1. Given information

• Direction of propagation: $\hat n = \frac{\hat i + \hat j}{\sqrt 2}$
• Polarization is along $\hat k$, so the electric field is along $\hat k$: $\vec E \parallel \hat k$

For a plane electromagnetic wave, the vectors $\vec E$, $\vec B$, and direction of propagation $\hat n$ are mutually perpendicular, and they satisfy $\vec B \propto \hat n \times \vec E$ if the phase is written as for propagation along $\hat n$.

2. Find the direction of magnetic field

Since $\hat n = \frac{\hat i + \hat j}{\sqrt 2}, \qquad \vec E \parallel \hat k$ we compute $\hat n \times \hat k = \frac{1}{\sqrt 2}(\hat i + \hat j) \times \hat k$ Using $\hat i \times \hat k = -\hat j, \qquad \hat j \times \hat k = \hat i$ we get $\hat n \times \hat k = \frac{1}{\sqrt 2}(-\hat j + \hat i)=\frac{\hat i - \hat j}{\sqrt 2}$ So, $\vec B \parallel \frac{\hat i - \hat j}{\sqrt 2}$

3. Check the phase factor

For a wave propagating along $\hat n = \frac{\hat i + \hat j}{\sqrt 2}$ the phase must be of the form $\cos(\omega t - \vec k\cdot \vec r)$ with $\vec k = k\hat n = k\frac{\hat i + \hat j}{\sqrt 2}$ Thus the magnetic field should have the same propagation phase corresponding to this direction.

4. Match with options

• Option A: direction $\dfrac{\hat i-\hat j}{\sqrt2}$, correct phase for propagation along $\dfrac{\hat i+\hat j}{\sqrt2}$.
• Option B: magnetic field along propagation direction, impossible.
• Option C: direction is opposite of required and phase sign corresponds to wrong propagation form.
• Option D: magnetic field parallel to electric field, impossible.

Therefore the correct option is: $\boxed{A}$

5. Comparison with stored answer

Stored correct answer: A

My derived answer: A

So they agree.