On the quantum theory of diffraction by an aperture and the Fraunhofer diffraction at large angles Bernard Fabbro
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Bernard Fabbro. On the quantum theory of diffraction by an aperture and the Fraunhofer diffraction at large angles. 2019. cea-01823747v2
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Bernard Fabbro∗ IRFU,CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France
(Dated: April 3, 2019) A theoretical model of diffraction based on the concept of quantum measurement is presented. It provides a general expression of the state vector of a particle after its passage through an aperture of any shape in a plane screen (diaphragm). This model meets the double requirement of compatibility with the Huygens-Fresnel principle and with the kinematics of the particle-diaphragm interaction. This led to assume that the diaphragm is a device for measuring the three spatial coordinates of the particle and that the transition from the initial state (incident wave) to the final state (diffracted wave) consists of two specific projections which occur successively in a sequence involving a transi- tional state. In the case of the diffraction at infinity (Fraunhofer diffraction), the model predicts the intensity of the diffracted wave over the whole angular range of diffraction (0◦ - 90◦). The predictions of the quantum model and of the theories of Fresnel-Kirchhoff (FK) and Rayleigh-Sommerfeld (RS1 and RS2) are close at small diffraction angles but significantly different at large angles, a region for which specific experimental studies are lacking. A measurement of the particle flow intensity in this region should make it possible to test the FK and RS1-2 theories and the suggested quantum model.
Keywords: Quantum measurement, particle-diaphragm interaction, Huygens-Fresnel princi- ple, Fraunhofer diffraction, large diffraction angles
I. INTRODUCTION passing through the aperture should be calculable in the framework of a ”purely quantum” theory of diffraction Diffraction has been the subject of numerous studies (i.e. a theory in which the calculations are made with that have led to significant progress in the knowledge state vectors in a Hilbert space). There are already of this phenomenon. However, this knowledge can theories of diffraction involving quantum mechanics be further improved because there is still research but most of them do not provide the expression of to be done both on the theoretical and experimental a quantum state. The first theory of this type dates levels. Let us examine the situation by considering the back to the beginnings of the history of quantum example of diffraction by a screen edge or by an aperture mechanics and treats the Fraunhofer diffraction by a in a plane screen (diaphragm) in the case where the grating by combining the concept of light quantum with diffracted wave is observed far enough beyond the screen. Bohr’s principle of correspondence [10]. Then, models involving quantum mechanics to calculate diffraction are Theoretical aspect. The intensity of the diffracted mostly those based on the formalism of path integrals wave is obtained most often by calculations based on [11–14] and those predicting quantum trajectories in the Huygens-Fresnel principle or on the resolution of the framework of hidden variables theories [15–17]. the wave equation with boundary conditions [1–9]. In Other models combine the resolution of the Schr¨odinger these theories, the amplitude of the diffracted wave as equation (or of the wave equation for photons) with the well as its intensity (modulus squared of the amplitude) Huygens-Fresnel principle [18–20]. Only one calculation are functions of the spatial coordinates. Therefore, in based explicitly on the concept of quantum measurement the context of quantum mechanics, the amplitude of - therefore involving state vectors in a Hilbert space - the diffracted wave can be identified with a function seems to have been done until now. It is due to Marcella proportional to the position wave function of the particle [21] and will be described and discussed in more detail after its passage through the aperture of the diaphragm. below. Apart from this attempt - and that of the present However, this position wave function is necessarily that paper - it seems that there is no other diffraction theory of the quantum state of the particle. So, in the above- that gives the expression of the quantum state of the mentioned theories, the calculation of the diffracted particle associated with a wave diffracted by an aperture. wave amplitude is equivalent to a direct calculation of a wave function without prior calculation of the quantum state from which this wave function should be derived. Experimental aspect. The area of observation of the In principle, the quantum state of the particle after wave diffracted by an aperture is the entire half-space beyond the diaphragm. In the case of diffraction at infinity (Fraunhofer diffraction), this corresponds to the whole diffraction angular range (0◦ - 90◦). How- ∗ [email protected] ever, it is well known that the Fresnel-Kirchhoff (FK) 2 and Rayleigh-Sommerfeld (RS1 and RS2) theories - ing to this postulate, the quantum measurement of the based on the Huygens-Fresnel principle and hereafter position results in a filtering of the position wave function called classical theories of diffraction - disagree in the of the initial state. In Ref. [21], this ”position filtering” region of large diffraction angles [1]. This means that is applied to a single coordinate in the plane of the di- at least two of these three theories do not describe aphragm and it is moreover implicitly assumed that the correctly the diffraction at large angles. However, from energy of the particle is conserved. Resuming the analy- a survey of the literature, it seems that no accurate sis of this question, we have been led to assume that the experimental study of the diffraction at large angles position filtering must correspond to a measurement of has been carried out so far. It is not surprising that the three spatial coordinates in order to make possible these measurements were not made when the classical compatibility with the Huygens-Fresnel principle. We theories appeared (late nineteenth century) because then show that the position measurement projects the such experiments were not feasible at the time due initial state on a state whose momentum wave function to the impossibility of having a sufficiently expanded is not compatible with the kinematics of the particle- dynamic range. Since then, technologies in optics have diaphragm interaction. This led us to assume that this made considerable progress (especially thanks to quasi- state is necessarily transitional and undergoes a specific monochromatic laser sources and accurate mesurements projection which corresponds to a measurement involv- of intensity using charge-coupled devices), so that it is ing an “energy-momentum filtering” of the momentum now possible to reconsider this question and achieve wave function. This makes it possible to obtain a final an experimental study (note however that the question state that is compatible with kinematics and that can be of large angles has been the subject of experimental associated with the diffracted wave. Finally, a general studies in the case of diffraction by two conductive expression of the final state is obtained for any shape of strips [22] and of theoretical studies using classical aperture. For the intensity in Fraunhofer diffraction, we diffraction theory to calculate non-paraxial propagation obtain an expression applicable to the whole diffraction in optical systems [23]). Moreover, if the aperture is in angle range (0◦- 90◦). The predictions of the quantum a non-reflective plane screen, we can assume that the model are in good agreement with those of the FK and intensity of the diffracted wave decreases continuously RS1-2 theories in the region of small angles. For the large to zero when reaching the transverse direction. The angles, the four predictions are different. Nevertheless, RS1 theory proves to be the only one that predicts the quantum and RS1 models both predict a decrease this decrease to zero and would therefore be the only in intensity to zero at 90◦ but, despite this, these two valid classical theory over the whole angular range of predictions are significantly different beyond 60◦. The diffraction. Measurements in the region of large angles situation is therefore as follows: there is an unexplored would therefore allow to confirm or refute this conjecture. angular range in which the diffraction has never been measured and several theories have different predictions for this range. Moreover, an accurate measurement of In response to the situation described above, we the particle flow (which is proportional to the intensity present a quantum model of diffraction giving the gen- of the diffracted wave) in the region of large angles is now eral expression of the quantum state of the particle as- possible in the context of current technologies in optics. sociated with the diffracted wave and making it possible Such an experimental test would therefore improve our to calculate the intensity in Fraunhofer diffraction over knowledge of diffraction. the whole angular range. It explicitly uses the postu- late of wave function reduction [24] by considering the diaphragm as a device for measuring the position of the particle passing through the aperture. In this model, The paper is organized as follows. In Sec. II, we de- the intensity of the diffracted wave in Fraunhofer diffrac- scribe the experimental context and the theoretical con- tion is calculated from the momentum wave function of ditions for which the presented model is applicable. In the state of the particle after the measurement (hereafter Sec. III, we express the amplitude of the diffracted wave called final state). The Huygens-Fresnel principle is not in Fraunhofer diffraction predicted by the classical theo- explicitly used as in the FK and RS1-2 theories but we ries in the case of an incident monochromatic plane wave. will see that it is underlying in the model. The approach Then, we introduce the appropriate diffraction angles for is inspired by that of Marcella [21] whose model predicts the calculation of the intensity up to the large angles. In the well-known formula in (sin X/X)2 of the Fraunhofer Sec. IV, we present the quantum model. We calculate diffraction by a slit. However, this result only applies to the final state associated with the diffracted wave and es- the small angles and other authors have furthermore con- tablish the quantum expression of the intensity in Fraun- cluded that the used method implicitly refers to classical hofer diffraction. In Sec. V, we compare the classical and optics [25]. In the present paper, we consider that these quantum predictions. We perform the calculation for a drawbacks do not come from the ”purely quantum” as- rectangular slit and a circular aperture and then we em- pect of the approach adopted by Marcella but from the phasize the interest of an experimental test for the large way in which the postulate of wave function reduction has diffraction angles. Finally, we conclude and suggest some been applied in the framework of this approach. Accord- developments in Sec. VI. 3
II. EXPERIMENTAL AND THEORETICAL In the quantum model presented hereafter, the spin CONTEXT is ignored. The predictions of the quantum model will therefore be compared with those of the scalar theory We consider an experimental setup which contains a of diffraction according to which the Huygens-Fresnel source of particles and a diaphragm of zero thickness, principle can be deduced from the integral Helmholtz- forming a perfectly opaque plane, with an aperture A. Kirchhoff theorem [1]. There are several versions of We suppose that the aperture can be of any shape and this scalar theory which differ by their assumed bound- possibly formed of non-connected parts (e.g. a system of ary conditions. The best known are the theories of slits). It is only assumed that its area is finite. We then Fresnel-Kirchhoff (FK) and Rayleigh-Sommerfeld (RS1 define the center of the aperture in the most convenient and RS2). These theories are prior to quantum mechan- way according to the shape (symmetry center or appro- ics. So, for simplicity, they will be called hereafter clas- priate center for a complicated shape). The device is in sical scalar theories of diffraction. a coordinate system (O; x,y,z) which origin O is located at the center of the aperture and which (Ox,Oy) plane is that of the diaphragm (Fig. 1). III. PREDICTIONS OF THE CLASSICAL SCALAR THEORIES OF DIFFRACTION FOR THE 0◦-90◦ ANGULAR RANGE diaphragm x aperture A detectors A. Amplitude of the diffracted wave source d s z According to the Huygens-Fresnel principle, the am- y O plitude of the diffracted wave is equal to the sum of the amplitudes of spherical waves emitted from the wave- front located at the diaphragm aperture. In the FK and RS1-2 theories, this sum is an integral whose ap- proximate expression can be calculated according to the FIG. 1. Schematic representation of experimental setup type of diffraction. Thus, in Fraunhofer diffraction, for (section in the horizontal plane (Ox,Oz)). a monochromatic plane wave of wavelength λ in normal incidence, the amplitude at a point of radius vector d The source is located on the z axis (normal incidence) beyond the diaphragm is expressed by [1, 2]: and the source-diaphragm distance, denoted s, is as- i exp [ik (s + d)] sumed to be large enough, so that the wave associated k,A (d) C U Cl. ≃ − 0 λ sd with any incident particle can be considered as a plane (1) wave at the level of the aperture. The observation point dx dy Ω(χ) dxdy exp ik x + y , is located at some distance d from the aperture, beyond × A − d d the diaphragm. Its position is denoted by the radius- ZZ vector d. where k =2π/λ is the wave number, C0 is a constant, χ The present study is limited to the case where the in- is the angle: cident wave is monochromatic. It is assumed that the χ χ(d) ( Oz , d ) , (2) wavelength λ is smaller than the size of the aperture and ≡ ≡ that the Fraunhofer diffraction is obtained at distance d. hereafter called deflection angle, and Ω(χ) is the obliquity In this case, the size of the aperture is negligible com- factor which is specific to the theory: pared to d, so that detectors can be arranged on a hemi- (1 + cos χ)/2 (FK) sphere of center O and radius d in order to measure the angular distribution of the particles that passed through Ω(χ) = cos χ (RS1) . (3) the aperture 1. 1 (RS2) The case where the size of the aperture is not negligible Substituting (3) into (1), we see that the values of the compared to the distance d corresponds to the Fresnel amplitudes predicted by the three theories are close when diffraction and is not addressed in this study (the reason the deflection angle χ is small and gradually diverge from for this will be explained in Sec. VI). each other when χ increases. This is a reason for limit- ing the application of these theories to the area of small angles. However, it seems that there is no specific crite- 1 The criterion for the Fraunhofer diffraction is: ∆2/(λd) ≪ 1 rion giving a quantitative determination of a limit angle (ditto for s), where ∆ ≡ max px2 + y2 represents the size of (e.g. a criterion similar to that involving the size of the (x,y)∈A the aperture [1, 3]. The assumption λ ≤ ∆ and this criterion aperture, the diaphragm-detector distance and the wave- imply that ∆ ≪ d (ditto for s). This relation and the criterion length - see footnote 1 - ). The classical theories actually are satisfied if d →∞ and s →∞ (diffraction at infinity). These predict the value of the amplitude for the whole angular latter conditions are sometimes used as criteria [2]. range. 4
Moreover, if the diaphragm is assumed to be non- y reflective, we can surmise that the amplitude must de- θ x = ( p , p ) crease continuously to zero at χ = 90◦. From (3), the z xz θ = ( p , p ) right-hand side of (1) is in general non-zero when χ = 90◦, y z yz except for the RS1 theory. The latter would then be the χ = ( p , p ) z most plausible classical theory for the large angles. p Since the distance d is large compared to the size of χ z the aperture, the directions of the wave vectors k of all the waves arriving at point of radius-vector d are close to x θ θ y that of this radius-vector. It is the same for the momenta x p of the associated particles, since p = ~k. Hence: p d d . (4) ≃ p FIG. 2. Variables (p,θx,θy) in the case of a momentum From (4), the deflection angle given by (2) is such that such that pz > 0. θx and θy are the diffraction angles and cos χ cos(Oz, p) = p /p and we also have: d /d χ is the deflection angle. pz, pxz and pyz are respectively z x the projections of the vector p on the z axis and on the p /p and≃ d /d p /p. Then, substituting into (1), we≃ x y y planes (x,z) and (y,z). can express the≃ amplitude in Fraunhofer diffraction as a function of distance d and momentum direction p/p: 1 ~ − (p,θx,θy) p,A p i p exp [i(p/ ) (s + d)] H px = p cos χ tg θx Class. d, C0 U p ≃ − 2π ~ sd = p (p,θx,θy) = py = p cos χ tg θy , (7) (5) pz p px py pz = p cos χ Ω arccos dxdy exp i~ x + y , × p A − p p where the deflection angle χ can be expressed as a func- ZZ tion of θ and θ : where the modulus p of the momentum is a parameter x y 1/2 (as k in (1)). χ χ(θ ,θ ) = arccos 1+tg 2θ + tg 2θ − (8) ≡ x y x y and is such that 0 χ
In summary, the action of PˆA,∆z corresponds to a mea- and we see that this leads to an undetermined value when ˆ 0 surement of the three coordinates (x,y,z) with an accu- ∆z = 0. We could then replace (13) by PL = 0 0 racy of the order of the size of the aperture A for the but substituting into (14) and then expressing the| waveih | 0 in transverse coordinates and an accuracy of the order of function, we get: z ψz = δ(z)exp i arg z ψz , ∆z for the longitudinal coordinate. In this context, ∆z 0 2 which implies that the p.d.f. z ψz is not defined. is a parameter. Besides the case where ∆z is finite, there However, it is possible to obtain an expression of the are the following limit cases corresponding to two oppo- p.d.f. at the limit ∆z = 0 by using the following method. site situations: First, we replace the projector Pˆ∆z by a ”generalized - if ∆z , the partial projector Pˆ∆z tends to the L → ∞ L projector” Rˆ ∆z called reduction operator and defined as: identity operator. The action of the projector Pˆ A,∆z then L corresponds to a measurement of (x, y) (transverse filter- ˆ ∆z ∆z RL dz FL (z) z z , (16) ing, accuracy given by the size of the aperture A) without ≡ R | ih | measurement of z (no longitudinal filtering, no accuracy). Z q ∆z - if ∆z 0 (∆z = 0), the action of Pˆ A,∆z concentrates where FL (z) is a positive function whose properties will the probability→ of presence6 of the particle in the region of be described below. Secondly, we replace (14) by: the wavefront at the aperture and, consequently, its longi- ˆ ∆z in ∆z RL ψz tudinal coordinate is z = 0 with almost infinite accuracy. ψz = . (17) So, at the time t1 of the change of state, the diffracted ψin Rˆ ∆z † Rˆ ∆z ψin z L L z wave corresponding to the final state of the particle is r about to be emitted from a volume including the wave- D E The formulation (17) ensures that the state ψ∆z is front at the aperture and its close vicinity. We are then z automatically normalized to 1. Substituting (16) into close to a situation consistent with the Huygens-Fresnel (17) and expressing the p.d.f., we find: principle. This case seems more plausible than the case ∆z for which such compatibility does not appear. ∆z in 2 → ∞ ∆z 2 FL (z) z ψz However, no value of ∆z should be rejected a priori. z ψz = . (18) ∆z in 2 At the limit ∆z = 0, a perfect compatibility with the dz′ FL (z ′) z′ ψ z R Huygens-Fresnel principle would be obtained provided Z that the probability density function (p.d.f.) of the lon- Contrary to (15), the equation (18) makes sense at the gitudinal coordinate of the particle at time t1 is equal to 0 limit ∆z = 0, provided that FL (z) = C(z) δ(z) where δ(z), where δ(z) is the Dirac distribution. However, it is C(z) is any function of arbitrary dimension. Indeed, not possible to obtain this result using (13) because the ∆z 2 in this case ( 18) implies: lim z ψz = δ(z), integral of the right-hand side is zero if ∆z = 0. Nev- ∆z 0 → ertheless, given the good agreement between the mea- whatever C(z). The easiest way is to choose C(z) = 1, 0 surements performed so far and the predictions of the dimensionless so that FL (z) = δ(z) and therefore is classical theories based on the Huygens-Fresnel principle, positive and normalized to 1. Then, since the passage this is worth looking for a way to treat this limit case. to the limit ∆z = 0 is continuous, it is logical to assume ∆z Fortunately, it turns out that this is possible provided, that FL (z) is positive and normalized to 1 whatever ∆z however, that the notion of projector is generalized. ∆z. We can therefore interpret FL (z) as the weight Consider a particle having a one-dimensional motion in 2 with which the filtering selects the value z ψz on the z-axis and suppose that the measurement of the of the initial p.d.f. at z. This function F ∆z(z) will be ˆ in L observable Z on this particle in the initial state ψz called longitudinal position filtering function . gives a result within the interval [ ∆z/2, +∆z/2]. Such − a measurement projects the initial state on the following 0 Since FL (z)= δ(z), the square root in the right-hand final state [24]: side of (16) is not defined and consequently (16) and (17) ˆ∆z in are not applicable when ∆z = 0. The passage to the ∆z PL ψz 2 ψz = , (14) limit can only be done for the p.d.f., by using (18) . 2 ψin Pˆ ∆z ψin ˆ ∆z ˆ ∆z z L z From (16), we have: RL = RL , contrary to the r 6 D E case of a projector. However, for the filtering function: ˆ ∆z ∆z where PL is the projector defined in (13). From (13) FL (z)=1/∆z if z [ ∆z/2, +∆z/2] or 0 otherwise and (14), the p.d.f. corresponding to the probability to ∈ − obtain a result within the interval [z,z + dz] is:
1 +∆z/2 − 2 ∆z 2 in 2 More generally, the case of a filtering function equal to the Dirac z ψz = dz′ z′ ψz distribution can be treated by performing the calculations with a ∆z/2 ! Z− (15) Gaussian of standard deviation σz and then passing to the limit 2 σz = 0 if it is possible. This is the case not only for the p.d.f. of z ψin if z [ ∆ z/2, +∆ z/2] z ∈ − Eq. (18) but also for the relative intensity whose expression will × be calculated later (subsection V B). ( 0 if z / [ ∆z/2, +∆z/2] ∈ − 7
(uniform filtering), the equations ( 13) and ( 16) lead where: ˆ ∆z 1/2 ˆ∆z to: RL = ∆z− PL . So, substituting into (17), we 1/2 A [ ∆z/2, +∆z/2] (22) see that the factor ∆z− is eliminated. Then, since A ≡ × − ˆ ∆z † ˆ ∆z ˆ ∆z PL PL = PL , the right-hand side of (17) is equal is the volume of transverse section A, of length ∆z, cen- tered at the origin O. The aperture A and the interval to the right-hand side of (14) and we finally obtain the ∆z [ ∆z/2, +∆z/2] are respectively called transverse aper- same state ψz as that of Eq. (14). Therefore, the − reduction operator can be used in place of the usual pro- ture and longitudinal aperture and the volume is called generalized 3D aperture (Fig. 3) (note that thereA exist jector in the particular case of a uniform filtering. We assume that this type of operator is appropriate for the calculations of diffraction based on the Huygens-Fresnel more general case of a non-uniform filtering. principle and involving another type of generalized aper- In particular, the longitudinal filtering could be non- ture [28, 29]). uniform for the following reason. The truncation made by ˆ∆z transverse position filtering function x the projector PL is similar to that made by the projector ˆ A PT involved in the transverse filtering. However, the two transverse filterings are probably not of the same type because the 2D aperture aperture is limited by a material edge in the transverse plane whereas there are no edges along the longitudinal direction. The longitudinal filtering could then be a non- z uniform filtering associated with a continuous filtering O y function forming a peak centered at z = 0 and of width generalized 3D ∆z. This peak is not necessarily symmetric with respect aperture to z = 0. The precise shape of the filtering function is part of the assumptions of the model. This shape can be important if ∆z is large but probably does not matter ∆z (longitudinal if ∆z is close to zero because the p.d.f. is then close 1D aperture) to the Dirac distribution (for simplicity, we will consider symmetric filtering functions centered at z = 0 in the following). longitudinal position filtering function The interval [ ∆z/2, +∆z/2] is a transmission interval that can be considered− as an ”aperture”. In the case of a FIG. 3. Example of generalized 3D aperture (section in the uniform filtering, it represents the region where the filter- (Ox,Oz) plane) and corresponding transverse and longitudi- ing function is non-zero. In the case of a non-uniform fil- nal position filtering functions, whose product is the (joint) tering, the filtering function can be non-zero everywhere position filtering function. (for example if it is a Gaussian). We are then led to de- fine more generally the aperture as the interval outside of From (16), (19) and (21), we can express the reduction which the integral of the filtering function is negligible. operator in the form: In summary, for the longitudinal coordinate, it is ˆ ∆z ˆ 3 assumed that the projector PL defined by ( 13) can RA d r F A(r) r r , (23) ≡ R3 | ih | ˆ ∆z be replaced by a reduction operator RL of the form (16). ZZZ q where F A(r) is the joint position filtering function (here- Using a similar generalization for the transverse coor- after called position filtering function) given by: dinates, we assume that the projector PˆA given by (11) T A ∆z F A(r) F (x, y) F (z). (24) can be replaced by a reduction operator of the form: ≡ T L A ˆ A A The function FT (x, y) is given by (20) and the function RT dxdy FT (x, y) xy xy , (19) ∆z ≡ R2 | ih | FL (z) forms a peak whose integral outside the interval ZZ q [ ∆z/2, +∆z/2] is negligible. From this and ( 24), it where F A(x, y) is the transverse position filtering func- T follows− that the integral of F (r) is negligible outside tion. Since the transverse filtering is assumed to be uni- A the volume defined by (22). form, this function is given by: A 1 A S(A)− if (x, y) A According to the Huygens-Fresnel principle, the F (x, y) = ∈ , (20) T 0 if (x, y) / A diffracted wave is emitted from the wavefront at the ∈ where S(A) is the area of A. aperture, which corresponds to the case ∆z = 0. Then, the case ∆z > 0 can be interpreted as follows: the Finally, we assume that the projector PˆA,∆z defined Huygens-Fresnel principle applies to several wavefronts contributing with different weights whose distribution is by (12) can be replaced by the reduction operator: ∆z the longitudinal position filtering function FL (z). A ∆z Rˆ A Rˆ Rˆ , (21) ≡ T ⊗ L 8
Following the analysis developed so far, we can formu- provided with the ”generalized 3D aperture” is: late the first assumption of the quantum model as follows: A ˆ in RA ψp0 (t1) ψA(t1) = , (26) Assumption 1. A diaphragm is a device for mea- ψin (t ) Rˆ † Rˆ ψin (t ) suring the three spatial coordinates of any particle p0 1 A A p0 1 passing through its aperture. rD E ˆ - Before the measurement, the initial state of the parti- where RA is the position reduction operator defined by cle corresponds to the incident wave propagating towards (23). Equation (26) expresses the final state only at the diaphragm. time t1. It is therefore necessary to express this state - After the measurement, the final state corresponds to for t >t1. Two observations allow to formulate two the diffracted wave propagating beyond the diaphragm. properties that the final state after time t1 must have. - During the measurement, the initial state undergoes First, no significant difference between the wavelength the action of the position reduction operator Rˆ A defined by (23). of the diffracted wave and that of the incident wave is observed in diffraction experiments with a diaphragm. The assumption 1 does not specify the characteristics Hence, denoting λ and p the wavelength and the momen- ˆ tum in the final state, we have: λ λ0 and consequently, of the state resulting from the action of the operator RA ≃ on the initial state. In fact, this state is not the final given the de Broglie relation: state associated with the diffracted wave. This question p p0. (27) will soon be examined (subsection IV D). This requires ≃ a second assumption that applies only in the case where This is consistent with kinematics. In the conditions the incident wave is monochromatic. of the experiment (microscopic object interacting with a macroscopic device), the particule transfers a very small part of its energy to the diaphragm, so that the C. Initial state associated with a monochromatic momentum modulus of the particle is almost conserved. incident wave. Therefore, the final state after time t1 must have the following property: To compare the predictions of the quantum model with (P1) The final state for t>t is close to an energy those of the classical scalar theories of diffraction pre- 1 eigenstate associated with the eigenvalue E(p ). sented in Sec. III, the appropriate initial state must cor- 0 respond to a scalar monochromatic plane wave in normal Secondly, we can assume that there is no diffracted incidence. Such a state is a momentum and energy eigen- wave returning from the aperture of the diaphragm to state (therefore a stationary state) of the form: the region where the source is located. This implies that in ~ the particle associated with the diffracted wave is always ψp0 (t) = exp [ (i/ ) E(p0) t] p0 , (25) − | i such that: where p0 = (p0x p0y p0z)=(00 p0) is the momentum pz > 0. (28) 2 2 2 of the incident particle and E(p0)= c m c + p0 is its energy. The position wave function of this initial state is Hence the property: 3/2 p the plane wave: (2π~)− exp (i/~)[ p .r E(p ) t ] { 0 − 0 } (where p0.r = p0z in the coordinate system of Fig. 1). (P2) The final state for t > t1 is such that its We assume that the state ψin (t) given by (25) can momentum wave function is zero for pz 0. p0 ≤ be used to describe the state of any free particle of mo- mentum p0 if the spin is not taken into account. In par- Let us verify whether the state ψA(t1) given by (26) ticular, we admit that this use is valid for the photon in can evolve towards a state having the properties (P1) and spite of the issues raised by the interpretation of the po- (P2). Substituting (24) into (23), then the obtained result sition wave function of a particle of zero mass [30–32]. In and (25) into (26) and given (20), we get the expression the following, the calculation of the relative intensity in of ψA(t1) . Then, we calculate the modulus squared Fraunhofer diffraction will require only momentum wave of the momentum wave function, that is the momentum functions, so this problem of interpretation will not arise. p.d.f. in the final state just after time t1. The result is independent of t1 and can be written in the form:
2 A p0,∆z p ψA(t1) = f (px,py) f (pz), (29) D. Transitional state. Energy-momentum filtering Px,Py Pz where:
A 2 1 According to the postulate of wave function reduction f (px,py) (2π~)− S(A)− in Px,Py applied to the initial state ψ (t1) and given the as- ≡ p0 2 (30) sumption 1, the state of the particle immediately after ~ dxdy exp [ (i/ )(pxx + pyy)] , the position quantum measurement by the diaphragm × − ZZA
9
p0,∆z 1 f (pz) (2π~)− tion” normalized to 1 and defined as: Pz ≡ 2 (31) ∆p δ1 sgn[pz] δ ( p p0) dz F ∆z(z) exp [(i/~)(p p )z] . F p0 (p) | |− , (34) L 0 z ≡ × R − 3 ′ ∆p Z d p′ δ1 sgn[pz ] δ ( p′ p0) q R3 e | |− A p0,∆z ZZZ We can verify that f (px,py) and f (pz ) are pos- Px,Py Pz where δ is the Kroenecker deltae and δ ∆p ( p p ) itive and normalized to 1. So, they are respectively the 1 sgn[pz] 0 is a function determined by kinematics and therefore| |− marginal p.d.f.s of the random variables (P , P ) and P x y z forming a peak centered at p = p , of width ∆p close to associated with the possible results of measurements of 0 e zero but non-zero. The width| | ∆p is non-zero because the the observables (Pˆ , Pˆ ) and Pˆ . Therefore, Eq. (29) ex- x y z particle energy is not strictly conserved. Consequently presses the fact that the random variables Px and Py are independent of the random variable P . However, this is δ ∆p ( p p ) is well-defined and therefore (33) makes z | |− 0 incompatible with the relation (27) which involves on the sense.q The interpretation of (34) is therefore as follows: contrary a strong dependence between the three compo- thee filtering function F p0 (p) selects all the possible values nents of the momentum since p0 is a fixed value. Thus, of the momentum components such that the momentum the spreading of the momentum components caused by ∆p modulus is close to p0 (term δ ( p p0), Eq. (27), the measurement of the spatial coordinates generally re- | |− property (P1)) and such that pz > 0 (term δ1 sgn[pz], Eq. sults in a spreading of the momentum modulus. So, the (28), property (P2)). e state ψ (t ) is not an energy eigenstate and, conse- p0, A 1 Then, to obtain the expression of ψ A(t1) as a quently, has not the property (P1). function of the initial state ψin (t ) , we substitute 0 p0 1 Moreover, from ( 31), the distribution f p ,∆z(p ) is Pz z (26) into (32). The result is the expression (35) below, none other than the modulus squared of the Fourier which is the second assumption of the quantum model. ∆z transform of FL (z). Therefore, if ∆z is small enough, p0,∆z Assumption 2. In the case of a monochromatic plane the width ∆pqz of f (pz) can be larger than p0 and the Pz wave in normal incidence (wavelength: λ = 2π~/p ), probability for p to be negative is then non-negligible. In 0 0 z the final state of the particle outgoing from the general- this case, (28) is not satisfied and consequently the state ized 3D aperture immediately after interacting at time ψ (t ) has not the property (P2). However, this state A 1 t with the diaphragmA is: can have the property (P2) if ∆z is large enough but, 1 p0, anyway, it has not the property (P1) because the proba- ψ A(t1) ˆ p0 ˆ in bilistic independence, mentioned above, between (Px, Py) R RA ψp0 (t1) = , (35) and Pz is effective whatever ∆z. in ˆ p0 ˆ † ˆ p0 ˆ in We are thus led to the following conclusion: the state ψp0 (t1) R R A R RA ψp0 (t1) ψA(t1) cannot evolve towards the final state associated rD E ˆ with the diffracted wave. It is a transitional state that where RA is the position reduction operator defined by ˆ p0 must be transformed, just after time t1, by means of a (23) and R is the energy-momentum reduction operator new projection to a state having the properties (P1) and defined by (33). (P2).
From the above conclusion, the appropriate final state E. Final state associated with the diffracted wave would be of the form: We now have all the ingredients to express the final Rˆ p0 ψ (t ) p0, A 1 state and to check if this state actually has the properties ψ A(t1) = , (32) (P1) and (P2) mentioned in subsection IV D. Substitut- ˆ p0 ˆ p0 ψA(t1) R † R ψA(t1) ing (23) and (33) into (35), and given (25) and (34), we
rD E see that the denominator of the right-hand side of (34) where Rˆ p0 is an energy-momentum reduction operator is eliminated and we are led to the following expression projecting on a state such that the momentum modulus of the final state at time t1: p.d.f. is close to δ(p p0) (hence the need for this operator p0, 1/2 ~ − ψ A(t1) = N A(∆p)− exp [ (i/ )E(p0) t1] to depend on p0, this point will be discussed hereafter in − (36) subsection IV F) and such that the momentum p.d.f. is 3 ∆p d pδ1sgn[pz] δ ( p p0) A(p p0) p , zero for pz 0. The expression of this reduction operator × R3 | |− F − | i must have≤ a form similar to (23). We assume that the ZZZ q appropriate operator is: where A(p p0) is thee Fourier transform of the square root ofF the position− filtering function:
p0 3 3/2 Rˆ d p F p0 (p) p p , (33) A(p p0) (2π~)− ≡ R3 | ih | F − ≡ ZZZ (37) p 3 ~ . p0 d r F A(r) exp [ (i/ )(p p0) r] where F (p) is an ”energy-momentum filtering func- × R3 − − ZZZ q 10
∆p and N A(∆p) is the normalization term: Since ∆p is close to zero and the function δ (p p0) is not under a square root in (43) and (44), we can− replace 3 ∆p it by the Dirac distribution δ(p p0) and thuse obtain an N A(∆p) d p δ1 sgn[pz] δ ( p p0) ≡ R3 | |− − ZZZ (38) approximate expression of the p.d.f.. Given the property: 2 δ(x x0)f(x)= δ(x x0)f(x0), this leads to: e A(p p0) . − − × F − p0, 1 2 f A (p,θx,θy) N A(0)− p0 δ(p p0) From (36), the final momentum p.d.f. is independent of P,Θx,Θy ≃ − (45) t1. So this p.d.f. can be written in the form: 2 Γ(θ ,θ ) A ( p(p ,θ ,θ ) p(p , 0, 0) ) . × x y F 0 x y − 0 p0, 2 p0, t1 : p ψ A(t1) fP A(p), (39) p0, ∀ ≡ Then the marginal p.d.f. fP A(p) is obtained by inte- grating (45) over θx and θy. Expressing N A(0) from (44) where P is the random vector (P x Py Pz). From (36) and (39), we have: applied to ∆p = 0, we see that this integration of (45) leads to an expression where N A(0) is eliminated. What p0, 1 ∆p remains is fP A(p) = N A(∆p)− δ1 sgn[pz] δ ( p p0) | |− (40) 2 p0, A(p p ) , f A(p) δ(p p0). (46) × eF − 0 P ≃ − Therefore we have p p in the final state which is ex- which can be considered as a p.d.f. defined in the half- 0 actly the kinematic constraint≃ expressed by (27). The space pz > 0, due to the factor δ1 sgn[pz ]. Now, such a p.d.f. can be transformed into a p.d.f. of the variables equation (46) confirms that the final state at time t1, p,θx,θy by using the change of variables formula asso- given by (36), is close to an energy eigenstate associated ciated with the one-to-one transformation defined by with the eigenvalue E(p0). Therefore, this final state is (6)-(7): H close to a stationary state, so that its temporal evolution can be expressed by: p0, f A (p,θx,θy) P,Θx,Θy p0, (41) t t : ψ A(t) 2 p0, 1 = p Γ(θx,θy) fP A( p(p,θx,θy) ), ∀ ≥ (47) p0, exp [ (i/~) E (p0) (t t1)] ψ A(t1) . ≃ − − where p(p,θx,θy) is given by (7) and
So, from (39), (40) and (47), we see that: 1 ∂ [px,py,pz] Γ (θ ,θ ) det p0, x y ≡ p2 ∂ [p,θ ,θ ] t t1, pz 0 : p ψ A(t) = 0. (48) x y ∀ ≥ ∀ ≤ (42) cos χ Finally, from (47) and (48), the final state ψp0, (t) A = 2 2 has the properties (P1) and (P2) mentioned in subsec- 1 sin θx sin θy − tion IV D and is therefore the appropriate final state is the angular factor (absolute value of the jacobian of that we can associate with the diffracted wave beyond 1 2 − divided by p ) which has been calculated from the diaphragm. (H 7) and ( 8). From ( 40) and using the fact that sgn[pz(p,θx,θy)] = 1 in the half-space z > 0 and that p0 = (0 0 p0) = p(p0, 0, 0), we can express F. Discussion p0, fP A (p(p,θx,θy)) and then apply the change of vari- p0, A The assumption 2 suggests to apply the postulate of ables formula (41) to get the p.d.f. fP,Θx,Θy (p,θx,θy). The result can be expressed in the form: wave function reduction by using ( 35) which involves p0 the product of operators Rˆ Rˆ A. From (23) and (33), p0, 1 2 ∆p A fP,Θx,Θy (p,θx,θy) = N A(∆p)− p δ (p p0) this product has two important features: (1) it is not − (43) 2 commutative and (2) it depends on the momentum Γ(θ ,θ ) A ( p(p,θ ,θ ) p(p , 0, 0) ) . × x y F x y − e 0 modulus p0 of the incident particle.
Then, to express N A(∆p), we apply the change of vari- p0 (1) Although Rˆ A and Rˆ do not commute, they do ables (px,py,pz) (p,θx,θy) in the integral of the right- not correspond to two incompatible observables. The hand side of (38).→ We obtain: operator Rˆ A corresponds effectively to the three position observables X,ˆ Y,ˆ Zˆ but the operator Rˆ p0 does not cor- ∞ 2 ∆p N A(∆p) = dp p δ (p p0) respond to the three observables associated with the mo- − Z0 mentum components. It corresponds to the modulus of +π/2 +π/2 e (44) the momentum. Due to kinematics, the measurement of dθx dθy Γ(θx,θy) the position of the particle causes a perturbation only on × π/2 π/2 Z− Z− the direction of its momentum and not on the modulus. 2 A ( p(p,θ ,θ ) p(p , 0, 0) ) . We can then admit that, although non-commutative, the × F x y − 0
11
p0 product Rˆ Rˆ A corresponds to two simultaneous mea- represents the ”measurement device only”. On the other in surements according to the following interpretation. The hand, in (35), the initial state ψp0 (t1) still represents ˆ p0 first measurement, associated with RA, is the measure- the incident particle but the operator Rˆ Rˆ A depends ment of the position of the particle in the initial state on p0 and consequently includes a piece of information in p0 ψp0 (t1) (see (26)). The second measurement, associ- on the incident particle. Therefore, the operator Rˆ Rˆ A ˆ p0 ated with R , can be interpreted as a measurement of is no longer a representation of the ”measurement device the energy and the momentum longitudinal component of only” but rather a representation of the system ”particle the particle in the transitional state ψA(t1) created by + measurement device in interaction”, so that there is the position measurement (see (32)). This measurement no longer clear separation between the particle and the gives the result E(p0) with near certainty for the energy measurement device. The initial state is transformed (property (P1), Eq. (46)-(47)) and the result pz > 0 for by an operator which depends on this initial state itself the momentum (property (P2), Eq. (48)). Although the (via Rˆ p0 ). From this point of vue, (35) seems more process consists in two successive changes of state, we appropriate than (26) because it better reflects the fact can consider that the two measurements are simultane- that the measurement device and the particle form an ous because the particle occupies the transitional state inseparable system at the time of their interaction. ψA(t1) only at time t1. In summary, the order of the operators in the prod- in (2) In ( 26), the initial state ψ (t ) and the ˆ p0 ˆ ˆ p0 p0 1 uct R RA and the fact that R depends on p0 are two ˆ reduction operator RA can be respectively considered as essential features of assumption 2. This assumption sug- representations of the incident particle and of the mea- gests to consider that the change of state of a particle surement device. Thus a sort of separation is established - initially in a momentum eigenstate - at the time of a between the particle and the measurement device in the quantum measurement of position by a diaphragm is not sense that, whatever the value of p0, the initial state is a single projection but proceeds according to the follow- always projected by the same operator Rˆ A which then ing sequence:
measurement of energy transitional initial state measurement and longitudinal final state state ≃ = of position component of momentum energy and - wave function - energy eigenstate, momentum quasi-monochromatic localized at results: E ≃ energy eigenstate, result: (x,y,z) ∈ the generalized diffracted wave monochromatic generalized 3D of the initial state propagating beyond 3D aperture, (due to kinematics) plane wave (λ0) aperture spreading in the diaphragm and and pz > 0 (particle propagating in energy and such that λ ≃ λ0. z axis direction exiting towards the momentum half-space z > 0)
The order of the measurements cannot be inverted be- G. Quantum model predictions for the Fraunhofer cause the associated operators do not commute. In spite diffraction of this non-commutativity, both measurements can be considered simultaneous if we admit that the particle oc- We now come to the expression of the relative inten- cupies the transitional state for a period of time which p0, sity. The marginal p.d.f. f A (θx,θy) is the p.d.f. that tends to zero. Θx,Θy the particle comes out of the aperture in the direction (θx,θy). In Fraunhofer diffraction and if the wavelength is smaller than the size of the aperture, the latter, seen from the detectors, can be considered pointlike (see Sec. Remark. If the order of the measurements is inverted, p0, II). In this case, f A (θx,θy) is none other than the the obtained state is the same as in the case of a posi- Θx,Θy p0 p0 tion filtering only. Indeed, replacing Rˆ Rˆ A by Rˆ ARˆ in directional intensity normalized to 1: in ˆ p0 in (35) is equivalent to replace ψp0 (t1) by R ψp0 (t1) p0, in (26). From (25) and (33), we have: Rˆ p0 ψin (t ) = I A(θx,θy) p0 1 QM F p0 (p ) ψin (t ) , which is expected since the initial +π/2 +π/2 0 p0 1 p0, state is an energy eigenstate associated with the eigen- dθx′ dθy′ IQMA(θx′ ,θy′ ) p π/2 π/2 (49) value E(p0 ). Hence, substituting into the right-hand side Z− Z− in p0 of (26) in place of ψp0 (t1) , we see that F (p0) is p0, ∞ p0, = f A (θx,θy) dp f A (p,θx,θy). eliminated and we recover ψ (t ) . Θx,Θy P,Θx,Θy A 1 p ≡ 0 Z
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From (49) (first equality), the relative intensity is: V. INTEREST AND POSSIBILITY OF AN EXPERIMENTAL TEST p0, p0, I(θ ,θ ) A fΘ ,AΘ (θx,θy) x y = x y . (50) p0, A. Comparison between quantum and classical I(0, 0) QM f A (0, 0) Θx,Θy formulae p0, A From (49) (second equality), we calculate fΘx,Θy (θx,θy) p0, Let us compare the quantum formula (55) to the clas- and f A (0, 0) by integrating (45) over p. After this Θx,Θy sical one (see (10)). From (20), (53) and (56), we can integration, given (27), it is convenient to replace p by 0 rewrite (10) in the form: p, this latter notation henceforth indicating the modulus of both the initial momentum and the final momentum, I(θ ,θ ) p,A x y Ω(χ)2 T A(p,θ ,θ ). (58) as in Sec. III where the conservation of the momentum I(0, 0) ≃ x y modulus is implicitly assumed. Then, substituting the Class. p, p, A A The comparison between the quantum formula (55) and expressions of fΘx,Θy (θx,θy) and fΘx,Θy (0, 0) into (50) modified with the substitution p p, we find: the classical formula (58) shows the following features: 0 → p, A I(θx,θy) A (i) The transverse diffraction term T (p,θx,θy) is the Γ(θx,θy) I(0, 0) QM ≃ same in the two formulae. 2 (51) A ( p(p,θx,θy) p(p, 0, 0) ) F − . (ii) The angular factors are different and have not × 2 A(0) the same origin. The factor Γ(θx,θy) of the quantum |F | 2 2 From (24) and since p0 p (see (27)), the Fourier trans- formula is equal to (cosχ)/(1 sin θx sin θy) (where χ form (37) can be expressed≃ in the form: is given by (8)) and is related− to the Jacobian of the A ∆z change of variables of the momentum p.d.f. (see (41) and A(p p ) (p ,p ) (p p), (52) F − 0 ≃ FT x y FL z − (42)) whereas the factor Ω(χ)2 of the classical formula where: is equal to the square of the obliquity factor involved in A 1 (p ,p ) (2π~)− the classical expression of the amplitude (see (1) and (3)). FT x y ≡ (53) A ~ ∆z dxdy FT (x, y) exp[ (i/ )(pxx + pyy)] , (iii) The longitudinal diffraction term L (p,χ) is × R2 − ZZ q present only in the quantum formula and is not predicted ∆z 1/2 by the classical theories. (p p) (2π~)− FL z − ≡ (54) ∆z ~ When the angles θx and θy are small, the angular fac- dz FL (z) exp [(i/ )(p pz)z] . × R − tors and the longitudinal diffraction term are all close Z q Quantum formula of the relative intensity in Fraun- to 1 so that the two formulae give similar results close to A hofer diffraction. Substituting (52) into (51) and ex- T (p,θx,θy). However, the results differ when the angles pressing p(p,θx,θy) and p(p, 0, 0) from (7), we obtain increase. The difference can be significant at large angles the quantum formula of the relative intensity in Fraun- and therefore observable experimentally as we shall see hofer diffraction for a monochromatic plane wave (λ = below. 2π~/p) in normal incidence on a generalized 3D aperture We consider the intensity variation in the plane A [ ∆z/2, +∆z/2]: (Ox,Oz) corresponding to θy = 0 and we set θ θx A≡ × − ≡ p, for brevity. Then (8) and (42) imply respectively: χ = θ I(θ ,θ ) A x y Γ(θ ,θ ) T A(p,θ ,θ ) L∆z(p,χ), (55) and Γ(θ, 0) = cos θ. Hence, (55) and (58) lead to (after I(0, 0) ≃ x y x y expressing Ω(θ) from (3)): QM where χ and Γ(θ ,θ ) are respectively given by (8) and p, x y I(θ, 0) A A ∆z (42), and ( cos θ T (p, θ, 0) L (p,θ), (59) I 0, 0) QM ≃ A 2 A T (p cos χ tg θx , p cos χ tg θy) T (p,θ ,θ ) F p,A x y ≡ A 2 (56) I(θ, 0) T (0, 0) F I(0, 0) Class. is the transverse diffraction term and (1 + cos θ)2/4 (FK) (60) 2 ∆z A ∆z L ( p (cos χ 1) ) cos2 θ T (p, θ, 0) , (RS1) , L (p,χ) F −2 (57) ≃ ≡ ∆z(0) 1 (RS2) FL is the longitudinal diffraction term . The rigth-hand sides where, from (20), (53) and (56): of (56) and (57) can be calculated using (53) and (54), 2 where F A(x, y) is given by (20) and the function F ∆z(z) 1 T L T A(p, θ, 0) = dxdy exp [ i(k sin θ)x] . (61) is part of the assumptions of the model. S(A)2 − ZZA
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B. Predictions for a slit and a circular aperture The longitudinal diffraction term is expressed by substi- 1/4 1/2 tuting (67) into (57). The factor (2/π) (σz/~) is Let us apply the formulae (59), (60) and (61) to the then eliminated and we get: cases of a rectangular slit and a circular aperture and to Lσz (p,θ) = exp 2σ 2k2 (1 cos θ)2 , (68) the case of a Gaussian longitudinal filtering. − z − which makes sense if σz = 0 (see footnote 2). a. Slit. Consider a rectangular slit R centered at (x, y) = (0, 0), such that its medians correspond to the Two examples of curves obtained from the formulae axis Ox (width = 2a) and Oy (height = 2b). In this case, ( 59) and ( 60) for two different realistic experimental we have: S(R) = 4ab and the integral of the right-hand cases are shown in Figs. 4 and 5 respectively corre- side of (61) is given by: sponding to the generalized 3D apertures R, σ R≡{ z} and C, σz : dxdy exp [ i(k sin θ)x ] C≡{ } R − ZZ +a +b (62) - Fig. 4: rectangular slit, (59) and (60) are applied for R = dx exp [ i(k sin θ)x ] dy. = = R, σz and T (p, θ, 0) is given by (63). a − b A R { } Z− Z− Integrating and then substituting into (61), we get: - Fig. 5: circular aperture, (59) and (60) are applied C for = = C, σz and T (p, θ, 0) is given by (65). sin(ak sin θ) 2 A C { } T R(p, θ, 0) = . (63) ak sin θ - In the two cases, Lσz (p,θ) is given by (68). b. Circular aperture. Consider a circular aperture C The main characteristic of these results is that the of radius r and centered at (x, y) = (0, 0). We have: curves are very close for small angles and diverge more S(C) = πr2 and the integral of the right-hand side of and more when θ increases until significant gaps are (61) is given by: reached, whatever the experimental parameters (shape and size of the aperture, wavelength). dxdy exp [ i(k sin θ)x ] C − ZZ +r +√r2 x2 − If σz = 0, the diffracted wave associated with the = dx exp [ i(k sin θ)x] dy (64) particle is emitted from the wavefront located at the r − √r2 x2 Z− r Z− − aperture, in accordance with the Huygens-Fresnel =4 dx r2 x2 cos [(k sin θ)x] . principle. The longitudinal diffraction term given by 0 − (68) is equal to 1. Then, the quantum values are as Z p This integral is known [33] (3.771.8, p.464). Hence, sub- large as possible and the difference between classical stituting into (61) after the integration, we obtain: predictions and quantum prediction is due exclusively to the angular factors 3. The QM values are smaller than 2J (rk sin θ) 2 T C(p, θ, 0) = 1 , (65) the values of the FK and RS2 theories and larger than rk sin θ the values of the RS1 theory. From (59) and (60), the ratios of the relative intensities QM/RS1 and RS2/QM where J1(x) is the Bessel function of order 1. are equal to 1/ cos θ. They already reach the value 2 c. Gaussian longitudinal filtering. The width ∆z of for θ = 60◦. Therefore, it is not necessary for the angle the longitudinal aperture can be deduced from the stan- to be very large to obtain a significant gap between the predictions of the quantum model and those of the dard deviation σz of the Gaussian filtering function and RS1-2 theories. On the other hand, the ratio FK/QM from the threshold α under which it is considered that 2 a quantity is negligible. For example, if the integral is equal to (1+ cos θ) /(4 cos θ) which is worth 9/8 for 2 θ = 60◦ and 2 for θ 80◦. The difference is significant of the Gaussian outside the interval is α = 10− , we ≃ have [34]: ∆z(σ , α) 5.16 σ . It is convenient to z ≃ z choose σz as parameter (instead of ∆z) and use the following notations: the subscript ∆z(σz, α) will be re- 3 Note that there is a formula, from classical optics, with the same placed by σz and the generalized 3D aperture = A angular factor as that of the quantum formula (59) (cos θ not [ ∆z(σ , α)/2, +∆z(σ , α)/2] will be denotedA A, σ ×. − z z { z} squared). This classical formula is based on the result of the So, let us assume that the longitudinal filtering function first rigorous calculation of the diffraction by a wedge, due to is Gaussian: Sommerfeld [2, 4] and completed by Pauli [5]. This result has been applied to the case of two wedges of zero angle placed op- σz √ 1 2 2 § FL (z) = (σz 2π)− exp z / 2σz . (66) posite one another to form a slit (see [8], 75, problem 1). The − obtained formula includes two terms: the right-hand side of (59) Substituting (66) into (54), we get [33](3.896.4, p. 514): (with T A(p,θ, 0) = T R(p,θ, 0) given by (63) and L∆z(p,θ) = 1) − plus a small additional term equal to [2ak cos(θ/2)] 2. However, 1/4 1/2 σz 2 σz 2 2 this small additional term is non-zero in the whole angular range (p p)= exp σ (k k ) . (67) ◦ ◦ FL z − π ~ − z − z 0 − 90 , so the intensity is not zero at the minima (see also [9]). 14
I(θ,0) I(θ,0) 1 1 I(0,0) Rectangular aperture Classical (FK) I(0,0) Circular aperture Classical (FK) 0.8 2a = 1e-05 m Classical (RS1) 0.8 2r = 4e-06 m Classical (RS1) 0.6 λ = 6.328e-07 m Classical (RS2) 0.6 λ = 5.3245e-07 m Classical (RS2) σ σ QM,GLF, z=0 QM,GLF, z=0 0.4 σ 0.4 σ QM,GLF, z=a/10 QM,GLF, z=r/19 0.2 0.2 0 0 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 ×10-3 ×10-3 3.5 4 3 3.5 2.5 3 2 2.5 1.5 2 1.5 1 1 0.5 0.5 0 0 15 20 25 30 35 40 45 50 55 15 20 25 30 35 40 45 50 55 ×10-3 ×10-3 0.5 0.25 0.4 0.2 0.3 0.15 0.2 0.1 0.1 0.05 0 0 55 60 65 70 75 80 85 90 55 60 65 70 75 80 85 90 θ (deg.) θ (deg.)
FIG. 4. Comparison of the relative intensities in Fraun- FIG. 5. Comparison of the relative intensities in Fraun- hofer diffraction as a function of the diffraction angle in the hofer diffraction as a function of the diffraction angle in horizontal plane for a rectangular slit of width 2a = 10 µm the horizontal plane for a circular aperture of diameter and a monochromatic incident wave corresponding to pho- 2r = 4 µm and a monochromatic incident wave correspond- tons of wavelength λ = 632.8 nm (Helium-Neon Laser). ing to photons of wavelength λ = 532.45 nm (KTP Laser). Five theoretical predictions are presented: three classi- The curves are different from those of Fig. 4 because the cal predictions corresponding to the theories of Fresnel- experimental parameters are different (shape and size of Kirchhoff (FK) and Rayleigh-Sommerfeld (RS1 and RS2) the aperture, wavelength). However, the gaps between and two quantum predictions corresponding to two values them are the same (except for σz = r/19) because they of the standard deviation σz associated with a Gaussian depend only on the diffraction angle. The value σz = r/19 longitudinal filtering (GLF, σz = 0 and σz = a/10). The was chosen so that the quantum curve is close to the RS1 value σz = 0 corresponds to the case where the diffracted curve. Moreover, for this value, the damping is less impor- wave associated with the particle is emitted from the wave- tant than that of Fig. 4 because the value r/19 is closer to front at the aperture whereas a non-zero value corresponds zero than the value a/10. to the case where several wavefronts contribute with differ- ent weights and this generates a damping increasing with θ and σz. curves and a significant gap can be obtained at not too large angles (Fig. 4). Coincidentally, the curves QM and RS1 can be very close but they cannot be exactly the only for the very large angles. So, the quantum model same everywhere because the factors cos θ and cos2θ are and the FK theory are in good agreement over a wide different (Fig. 5). Moreover, applying (68) to the case angular range. σz and then substituting into (59), we see that the intensity→ ∞ tends to a peak of zero width at θ = 0. So, the If σz > 0, the diffracted wave associated with the quantum model predicts no diffraction at all in this case. particle is emitted from several wavefronts which con- tribute with different weights given by the distribution In summary, the quantum formula leads to different σz FL (z). The longitudinal diffraction term is strictly less predictions according to the value of σz and these pre- than 1. So the quantum relative intensity, maximum dictions are different from that of the classical theories. for σz = 0, undergoes a damping which increases as a Therefore, a measurement of the relative intensity in the function of θ and σz. When σz is increasing from zero, region of large diffraction angles - where the gaps between the quantum values eventually pass below the values of the different predictions are significant - would be an ap- the RS1 theory. If σz is large enough, the quantum curve propriate experimental test of both the classical theories decreases globally much more rapidly than the classical and the quantum model. 15
VI. CONCLUSION Kirchhoff theory over a wide angular range. As ∆z in- creases, the damping reduces the diffraction at smaller and smaller angles and even removes it completely if The diffraction of the wave associated with a particle ∆z . A study performed in the case of the diffrac- arriving on a diaphragm has been calculated by means tion→ by ∞ a rectangular slit and by a circular aperture shows of a theoretical model based exclusively on quantum me- that a measurement of the relative intensity in the region chanics. This model considers the diffraction as the con- of large angles should make it possible to compare these sequence of a quantum measurement of the position of predictions to experimental data and consequently to dis- the particle passing through the diaphragm aperture. It criminate between the different theories. is based on two main assumptions. The quantum model presented here provides an ex- According to the first assumption, the diaphragm is pression of the quantum state of the particle associated considered as a measurement device not only of the trans- with the wave diffracted by an aperture of finite area. verse coordinates but also of the longitudinal coordinate It is then possible to calculate the Fraunhofer diffrac- of the particle, so that the initial state is projected on a tion, from the momentum wave function of this state, state whose position wave function is localized in a ”gen- in the case of a monochromatic plane wave in normal eralized 3D aperture” including the 2D aperture of the incidence. Another application can be considered. The diaphragm. The longitudinal size ∆z of this 3D aper- quantum model is a natural framework for treating the ture can be considered as the width of a ”longitudinal case of a non-monochromatic incident wave. This re- aperture”. It is a parameter whose value is between zero quires applying the model by replacing the initial state and infinity. If ∆z = 0, the diffracted wave is emit- used in the present paper (momentum eigenstate) by a ted from the wavefront at the aperture, which is consis- state corresponding to a wave packet. It will then be tent with the Huygens-Fresnel principle. If ∆z > 0, it necessary to modify the second assumption of the model can be assumed that the Huygens-Fresnel principle ap- that currently applies only to the monochromatic case. plies to several wavefronts which contribute with differ- Finally, the question of calculating Fresnel diffraction ent weights according to some distribution whose integral from the suggested quantum model arises. For a parti- outside the longitudinal aperture is negligible and whose cle whose quantum state is represented by a state vec- shape is part of the assumptions of the model. Due to tor of the Hilbert space, the momentum wave function the position measurement, the momentum wave function is the Fourier transform of the position wave function. of the state resulting from the projection of the initial Now, when the Huygens-Fresnel principle is applied in state is incompatible with the kinematics of the particle- the framework of the classical theory to calculate the diaphragm interaction, so that this state is necessarily amplitude of the diffracted wave, this Fourier transform a transitional state. This is the reason of the second is obtained only in the particular case of Fraunhofer assumption which suggests, in the case of a monochro- diffraction, when it is possible to neglect the terms that matic incident wave, that a specific operator projects must be taken into account in the case of Fresnel diffrac- the transitional state on the final state corresponding to tion. These characteristic terms of Fresnel diffraction are the diffracted wave propagating beyond the diaphragm. therefore totally absent from the formalism of the state Thus, the quantum model describes the change of state vectors and their associated wave functions. So, the sug- by two quasi-simultaneous successive projections. gested quantum model, based on this formalism, is not The quantum model provides a general expression of applicable in principle to the case of Fresnel diffraction. the quantum state associated with the diffracted wave, leading in particular to a new formula of the relative in- tensity in Fraunhofer diffraction in the case of an incident ACKNOWLEDGEMENTS monochromatic plane wave. This quantum formula is ap- plicable to the whole angular range (0◦ - 90◦). Compari- I would like to thank M. Besan¸con, F. Charra and son with the theories of Fresnel-Kirchhoff and Rayleigh- P. Roussel for helpful suggestions and comments on the Sommerfeld shows that the predictions are close for small manuscript. I am also thankful to the team of the diffraction angles but significantly different for large an- IRAMIS-CEA/SPEC/LEPO for useful discussions on the gles. Moreover, the quantum model predicts a damping feasibility of an experiment to measure the diffraction of the intensity, increasing with angle and parameter ∆z. at large angles. This work was supported by the CEA. If ∆z = 0, there is no damping and the prediction of The figures were carried out using the ROOT software the quantum model is very close to that of the Fresnel- (https://root.cern.ch/).
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