Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Rigorous expressions of Huygens’ principle in scalar theory: reply

Malong Fu and Yang Zhao

Open Access Open Access

Abstract

The Green’s function is an auxiliary function that needs to satisfy different conditions according to the practical mathematical models applied to the problems. The required conditions do not have to be Helmholtz equation. Two possible common conditions are summarized in this reply, i.e., the Maxwell reciprocity and the corresponding homogeneous boundary conditions.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Two possible common conditions for the Green’s function

In [1], many versions of Green’s function are constructed to obtain the solution of the corresponding mathematical problems. Several representative examples are summarized from [1] to show that satisfying the Helmholtz equation is not a necessity to the Green’s function. In those cases, x, $\rho $, $\theta ,\; \; t$ are variables, a and b are the boundaries, $\alpha ,\; \; \beta ,c\; $ are arbitrary constants, h and g are arbitrary functions, f is the function to represent the source, u is the function to be solved, G is the Green’s function.

Case 1, summarized from pages 393–394 of [1]:

For a general inhomogeneous problem:

$${L(u )= f(x ),}$$
where L is the Sturm-Liouville operator. If the boundary conditions are Dirichlet and homogeneous, i.e.,
$${u(a )= u(b )= 0.}$$

Then, the Green’s function $G({x,{x_s}} )$ should satisfy Eqs. (3), (4) and (5) simultaneously.

$${L[{G({x,{x_s}} )} ]= \delta ({x - {x_s}} ),}$$
$${G({x,{x_s}} )= G({{x_s},x} ),}$$
$${G({a,{x_s}} )= G({b,{x_s}} )= 0.}$$

In such a general case, the Green’s function should satisfy an inhomogeneous equation, i.e., Eq. (3), consisting of a homogeneous equation, i.e., $L[{G({x,{x_s}} )} ]= 0$, and a delta source $\delta ({x - {x_s}} )$. Besides, the Green’s function must meet the boundary conditions defined by Eq. (5). Meanwhile, the Green’s function shows the Maxwell reciprocity as given by Eq. (4). With such conditions, the solution of the problem can be obtained as

$${u(x )= \mathop \smallint \nolimits_a^b f({{x_0}} )G({x,{x_0}} )d{x_0}.}$$

It can be found, the Green’s function $G({x,{x_s}} )$ in this case represents the response at coordinate x induced by the point source at ${x_s}$.

Case 2, summarized from pages 397–398 of [1]:

For the problem:

$${\frac{{{\textrm{d}^2}u}}{{\textrm{d}{x^2}}} = f(x ),}$$
with the Dirichlet boundary conditions:
$${u(a )= \alpha ,u(b )= \beta .}$$

Then the Green’s function should satisfy Eqs. (4), (5) and (9) simultaneously.

$${\frac{{{\textrm{d}^2}G({x,{x_s}} )}}{{\textrm{d}{x^2}}} = \delta ({x - {x_s}} ).}$$

Similar with Case 1, the Green’s function features with the Maxwell reciprocity governed by Eq. (4), satisfies the homogeneous boundary conditions given by Eq. (5), and conforms with an inhomogeneous boundary condition which consists of a homogeneous equation and a delta source as defined by Eq. (9). Then a different version of the solution is obtained as:

$${u(x )= \mathop \smallint \nolimits_a^b f({{x_0}} )G({x,{x_0}} )d{x_0} + \beta {{\left. {\frac{{{\textrm{d}\;}G({x,{x_0}} )}}{{\textrm{d}{x\;}}}} \right|}_{{x_0} = b}} - \alpha {{\left. {\frac{{{\textrm{d}\;}G({x,{x_0}} )}}{{\textrm{d}{x\;}}}} \right|}_{{x_0} = a}}.}$$

It is obviously that it reveals the influence of the inhomogeneous boundary as indicated by the last two terms of Eq. (10).

Case 3, summarized from pages 411–412 of [1]:

For the problem:

$${\frac{{{\textrm{d}^2}u}}{{\textrm{d}{x^2}}} = f(x ),}$$
with the Neumann boundary conditions:
$${\frac{{{\textrm{d}\;}u}}{{\textrm{d}{x\;}}}(a )= \frac{{{\textrm{d}\;}u}}{{\textrm{d}{x\;}}}(b )= 0.}$$

Then, the Green’s function has to satisfy Eqs. (4), (13) and (14) simultaneously to solve Eq. (11).

$${\frac{{{\textrm{d}^2}G({x,{x_s}} )}}{{\textrm{d}{x^2}}} = \delta ({x - {x_s}} )- \frac{1}{{b - a}},}$$
$${{{\left. {\frac{{{\textrm{d}\;}G({x,{x_0}} )}}{{\textrm{d}{x\;}}}} \right|}_{{x_0} = a}}{{\left. { = \frac{{{\textrm{d}\;}G({x,{x_0}} )}}{{\textrm{d}{x\;}}}} \right|}_{{x_0} = b}} = 0.}$$

Different from Case 1 and Case 2, Eq. (13) is an inhomogeneous equation consisting of a homogeneous equation, a delta source and an extra term, i.e., $1/({b - a} )$. Nevertheless, the Green’s function still has the property of Maxwell reciprocity but has to meet a different set of homogeneous boundary conditions as defined by Eq. (14).

Case 4, summarized from pages 422–423, 430–432 of [1]:

For the two-dimension problem:

$${{\nabla ^2}u({\rho ,\theta } )= f({\rho ,\theta } ),}$$
with the Dirichlet condition:
$${u({a,\theta } )= h(\theta ).}$$

The Green’s function must satisfy Eqs. (4), (17) and (18) simultaneously to solve the problem.

$${{\nabla ^2}G({\rho ,\theta ;{\rho_s},{\theta_s}} )= \delta ({\rho - {\rho_s}} )\delta ({\theta - {\theta_s}} )- \delta ({\rho - \rho_s^\ast } )\delta ({\theta - \theta_s^\ast } ),}$$
$${G({a,\theta ;{\rho_s},{\theta_s}} )= 0.}$$

Again, besides the Maxwell reciprocity and the corresponding homogeneous boundary condition respectively defined by Eqs. (4) and (18), Eq. (17) that the Green’s function must satisfy is not merely an inhomogeneous equation that consists of a homogeneous equation and a delta source but an extra delta source is added. Then the solution can be obtained:

$${u({\rho ,\theta } )= {\int\!\!\!\int }f({\rho ,\theta } )G({a,\theta ;{\rho_s},{\theta_s}} ){\rho _s}d{\rho _s}d{\theta _s} + \mathop \smallint \nolimits_0^{2\pi } h({{\theta_s}} ){{\left. {\frac{{\partial G({a,\theta ;{\rho_s},{\theta_s}} )}}{{\partial {\rho_s}}}} \right|}_{{\rho _s} = a}}ad{\theta _s}.}$$

Due to the last term of the right side of Eq. (19), the Green’s function could not be interpreted as the response induced by a point source.

Case 5, summarized from pages 515–516 of [1]:

For the wave equation:

$${\frac{{{\partial ^2}u}}{{\partial {t^2}}} - {c^2}{\nabla ^2}u = 0,}$$
subject to the initial conditions:
$${{{ u |}_{t = 0}} = 0\; and\; {{\left. {\frac{{\partial u}}{{\partial t}}} \right|}_{t = 0}} = \delta ({{\boldsymbol x} - {{\boldsymbol x}_0}} ).}$$

Only if the Green’s function satisfies the homogeneous boundary conditions, then the solution can be derived:

$${u = G.}$$

Since the problem has no source term as in Eq. (20), the Green’s function has nothing to do with the source. Substituting Eq. (22) into Eq. (20), then the Green’s function satisfies

$${\frac{{{\partial ^2}G}}{{\partial {t^2}}} - {c^2}{\nabla ^2}G = 0.}$$

In this case, the Green’s function needs to satisfy a homogeneous function. In addition, it still has the Maxwell reciprocity and has to satisfy the corresponding homogeneous boundary conditions.

Although Eq. (3) is a condition that the Green’s function must follow for most of the cases, the Green’s functions corresponding to different problems are quite different and don’t have to always follow Eq. (3), e.g., Case 3, Case 4, and Case 5. The conditions that the Green’s function must satisfy are decided by the definite conditions of the mathematical model. Thus, the Green’s function is an auxiliary function to solve the problem and may have no explicit meaning in physics. Specially, if the problem is defined by an ordinary differential equation which is inhomogeneous and subject to the homogeneous Dirichlet boundary conditions, then the constructed Green’s function to solve such problem must satisfy the inhomogeneous equation consisting of a homogeneous differential equation and a delta-source. In this case, the Green’s function could represent the impulse response induced by a point source.

Nevertheless, two possible common characteristics of the Green’s function are summarized in this reply, i.e., the Maxwell reciprocity defined by Eq. (4) and the corresponding homogeneous boundary conditions, to cover a broad range of problems. The two common characteristics are based on the observations on many problems in the literatures [1], it is not theoretically proved in this reply. Thus, the summarized two common characteristics may not fit for other unknow problems.

In our previous paper [2], the original mathematical model is a homogeneous wave equation:

$$\begin{array}{c} {\left\{ {\begin{array}{l} {{\nabla^2}u - \frac{1}{v}\frac{{{\partial^2}}}{{\partial {t^2}}}u = 0}\\ {u({{P_0},t} )= g({{x_0},{y_0},{z_0},t} ),\frac{\partial }{{\partial t}}u({{P_0},t} )= \dot{g}({{x_0},{y_0},{z_0},t} ),({{x_0},{y_0},{z_0}} )\in {S_0}} \end{array}} \right.,} \end{array}$$
where the definite conditions are special inhomogeneous Cauchy boundary conditions. As a result, the proposed Green’s function adopted in [2] to solve the corresponding Helmholtz equation, does not have to satisfy the inhomogeneous Helmholtz equation consisting of a homogeneous Helmholtz equation and a delta-source. Obviously, the Green’s function in [2] cannot represent the field induced by a point source. The corresponding homogeneous boundary conditions and the Maxwell reciprocity are met by the Green’s function proposed in [2].

2. Green’s function in scalar theory

Our work is the extension of the Rayleigh-Sommerfeld diffraction formula (RSDF), i.e., Eq. (2) of [2]. The Green’s function of the RSDF is the Eq. (12) of [2], which also could be found in Sommerfeld’s lecture and other publications [3–5]:

$${G({{P_x};{P_1},{P_2}} )= \frac{{\exp ({\textrm{i}k{r_{x1}}} )}}{{{r_{x1}}}} - \frac{{\exp ({\textrm{i}k{r_{x2}}} )}}{{{r_{x2}}}}.}$$

Or change it to the format in [6]:

$${G({{P_x};{P_1},{P_2}} )= \frac{1}{{|{{\boldsymbol r} - {{\boldsymbol r}_\mathbf{0}}} |}}\exp ({\textrm{i}k|{{\boldsymbol r} - {{\boldsymbol r}_{\mathbf{0}}}} |} )- \frac{1}{{|{{\boldsymbol r} - {\boldsymbol r}_{\mathbf{0}}^{{\prime}}} |}}\exp ({\textrm{i}k|{{\boldsymbol r} - {\boldsymbol r}_{\mathbf{0}}^{{\prime}}} |} ).}$$

Equation (26) is in fact a composition of two spherical waves, each of which satisfies the Helmholtz equation at almost all points except its origin point which is a singularity. Then the Green’s function satisfies:

$${{\nabla ^2}G + {k^2}G ={-} 4\pi [{\delta ({{\boldsymbol r} - {{\boldsymbol r}_{\mathbf{0}}}} )+ \delta ({{\boldsymbol r} - {\boldsymbol r}_{\mathbf{0}}^{{\prime}}} )} ],}$$
where ${\boldsymbol r\; }$ is the position vector of the observation point. Due to the right term of Eq. (27), the Green’s function adopted by Sommerfeld satisfies the Helmholtz equation only in the solution region which is the source-free region with the observation point removed. The reason is simple: The Green’s function is a constructed auxiliary function to solve the problem, without necessity to satisfy the physical meaning depicted by the Helmholtz equation.

In scalar theory, only the light disturbance u explained in [4] can be detected. As a result, u must satisfy the wave equation in the source-free domain. Owing to the Fourier transform relation between u and the complex amplitude U defined in [7], U must satisfy the Helmholtz equation in the source-free domain to represent the monochromatic light field according to the Eq. (6) of [2], which is completely guaranteed in [2].

3. Conclusions

In summary, the Green’s function is a constructed auxiliary function to solve the related mathematical model. For different physics problems, the Green’s functions vary. Therefore, the Green’s function is not necessary to satisfy a fixed set of conditions. Since the mathematical model of the Huygens principle is subject to the required Cauchy boundary conditions, the Green’s function adopted to solve such model does not have to satisfy the Helmholtz equation or the equation consisting of the Helmholtz equation and a delta-source. Thus, the adopted Green’s function does not represent the monochromatic light field or other physical meanings.

Funding

Startup Fund for Young Faculty at Shanghai Jiao Tong University (SFYF at SJTU) (21X010500725).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. R. Haberman, Applied Partial Differential Equations: with Fourier Series and Boundary Value Problems, (Pearson Education, 2004, 4th ed), pp. 380–533.

2. M. Fu and Y. Zhao, “Rigorous expression of Huygens’ principle in scalar theory,” Opt. Express 29(4), 6257–6270 (2021). [CrossRef]  

3. A. Sommerfeld, “The theory of diffraction,” in Optics, Lecture on Theoretical Physics, (Academic, 1954), pp. 179–266.

4. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, (Pergamon, 1980, 6th ed), pp. 412–425

5. J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1996, 2nd ed), pp. 32–62.

6. M. Charnotskii, “Rigorous expressions of Huygens’ principle in scalar theory: comment,” Opt. Express 29(18), 28953 (2021). [CrossRef]  

7. D. Griffiths, Introduction to Electrodynamics, (Prentice-Hall, 1999, 3th ed), pp. 368–369.

Data availability

No data were generated or analyzed in the presented research.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (27)

Equations on this page are rendered with MathJax. Learn more.

L ( u ) = f ( x ) ,
u ( a ) = u ( b ) = 0.
L [ G ( x , x s ) ] = δ ( x − x s ) ,
G ( x , x s ) = G ( x s , x ) ,
G ( a , x s ) = G ( b , x s ) = 0.
u ( x ) = ∫ a b ⁡ f ( x 0 ) G ( x , x 0 ) d x 0 .
d 2 u d x 2 = f ( x ) ,
u ( a ) = α , u ( b ) = β .
d 2 G ( x , x s ) d x 2 = δ ( x − x s ) .
u ( x ) = ∫ a b ⁡ f ( x 0 ) G ( x , x 0 ) d x 0 + β d G ( x , x 0 ) d x | x 0 = b − α d G ( x , x 0 ) d x | x 0 = a .
d 2 u d x 2 = f ( x ) ,
d u d x ( a ) = d u d x ( b ) = 0.
d 2 G ( x , x s ) d x 2 = δ ( x − x s ) − 1 b − a ,
d G ( x , x 0 ) d x | x 0 = a = d G ( x , x 0 ) d x | x 0 = b = 0.
∇ 2 u ( ρ , θ ) = f ( ρ , θ ) ,
u ( a , θ ) = h ( θ ) .
∇ 2 G ( ρ , θ ; ρ s , θ s ) = δ ( ρ − ρ s ) δ ( θ − θ s ) − δ ( ρ − ρ s ∗ ) δ ( θ − θ s ∗ ) ,
G ( a , θ ; ρ s , θ s ) = 0.
u ( ρ , θ ) = ∫ ∫ f ( ρ , θ ) G ( a , θ ; ρ s , θ s ) ρ s d ρ s d θ s + ∫ 0 2 π ⁡ h ( θ s ) ∂ G ( a , θ ; ρ s , θ s ) ∂ ρ s | ρ s = a a d θ s .
∂ 2 u ∂ t 2 − c 2 ∇ 2 u = 0 ,
u | t = 0 = 0 a n d ∂ u ∂ t | t = 0 = δ ( x − x 0 ) .
u = G .
∂ 2 G ∂ t 2 − c 2 ∇ 2 G = 0.
{ ∇ 2 u − 1 v ∂ 2 ∂ t 2 u = 0 u ( P 0 , t ) = g ( x 0 , y 0 , z 0 , t ) , ∂ ∂ t u ( P 0 , t ) = g ˙ ( x 0 , y 0 , z 0 , t ) , ( x 0 , y 0 , z 0 ) ∈ S 0 ,
G ( P x ; P 1 , P 2 ) = exp ⁡ ( i k r x 1 ) r x 1 − exp ⁡ ( i k r x 2 ) r x 2 .
G ( P x ; P 1 , P 2 ) = 1 | r − r 0 | exp ⁡ ( i k | r − r 0 | ) − 1 | r − r 0 ′ | exp ⁡ ( i k | r − r 0 ′ | ) .
∇ 2 G + k 2 G = − 4 π [ δ ( r − r 0 ) + δ ( r − r 0 ′ ) ] ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use