the film is edge-emitted in all angles in the plane perpendicular to the film plane and including the light propagation direction in the film since the film thickness is smaller than the diffracting limit (df < k; k wavelength of light in vacuum, full divergence angle is Dh ¼ 2p). Here the angle and wavelength dependent transmission of fluorescence light from a luminescent thin film into the substrate (bottom cladding) and into the air (top cladding) is determined theoretically by applying the Fresnel equations of reflection and transmission to multiple beam reflection and transmission in an asymmetric waveguide structure [27– 30]. The dependence of the constructive and destructive Fabry–Perot like interference on the refractive index allows the determination of the refractive indices of thin films for s-polarised and p-polarised light. The sub-laser-threshold leakymode emission behaviour is studied for molecular 4-methyl-TPD thin films [31,32] and polymeric TPD(4M)-MEH-P-PPV thin films [31,32].
2. Theory The light emission from an emitting thin film on a substrate is studied. The emission of light to the air side and to the substrate side is considered. The
A. Penzkofer et al. / Optics Communications 229 (2004) 279–290
air-side thickness and the substrate thickness are unlimited. Any absorption and any amplification by stimulated emission is neglected. A schematic drawing is shown in Fig. 1(a). S-polarised light (TE mode propagation, electric field vector parallel to film plane) and p-polarised light (TM mode propagation, electric field vector in plane of propagation perpendicular to film plane) are considered. The situation of reflection and refraction at an interface, J, separating the media i and j with refractive indices ni > nj is illustrated in Fig. 1(b). The amplitude coefficients of reflection, rij , and transmission, tij , of the electrical field strength at an angle of incidence, ui , less than the angle of total internal reflection, ui;t , and at an angle of refraction, uj , are given by [27–29]
281
rijs ¼
ni cosðui Þ nj cosðuj Þ ; ni cosðui Þ þ nj cosðuj Þ
ð1aÞ
rijp ¼
nj cosðui Þ ni cosðuj Þ ; nj cosðui Þ þ ni cosðuj Þ
ð1bÞ
tijs ¼
2ni cosðui Þ ; ni cosðui Þ þ nj cosðuj Þ
ð2aÞ
tijp ¼
2ni cosðui Þ ; nj cosðui Þ þ ni cosðuj Þ
ð2bÞ
whereby
ui ¼ arcsin
nj sin uj : ni
ð3Þ
The superscripts s and p stand for s- and ppolarised light, respectively. For ni > nj the angle of total internal reflection, ui;t , is nj ui;t ¼ arcsin : ð4Þ ni For parallel polarised light the Brewster angle, uj;B , in medium j is ni uj;B ¼ arctan : ð5Þ nj In the case of ni > nj and ui > ui;t the electric field amplitude transmission coefficients are tijs ¼ tijp ¼ 0;
ð6Þ
and the amplitude reflection coefficients are [27–29] rijl ¼ exp 2i/lij ; l ¼ s; p ð7Þ with the phase shifts 2h i1=2 3 2 2 2 n sin ðu Þ n i i j 6 7 /sij ¼ arctan 4 5; ni cosðui Þ
ð8aÞ
and Fig. 1. (a) Schematic of ray path through thin film on substrate caused by light emission in film. Transmissions at interface A to air and at interface B to the substrate are indicated. (b) Detailed drawing of ray path by direct transmission at interface J and after reflection at J and I.
2 h i1=2 3 2 2 2 n i ni sin ðui Þ nj 6 7 /pij ¼ arctan 4 5: n2j cosðui Þ
ð8bÞ
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For ui < ui;t , the phase changes at the interfaces by light refraction are 0 for rijl P 0; /lij ¼ l ¼ s; p ð9Þ p for rijl < 0; rijl
as is seen by taking the phase part of ¼ jrijl j expði/lij Þ. The phase change between equal wave fronts of a directly refracted ray at interface, J, and a ray refracted at J after reflections at J and I (see Fig. 1(b)) is given by /path ¼ 2s1 ki s3 kj ;
ð10Þ
where s1 ¼
df ; cosðui Þ
ð10aÞ
s2 ¼ df tanðui Þ;
ð10bÞ
s3 ¼ 2s2 sinðuj Þ ¼ 2df tanðui Þ sinðuj Þ;
ð10cÞ
kl ¼ nl 2p~ m ¼ nl 2p=k;
ð10dÞ
l ¼ i; j;
l ¼ Tij;J
Ij nj cosðuj Þ l 2 jt j : ¼ Ii ni cosðui Þ ij;J
ð13Þ
3. Simulations In the simulations a luminescent film on a transparent substrate is considered. The transmission from the film to the air at the film–air interface, A, and the transmission from the film to the substrate at the film–substrate interface, B, are simulated (Fig. 1(a)). The phase changes are determined. The refractive indices are fixed to na ¼ 1 for the air, nf ¼ 1:7 for the film, and ns ¼ 1:5 for the substrate. The angles of transmission, ua (at air side) and us (at substrate side), the film thickness, df , and the wavelength, k, are varied. In Fig. 2 the intensity reflectance at the film–air 2 interface, RA;s ¼ jrihs j of s-polarised light, RA;p ¼ p 2 jrih j of p-polarised light, and the intensity reflectance at the film–substrate interface, RB;s ¼ jrijs j2 of 2 s-polarised light, RB;p ¼ jrijp j of p-polarised light is displayed. The zero reflectance at Brewster angle for p-polarised light, and the rise in reflectance
kl is the wave vector in medium l; m~ is the wave number in vacuum, and k is the wavelength in vacuum. The total phase difference between a beam being reflected once at both interfaces, J and I, before transmitted at J and a beam directly transmitted at interface J is D/lJ ¼ /lJIJ /lJ ¼ /lij þ /lih þ /path :
ð11Þ
Constructive interference occurs if D/lJ ¼ 2pm;
m ¼ 0; 1; 2; 3; . . . ;
ð12aÞ
while destructive interference occurs if D/lJ ¼ 2pðm þ 1=2Þ;
m ¼ 0; 1; 2; 3; . . . ;
ð12bÞ
m is the interference order. l , at inThe total electric field transmission, tij;J terface J is obtained by the geometric row of the direct transmission and the transmissions after multiple reflection according to l tij;J ¼
tijl : 1 expði/path Þrijl rihl
ð12Þ
The intensity transmission from i to j at interface J is given by [27,28]
Fig. 2. Single-interface intensity reflectance versus angle of incidence for propagation from optical denser to optical thinner medium. Situations are presented for s- and p-polarised light at film–air interface, A, and at film–substrate interface, B. Applied refractive indices are na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5.
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towards 1 near the critical angle of total internal reflection are clearly seen. In Figs. 3 and 4 the emission behaviours from film to air and from film to substrate at normal incidence ðua ¼ us ¼ 0) are illustrated. In Fig. 3 the film thickness is varied at constant wavelength, while in Fig. 4 the wavelength is varied at constant film thickness. There is 2p phase difference between s- (TE) and p- (TM) polarised light which gives no difference in the transmission of s- and p-polarised light. The transmission modulates with film thickness (Fig. 3(a)) and wavelength (Fig. 4(a)) due to constructive and destructive multi-beam interference at the interfaces. The transmission at the film– air interface is lower than at the film–substrate interface because of the larger refractive index step at the film-air border than at the film-substrate border. The modulation depths are moderate since the single-path reflections are small (0.06722 at interface A, and 0.003906 at interface B). At constructive interference the intensity transmission to
Fig. 3. Simulation of thickness dependence of transmission at film–air interface, TA , and at film–substrate interface, TB , in the case of normal incidence, uf ¼ 0. Used parameters: Wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions (no difference between s- and p-polarised light). (b) Phase differences, D/s and D/p , between double reflected beam before transmission and directly transmitted beam (Eq. (11), no differences at interfaces A and B). The interference order m is indicated.
283
Fig. 4. Simulation of wavelength dependence of transmission at film–air interface, TA , and at film–substrate interface, TB , in the case of normal incidence, uf ¼ 0. Used parameters: Film thickness df ¼ 400 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions (no difference between s- and ppolarised light). (b) Phase differences, D/s and D/p , between double reflected beam before transmission and directly transmitted beam (Eq. (11), no differences at interfaces A and B). The interference order m is indicated.
the substrate increases beyond 100% because of directional collection (light emitted to A and partially reflected there constructively interferes with light directly emitted to B). In Figs. 5 and 6 the emission behaviours from film to air and from film to substrate at grazing angle of transmission ðua ¼ us ¼ 89°) are illustrated. In Fig. 5 the film thickness is varied at constant wavelength, while in Fig. 6 the wavelength is varied at constant film thickness. There is p phase difference between s- (TE) and p- (TM) polarised light at the film–air interface A since the angle of incidence is above the Brewster angle at interface A and below the Brewster angle at interface B. Therefore, the s- and p-polarised light transmitted at A to air are 180° out of phase (where TA;s is maximal there TA;p is minimal, and vice versa). The transmission at A is small since the grazing transmission corresponds to near total internal reflection where nearly all light is reflected at A, and at the film–substrate interface B the
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Fig. 5. Simulation of thickness dependence of transmission and phase change at film–air interface A and at film–substrate interface B in the case of grazing angles of transmission, uh ¼ ua ¼ 89° and uj ¼ us ¼ 89°. Used parameters: Wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions. (b) Phase differences. Interference order m is indicated.
reflection is small (far away from angle of total internal reflection). At grazing angle of transmission at the film–substrate interface there occurs strong multiple-beam interference since all light is reflected at interface A and nearly all light is reflected at interface B. Over wide thickness ranges (Fig. 5(a)) and over wide wavelength ranges (Fig. 6(a)) the grazing angle emission is suppressed, and over small thickness and wavelength ranges the grazing angle transmission is strongly enhanced (Fabry–Perot effect). The intensity transmission at constructive resonance increases beyond 1 because of angular compression of the emission towards the forward direction (ratio of cross-sectional area in film to cross-sectional area in substrate is cosðuf Þ= cosðus Þ, with cosðus Þ ! 0 for us ! 90°). In Fig. 7 the dependences of the phase difference, D/A , and of the transmission, TA , on the angle of transmission, ua , at the film–air interface, A, are illustrated. The applied parameters are k ¼ 540 nm, df ¼ 400 nm, na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5. Below the Brewster angle ðua < uA;Br Þ
Fig. 6. Simulation of wavelength dependence of transmission and phase change at film–air interface A and at film–substrate interface B in the case of grazing angles of transmission, uh ¼ ua ¼ 89° and uj ¼ us ¼ 89°. Used parameters: Film thickness df ¼ 400 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions. (b) Phase differences. Interference order m is indicated.
Fig. 7. Simulation of angle dependence of transmission at film– air interface A (angle of transmission ua Þ. Used parameters: Film thickness df ¼ 400 nm; wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions, TA , of s- and p-polarised light. (b) Phase differences, D/A , of s- and ppolarised light. uA;Br indicates Brewster angle at air–film interface A. Interference order, m, is indicated.
A. Penzkofer et al. / Optics Communications 229 (2004) 279–290
the phase change between s- and p-polarised light is 2p. This phase change reduces to p above the Brewster angle. The transmission of the p-polarised light is 100% at the Brewster angle (Fig. 7(a)). The transmission approaches zero at high angle of transmission (total internal reflection at A). The interface B has only weak influence on the transmission at A because of only weak reflection at B for the involved angle range in the film. In Fig. 8 the dependences of the phase difference, D/B , and of the transmission, TB , on the angle of transmission, us , at the film–substrate interface B are illustrated. The applied parameters are k ¼ 540 nm, df ¼ 400 nm, na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5. Below the Brewster angle at interface A ðua < uA;Br Þ the phase change between s- and p-polarised light is 2p (corresponding us position is denoted as Br,A in Fig. 8(b)). This phase change reduces to p between the Brewster angle and the angle of total internal reflection at A ðuA;Br < ua < uA;t Þ. Then the phase change between s- and p-
Fig. 8. Simulation of angle dependence of transmission at film– substrate interface B (angle of transmission us ). Used parameters: Film thickness df ¼ 400 nm; wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. (a) Transmissions, TB , of s- and p-polarised light. (b) Phase differences, D/B , of s- and p-polarised light. Br,A indicates us angle position for Brewster angle at air–film interface A; TR,A indicates us angle position for angle of total reflection at film–air interface; Br,B indicates us;Br the Brewster angle at the film–substrate interface B.
285
polarised light increases due to the phase change at A above uA;t until the Brewster angle at the film– substrate interface B is reached, where the phase difference is reduced by p. The complex changes of D/B influence the intensity transmission, TB , from film to substrate (Fig. 8(a)). The transmission of the p-polarised light is 100% at the Brewster angle us ¼ uB;Br (Fig. 8(a)). The transmission has maxima whenever the phase differences D/B approach towards integer multiples, m, of 2p, and the transmission has valleys around uneven integers of p. The explicit transmission, TB ðus Þ, depends strongly on the explicit values of the film thickness, the wavelength, and the refractive indices of film and substrate. In Fig. 9 the transmission peaks, Tpeak , and the corresponding wavelengths, kpeak , due to constructive interference at the film–substrate interface B for s- and p-polarised light are presented versus angle of transmission us . The fixed parameters are df ¼ 400 nm, na ¼ 1, nf ¼ 1:7, and ns ¼ 1:5. The interference order is m ¼ 2. The wavelengths of constructive interference shift to
Fig. 9. Simulation of angle dependence of peak transmission, Tpeak , (a) and corresponding peak wavelength, kpeak , (b) for emission at film–substrate interface B (angle of transmission us Þ. Order of interference m ¼ 2. Used parameters: Film thickness df ¼ 400 nm; wavelength k ¼ 540 nm; refractive indices na ¼ 1, nf ¼ 1:7, ns ¼ 1:5. Curves are shown for s- (TE) and p- (TM) polarised light.
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shorter wavelength with increasing angle of transmission due to the angle dependence of /path (Eq. (10)). The constructive transmission resonance becomes sharper and the transmission peak higher with rising angle of transmission since the reflectance at the interface A is 100% for us P arcsin½sinðuA;t Þnf =ns and the reflectance at the interface B approaches 100% as us approaches 90° (situation of total internal reflection at B). This is the situation of a Fabry–Perot interferometer of high finesse [27–29] (At resonance more and more light becomes transmitted as us approaches 90°, and the intensity transmission rises towards infinity because of cross-sectional contraction towards zero. In reality a singularity does not occur because of increasing diffraction with cross-sectional contraction). The dependence of the wavelength position, kpeak , of peak constructive transmission at grazing emission to the substrate on the film refractive index, nf , is illustrated in Fig. 10. Curves are presented for interference orders m ¼ 1, m ¼ 2, and m ¼ 3, film thickness of df ¼ 400 nm, and a filmsubstrate angle of transmission of us ¼ 89:5°, both
for s- and p-polarised light. The fixed refractive indices are na ¼ 1, and ns ¼ 1:5. There is a strong dependence of kpeak on nf . For grazing emission, us > 85°, the position kpeak depends only weakly on the actual value of us ðkpeak shifts approximately 2 nm to shorter wavelengths for us ¼ 85° compared to us ¼ 90°). The wavelength of constructive interference, kpeak , is proportional to the film thickness. The strong dependence of kpeak on nf at grazing angle of transmission may be used to measure accurately the refractive indices of thin films for s- and p-polarised light.
4. Relation of leaky-mode emission to wave-guiding The occurrence of grazing angle emission of optically pumped luminescent polymeric thin films on substrates with film thicknesses below and above the critical thickness of wave-guiding was already reported in Section 1. For wave-guiding light of wavelength, kL , in asymmetric thin films the film thickness, df , has to be larger than a critical thickness, ‘crit , which is given by [1] 2 1=2 kL ns 1 ‘crit ¼ arctan : ð14Þ n2f n2s 2pðn2f n2s Þ Light of any wavelength, k < kL , is guided for uf > uB;t , each wavelength at a particular angle, uf ðkÞ > uB;t , for constructive interference (see for example Eq. (8) in [33]). In the case of our simulations with k ¼ kASE;max ¼ 540 nm, nf ¼ 1:7, and ns ¼ 1:5 it is ‘crit ¼ 127:53 nm. Having a film of thickness df light is wave-guided for k < kC , the cut-off wavelength, which is obtained by re-writing Eq. (14) to kC ¼
Fig. 10. Simulation of the dependence of the wavelength, kpeak , of constructive grazing-angle emission from film to substrate on the film refractive index, nf . The fixed parameters are na ¼ 1, ns ¼ 1:5, us ¼ 89:5°, and df ¼ 400 nm. The interference orders are m ¼ 1, 2, and 3. The solid curves apply to s-polarised light, and the dashed curves apply to p-polarised light.
2pðn2f n2s Þdf 2 1=2 : ns 1 arctan n2f n2s
ð15Þ
For df > ‘crit amplified spontaneous emission (ASE) occurs above a threshold pump pulse energy density, w0L;th , with wavelength of maximum amplitude, kASE;max < kC , determined by the position of maximum effective stimulated emission cross-section [26].
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For grazing-angle leaky-mode emission the wavelength of constructive peak emission, kpeak ðus Þ, is fixed by the Fabry–Perot constructive interference condition, D/B ¼ 2pm (m integer, see Eq. (12a)). Using df ¼ ‘crit ¼ 127:53 nm for nf ¼ 1:7, ns ¼ 1:5, and na ¼ 1, and k ¼ kASE;max ¼ 540 nm, we calculate kpeak;s (us ¼ 89°) ¼ 292.5 nm (m ¼ 1), ¼ 120 nm (m ¼ 2), and kpeak;p (us ¼ 89°) ¼ 354 nm (m ¼ 1), ¼ 129 nm (m ¼ 2) at the critical film thickness. The calculated examples show that at the critical film thickness, ‘crit , the wavelength, kpeak , of constructive leaky-mode emission is below the wavelength of amplified spontaneous emission, kASE;max . If kpeak is outside the range of the fluorescence emission of the film then no constructive leaky-mode emission occurs. In wave-guiding thin films the effective interaction length for amplified spontaneous emission, ‘ASE , is approximately given by the absorption length, ‘abs ¼ 1=af ðkASE;max Þ, where af is the absorption coefficient of the luminescent film material at the wavelength of peak amplified spontaneous emission [26], i.e., 1 ‘ASE ¼ ‘abs ¼ : ð16Þ af ðkASE;max Þ In the case of grazing-angle in-substrate leakymode emission the effective interaction length of amplification of spontaneous emission is the shorter of the absorption length, ‘abs , and the leaky-mode length, ‘lm ¼ 2df =½cosðuf Þð1 RB;l Þ l ¼ s, p; i.e., ‘ASE ¼ minð‘abs ; ‘lm Þ: ð17Þ Except very near to the angle of total internal reflection, uB;t , (us very near to 90°), in good lasing materials it will be ‘abs > ‘lm and the amplification length will be longer for wave-guided emission than for leaky-mode emission. The wave-guided travelling-wave lasing will have the lower threshold than the leaky-mode lasing, and the waveguided travelling-wave laser action will suppress efficient leaky-mode laser action. In wave-guiding thin films with grating structure the constructive interference condition is given by the grating equation (Bragg condition) [27–29]. This interference forces the broad-band travelling-wave laser operation into the narrow-band wave-guided distributed feed-back laser operation [33–36].
287
5. Experimental sub-laser-threshold leaky-mode emission The modification of the spontaneous emission in grazing-angle direction due to multiple-beam interference (micro-cavity effect [5–7]) has been reported shortly in [31] for the laser-active molecule 4-methyl-TPD [31,32] on a glass substrate. Here more detailed results are given for 4-methylTPD and the laser-active polymer TPD(4M)MEH-P-PPV [31,32]. The structural formulae of the molecule and the polymer are shown in Fig. 11. In Fig. 12 results are presented for two 4methyl-TPD thin films of different thickness on glass substrates. In Fig. 12a the film thickness is df 360 nm. The dotted line shows the fluorescence spectrum, SF;n ðkÞ, emitted normal to the film surface where interference effects are negligible (see Fig. 3). A cavity-free spontaneous emission spectrum is observed. The solid curve shows the edge emitted light spectrum, SF;gr ðkÞ, well below the threshold of wave-guided travelling-wave lasing (w0L ¼ 5 lJ cm2 , w0L;th 20 lJ cm2 [31]). The spectrum was measured without a polarizer in the detection path. The collection angle of edge emitted light was approximately 10°. The edge-emitted spectrum looks quite different from the normal surface emitted spectrum (micro-cavity modification of spontaneous emission). The peak at 435 nm is interpreted as TM mode resonance
CH3
CH3
N
N
4-methyl-TPD CH3
CH3 O
N
N
H C
C H
n
H3CO
TPD(4M)-MEH-P-PPV
Fig. 11. Structural formulae of 4-methyl-TPD and TPD(4M)MEH-P-PPV (for details see [31,32]).
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Fig. 12. Normalized absorption spectrum, ra ðkÞ=ra;max , normalized spontaneous emission spectra S F =S F;max and normalized wave-guided amplified spontaneous emission spectra S ASE =S ASE;max of two 4-methyl-TPD films on glass substrates. Exposed film area by picosecond pump laser is 0.19 mm 5 mm (for details see [31,32]). (a) Film thickness df ¼ 360 nm. Curves are explained in the text. (b) Film thickness df ¼ 195 nm. Curves are explained in the text.
corresponding to a film refractive index of nf;p ¼ 1:803. The peak at 412 nm is interpreted as TE mode resonance corresponding to a film refractive index of nf;s ¼ 1:820 (substrate refractive angle ns ¼ 1:53). The dash-dotted curve (TE polarised edge-emission collected) and the dashedtriple-dotted curve (TM polarised edge emission collected) were obtained well above laser threshold (w0L ¼ 160 lJ cm2 ). They show the wave-guided amplified spontaneous emission spectra. The spectral peak occurring at the first vibronic peak of the normal spontaneous emission spectrum is determined by the position of the highest effective stimulated emission cross-section. The position of the highest normal spontaneous emission does not have the highest gain because of residual groundstate absorption. The shape of the absorption cross-section spectrum, ra ðkÞ, is shown by the dashed curve in Fig. 12(a). In Fig. 12(b) the spontaneous emission behaviour is shown for a 4-methyl-TPD film of df 195 nm film thickness. The dotted line shows the
fluorescence emitted normal to the surface, SF;n ðkÞ, and the solid curve shows the edge-emitted fluorescence, SF;gr ðkÞ, at a pump pulse energy density of w0L ¼ 7:8 lJ cm2 . The wavelength of the edge emission peak is determined by the absorption edge of the polymer and not by the occurrence of constructive grazing-angle interference (resonance is expected in the absorption band). In Fig. 13 results are shown for a 430 nm thick TPD(4M)-MEH-P-PPV film on a glass substrate. The dotted curve shows the normal-incidence spontaneous emission spectrum, SF :n ðkÞ. The solid curve displays the s-polarised leaky-mode edge emission, SF ;s;gr ðkÞ, below laser threshold (w0L ¼ 15 lJ cm2 , w0L;th 17 lJ cm2 [31]). The emission peak around 540 nm corresponds to a refractive index of nf;s (540 nm) 1.871 using a substrate refractive index of ns ¼ 1:518. The dashed curve gives the p-polarised leaky-mode edge emission, SF;p;gr ðkÞ ðw0L ¼ 10 lJ cm2 ). The emission peak around 590 nm fits the refractive index of nf;p (590 nm) 1.867 The dash-dotted curve shows the wave-guided amplified spontaneous emission, SASE ðkÞ, well above laser threshold (w0L 200 lJ cm2 ). The ASE peak fits to the emission peak of
Fig. 13. Normalized absorption spectrum, ra ðkÞ=ra;max , normalized spontaneous emission S F =S F;max and normalized waveguided amplified spontaneous emission spectra S ASE =S ASE;max of a TPD(4M)-MEH-P-PPV film on a glass substrate. Film thickness, df ¼ 430 nm. Curves are explained in the text.
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the normal (cavity-free) spontaneous emission. The long-wave-length absorption cross-section spectrum, ra ðkÞ, normalized to ra (360 nm) ¼ 1.63 1016 cm2 [31], is shown by the dashed curve. It indicates that the absorption edge does not influence the wavelength peak positions of the s- and p-polarised grazing-angle spontaneous emission for the applied film thickness.
6. Conclusions The light transmission of luminescent films on transparent substrates of lower refractive index than the films was studied theoretically. The grazing-angle constructive and destructive TE and TM light emission into the substrate was analysed in detail. The micro-cavity behaviour was worked out. The results were set in context with the waveguided amplified spontaneous emission (travellingwave laser action). The leaky-mode emission was discussed for luminescent organic neat films on transparent substrates, and results were presented for an organic molecular thin film (4-methyl-TPD) and an organic polymeric thin film (TPD(4M)-MEH-P-PPV). The presented theory applies to all transparent films on transparent substrates with higher film refractive index than substrate refractive index. The wavelength selective constructive leakymode emission was shown to allow the determination of the refractive indices of luminescent thin films for TE and TM polarised light.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
[15]
[16] [17] [18]
[19] [20] [21] [22] [23] [24] [25]
Acknowledgements
[26]
The authors thank Prof. Ilmo Sildos, University of Tartu, Estonia, for stimulating discussions.
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